[ { "id": "Set_theory_and_logic_0000", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "3", "keywords": [ "proof\u2019", "\u2018interval notation\u2019", "\u2018complement\u2019" ], "problem_v1": "When expressing sets, you may write \u201cinf\u201d for infinity and \u201cU\u201d for union. Write the complement of the given set in interval notation: $(5, \\infty)$ [ANS]\nWrite the complement of the given set in interval notation: $(-\\infty,-3]$ [ANS]\nWrite the complement of the given set in interval notation: $(-3, 5]$ [ANS]", "answer_v1": [ "(-infinity,5]", "(-3,infinity)", "(-infinity,-3] U (5,infinity)" ], "answer_type_v1": [ "INT", "INT", "INT" ], "options_v1": [ [], [], [] ], "problem_v2": "When expressing sets, you may write \u201cinf\u201d for infinity and \u201cU\u201d for union. Write the complement of the given set in interval notation: $(1, \\infty)$ [ANS]\nWrite the complement of the given set in interval notation: $(-\\infty,-1]$ [ANS]\nWrite the complement of the given set in interval notation: $(-1, 1]$ [ANS]", "answer_v2": [ "(-infinity,1]", "(-1,infinity)", "(-infinity,-1] U (1,infinity)" ], "answer_type_v2": [ "INT", "INT", "INT" ], "options_v2": [ [], [], [] ], "problem_v3": "When expressing sets, you may write \u201cinf\u201d for infinity and \u201cU\u201d for union. Write the complement of the given set in interval notation: $(2, \\infty)$ [ANS]\nWrite the complement of the given set in interval notation: $(-\\infty,-3]$ [ANS]\nWrite the complement of the given set in interval notation: $(-3, 2]$ [ANS]", "answer_v3": [ "(-infinity,2]", "(-3,infinity)", "(-infinity,-3] U (2,infinity)" ], "answer_type_v3": [ "INT", "INT", "INT" ], "options_v3": [ [], [], [] ] }, { "id": "Set_theory_and_logic_0001", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "3", "keywords": [ "proof\u2019", "\u2018interval notation\u2019", "\u2018union\u2019", "\u2018intersection\u2019" ], "problem_v1": "What is $(5, 12) \\cap (8, 16]$? [ANS]\nWhat is $(5, 12) \\cup (8, 16]$? [ANS]", "answer_v1": [ "(8,12)", "(5,16]" ], "answer_type_v1": [ "INT", "INT" ], "options_v1": [ [], [] ], "problem_v2": "What is $(1, 7) \\cap (6, 9]$? [ANS]\nWhat is $(1, 7) \\cup (6, 9]$? [ANS]", "answer_v2": [ "(6,7)", "(1,9]" ], "answer_type_v2": [ "INT", "INT" ], "options_v2": [ [], [] ], "problem_v3": "What is $(2, 8) \\cap (6, 11]$? [ANS]\nWhat is $(2, 8) \\cup (6, 11]$? [ANS]", "answer_v3": [ "(6,8)", "(2,11]" ], "answer_type_v3": [ "INT", "INT" ], "options_v3": [ [], [] ] }, { "id": "Set_theory_and_logic_0002", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "2", "keywords": [ "interval", "set", "intersection", "union" ], "problem_v1": "Let $S=[-4,5)$, $T=[2,7]$, and $W=(-\\infty,-1)$. For each intersection or union, choose the correct notation for the resulting interval.\n$\\begin{array}{cccccccccccccccc}\\hline & [ANS] 1. S\\cup W [ANS] 2. S\\cap W [ANS] 3. S\\cup T [ANS] 4. T\\cap W & [ANS] & 1. & S\\cup W & [ANS] & 2. & S\\cap W & [ANS] & 3. & S\\cup T & [ANS] & 4. & T\\cap W & & A. \\emptyset B. (-\\infty,5) C. [-4,7] D. [-4,-1) \\\\\\hline [ANS] & 1. & S\\cup W \\\\\\hline [ANS] & 2. & S\\cap W \\\\\\hline [ANS] & 3. & S\\cup T \\\\\\hline [ANS] & 4. & T\\cap W \\\\\\hline\\end{array}$", "answer_v1": [ "B", "D", "C", "A" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $S=[-10,-3)$, $T=[-5,1]$, and $W=(-\\infty,-6)$. For each intersection or union, choose the correct notation for the resulting interval.\n$\\begin{array}{cccccccccccccccc}\\hline & [ANS] 1. S\\cap W [ANS] 2. S\\cap T [ANS] 3. T\\cap W [ANS] 4. S\\cup T & [ANS] & 1. & S\\cap W & [ANS] & 2. & S\\cap T & [ANS] & 3. & T\\cap W & [ANS] & 4. & S\\cup T & & A. [-5,-3) B. \\emptyset C. [-10,-6) D. [-10,1] \\\\\\hline [ANS] & 1. & S\\cap W \\\\\\hline [ANS] & 2. & S\\cap T \\\\\\hline [ANS] & 3. & T\\cap W \\\\\\hline [ANS] & 4. & S\\cup T \\\\\\hline\\end{array}$", "answer_v2": [ "C", "A", "B", "D" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $S=[-8,0)$, $T=[-3,1]$, and $W=(-\\infty,-5)$. For each intersection or union, choose the correct notation for the resulting interval.\n$\\begin{array}{cccccccccccccccc}\\hline & [ANS] 1. S\\cap W [ANS] 2. S\\cap T [ANS] 3. S\\cup T [ANS] 4. S\\cup W & [ANS] & 1. & S\\cap W & [ANS] & 2. & S\\cap T & [ANS] & 3. & S\\cup T & [ANS] & 4. & S\\cup W & & A. [-3,0) B. [-8,-5) C. (-\\infty,0) D. [-8,1] \\\\\\hline [ANS] & 1. & S\\cap W \\\\\\hline [ANS] & 2. & S\\cap T \\\\\\hline [ANS] & 3. & S\\cup T \\\\\\hline [ANS] & 4. & S\\cup W \\\\\\hline\\end{array}$", "answer_v3": [ "B", "A", "D", "C" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Set_theory_and_logic_0003", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "2", "keywords": [ "set theory", "union", "intersection", "Set", "Intersection", "Union", "Containment", "Complement" ], "problem_v1": "$A=\\lbrace 1,3,5 \\rbrace, B=\\lbrace 2,3 \\rbrace$ Check ALL elements of the following sets:\n(a) $A \\cap B$ [ANS] A. 2 B. 1 C. 4 D. 5 E. 3\n(b) $A \\cup B$ [ANS] A. 2 B. 3 C. 1 D. 4 E. 5\n(c) $A \\setminus B$ [ANS] A. 5 B. 3 C. 1 D. 2 E. 4", "answer_v1": [ "E", "ABCE", "AC" ], "answer_type_v1": [ "MCS", "MCM", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "$A=\\lbrace 1,3,5 \\rbrace, B=\\lbrace 2,3 \\rbrace$ Check ALL elements of the following sets:\n(a) $A \\cap B$ [ANS] A. 5 B. 4 C. 3 D. 2 E. 1\n(b) $A \\cup B$ [ANS] A. 4 B. 5 C. 2 D. 3 E. 1\n(c) $A \\setminus B$ [ANS] A. 5 B. 2 C. 4 D. 1 E. 3", "answer_v2": [ "C", "BCDE", "AD" ], "answer_type_v2": [ "MCS", "MCM", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "$A=\\lbrace 1,3,5 \\rbrace, B=\\lbrace 2,3 \\rbrace$ Check ALL elements of the following sets:\n(a) $A \\cap B$ [ANS] A. 1 B. 2 C. 3 D. 4 E. 5\n(b) $A \\cup B$ [ANS] A. 3 B. 5 C. 4 D. 1 E. 2\n(c) $A \\setminus B$ [ANS] A. 5 B. 3 C. 1 D. 4 E. 2", "answer_v3": [ "C", "ABDE", "AC" ], "answer_type_v3": [ "MCS", "MCM", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0004", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "3", "keywords": [ "set theory", "subset", "union", "intersection" ], "problem_v1": "Let $\\ U=$ Universal set $=\\{0, \\ 1, \\ 2, \\ 3, \\ 4, \\ 5, \\ 6, \\ 7, \\ 8, \\ 9\\}$, $\\ A=\\{1, \\ 2, \\ 4, \\ 5, \\ 7, \\ 8\\}$, $\\ $ and $\\ B=\\{4, \\ 5, \\ 6, \\ 7\\}$. List the elements of the following sets in the increasing order: $A'=\\{$ [ANS], $\\ $ [ANS], $\\ $ [ANS], $\\ $ [ANS] $\\}$ $(A \\cup B)'=\\{$ [ANS], $\\ $ [ANS], $\\ $ [ANS] $\\}$ $(A \\cup B')'=\\{$ [ANS] $\\}$ $A \\cap B'=\\{$ [ANS], $\\ $ [ANS], $\\ $ [ANS] $\\}$", "answer_v1": [ "0", "3", "6", "9", "0", "3", "9", "6", "1", "2", "8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $\\ U=$ Universal set $=\\{0, \\ 1, \\ 2, \\ 3, \\ 4, \\ 5, \\ 6, \\ 7, \\ 8, \\ 9\\}$, $\\ A=\\{0, \\ 2, \\ 3, \\ 4, \\ 8, \\ 9\\}$, $\\ $ and $\\ B=\\{0, \\ 1, \\ 2, \\ 9\\}$. List the elements of the following sets in the increasing order: $A'=\\{$ [ANS], $\\ $ [ANS], $\\ $ [ANS], $\\ $ [ANS] $\\}$ $(A \\cup B)'=\\{$ [ANS], $\\ $ [ANS], $\\ $ [ANS] $\\}$ $(A \\cup B')'=\\{$ [ANS] $\\}$ $A \\cap B'=\\{$ [ANS], $\\ $ [ANS], $\\ $ [ANS] $\\}$", "answer_v2": [ "1", "5", "6", "7", "5", "6", "7", "1", "3", "4", "8" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $\\ U=$ Universal set $=\\{0, \\ 1, \\ 2, \\ 3, \\ 4, \\ 5, \\ 6, \\ 7, \\ 8, \\ 9\\}$, $\\ A=\\{1, \\ 2, \\ 3, \\ 4, \\ 5, \\ 6\\}$, $\\ $ and $\\ B=\\{2, \\ 3, \\ 6, \\ 9\\}$. List the elements of the following sets in the increasing order: $A'=\\{$ [ANS], $\\ $ [ANS], $\\ $ [ANS], $\\ $ [ANS] $\\}$ $(A \\cup B)'=\\{$ [ANS], $\\ $ [ANS], $\\ $ [ANS] $\\}$ $(A \\cup B')'=\\{$ [ANS] $\\}$ $A \\cap B'=\\{$ [ANS], $\\ $ [ANS], $\\ $ [ANS] $\\}$", "answer_v3": [ "0", "7", "8", "9", "0", "7", "8", "9", "1", "4", "5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0005", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "2", "keywords": [ "inequalities" ], "problem_v1": "Let $A=\\{2, 3, 6, 7, 8\\}$, $B=\\{5, 6, 7, 8\\}$. Find the following sets in list form. Separate elements with commas. If there are no elements in the set, enter \"NONE\". a) $A \\cap B=$ [ANS]\nb) $A \\cup B=$ [ANS]", "answer_v1": [ "(6,7,8)", "(2,3,5,6,7,8)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Let $A=\\{2, 4, 5, 10, 11\\}$, $B=\\{2, 3, 5, 11\\}$. Find the following sets in list form. Separate elements with commas. If there are no elements in the set, enter \"NONE\". a) $A \\cap B=$ [ANS]\nb) $A \\cup B=$ [ANS]", "answer_v2": [ "(2,5,11)", "(2,3,4,5,10,11)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Let $A=\\{1, 2, 5, 6, 7\\}$, $B=\\{2, 6, 7, 10\\}$. Find the following sets in list form. Separate elements with commas. If there are no elements in the set, enter \"NONE\". a) $A \\cap B=$ [ANS]\nb) $A \\cup B=$ [ANS]", "answer_v3": [ "(2,6,7)", "(1,2,5,6,7,10)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Set_theory_and_logic_0006", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "2", "keywords": [ "algebra", "union", "intersection", "set" ], "problem_v1": "Match the indicated sets below with the letters labeling their equivalent sets if $A=\\lbrace 1,2,3,4,5,6 \\rbrace$, $B=\\lbrace 2,4,6,8 \\rbrace$, $C=\\lbrace 7,8,9,10 \\rbrace$. [ANS] 1. $A\\cap B$ [ANS] 2. $A\\cup C$ [ANS] 3. $B\\cup C$ [ANS] 4. $B\\cap C$ [ANS] 5. $A\\cup B$\nA. $\\lbrace 1,2,3,4,5,6,7,8,9,10 \\rbrace$ B. $\\lbrace 8 \\rbrace$ C. $\\lbrace 2,4,6,7,8,9,10 \\rbrace$ D. $\\lbrace 2,4,6 \\rbrace$ E. $\\lbrace 1,2,3,4,5,6,8 \\rbrace$", "answer_v1": [ "D", "A", "C", "B", "E" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Match the indicated sets below with the letters labeling their equivalent sets if $A=\\lbrace 1,2,3,4,5,6 \\rbrace$, $B=\\lbrace 2,4,6,8 \\rbrace$, $C=\\lbrace 7,8,9,10 \\rbrace$. [ANS] 1. $A\\cup B$ [ANS] 2. $B\\cap C$ [ANS] 3. $B\\cup C$ [ANS] 4. $A\\cup C$ [ANS] 5. $A\\cap B$\nA. $\\lbrace 8 \\rbrace$ B. $\\lbrace 1,2,3,4,5,6,8 \\rbrace$ C. $\\lbrace 2,4,6,7,8,9,10 \\rbrace$ D. $\\lbrace 2,4,6 \\rbrace$ E. $\\lbrace 1,2,3,4,5,6,7,8,9,10 \\rbrace$", "answer_v2": [ "B", "A", "C", "E", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Match the indicated sets below with the letters labeling their equivalent sets if $A=\\lbrace 1,2,3,4,5,6 \\rbrace$, $B=\\lbrace 2,4,6,8 \\rbrace$, $C=\\lbrace 7,8,9,10 \\rbrace$. [ANS] 1. $B\\cup C$ [ANS] 2. $A\\cap B$ [ANS] 3. $A\\cup B$ [ANS] 4. $B\\cap C$ [ANS] 5. $A\\cup C$\nA. $\\lbrace 2,4,6 \\rbrace$ B. $\\lbrace 1,2,3,4,5,6,7,8,9,10 \\rbrace$ C. $\\lbrace 8 \\rbrace$ D. $\\lbrace 1,2,3,4,5,6,8 \\rbrace$ E. $\\lbrace 2,4,6,7,8,9,10 \\rbrace$", "answer_v3": [ "E", "A", "D", "C", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0007", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "2", "keywords": [], "problem_v1": "Let $W=\\lbrace a,b,c,d,e\\rbrace$, $X=\\lbrace a,c,f\\rbrace$\nDetermine $W-X$ [ANS] A. $\\lbrace 1,2,3$ B. $\\lbrace\\rbrace$ C. $\\lbrace 1,2,3,4,5\\rbrace$ D. $\\lbrace 0,0,0,4,5\\rbrace$ E. $\\lbrace b,d,e\\rbrace$\nDetermine $X-W$ [ANS] A. $\\lbrace f\\rbrace$ B. $\\lbrace 0,0,0\\rbrace$ C. $\\lbrace 4,5\\rbrace$ D. $\\lbrace 1,2,3\\rbrace$ E. $\\lbrace 1,2,3,4,5\\rbrace$", "answer_v1": [ "E", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Let $W=\\lbrace a,b,c,d,e\\rbrace$, $X=\\lbrace a,c,f\\rbrace$\nDetermine $W-X$ [ANS] A. $\\lbrace 0,0,0,4,5\\rbrace$ B. $\\lbrace 1,2,3,4,5\\rbrace$ C. $\\lbrace b,d,e\\rbrace$ D. $\\lbrace 1,2,3$ E. $\\lbrace\\rbrace$\nDetermine $X-W$ [ANS] A. $\\lbrace 0,0,0\\rbrace$ B. $\\lbrace 4,5\\rbrace$ C. $\\lbrace 1,2,3\\rbrace$ D. $\\lbrace f\\rbrace$ E. $\\lbrace 1,2,3,4,5\\rbrace$", "answer_v2": [ "C", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Let $W=\\lbrace a,b,c,d,e\\rbrace$, $X=\\lbrace a,c,f\\rbrace$\nDetermine $W-X$ [ANS] A. $\\lbrace\\rbrace$ B. $\\lbrace 1,2,3$ C. $\\lbrace b,d,e\\rbrace$ D. $\\lbrace 1,2,3,4,5\\rbrace$ E. $\\lbrace 0,0,0,4,5\\rbrace$\nDetermine $X-W$ [ANS] A. $\\lbrace 1,2,3,4,5\\rbrace$ B. $\\lbrace 0,0,0\\rbrace$ C. $\\lbrace 1,2,3\\rbrace$ D. $\\lbrace 4,5\\rbrace$ E. $\\lbrace f\\rbrace$", "answer_v3": [ "C", "E" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0008", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "2", "keywords": [ "Probability", "set theory", "subset", "union", "intersection" ], "problem_v1": "Let $A=\\{1, 2, 4, 5, 7\\}$, $B=\\{3, 4, 5, 7\\}$, $C=\\{0, 1, 3, 4, 6, 8\\}$. List the elements of the following sets. If there is more than one element write them separated by commas. $A \\cap B=\\{$ [ANS] $\\}$ $A \\cup B=\\{$ [ANS] $\\}$ $(B \\cup C) \\cap A=\\{$ [ANS] $\\}$ $B \\cup (C \\cap A)=\\{$ [ANS] $\\}$", "answer_v1": [ "(4, 5, 7)", "(1, 2, 3, 4, 5, 7)", "(1, 4, 5, 7)", "(1, 3, 4, 5, 7)" ], "answer_type_v1": [ "UOL", "UOL", "UOL", "UOL" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $A=\\{2, 3, 4, 7, 8\\}$, $B=\\{1, 2, 4, 8\\}$, $C=\\{0, 1, 5, 6, 7, 8\\}$. List the elements of the following sets. If there is more than one element write them separated by commas. $A \\cap B=\\{$ [ANS] $\\}$ $A \\cup B=\\{$ [ANS] $\\}$ $(B \\cup C) \\cap A=\\{$ [ANS] $\\}$ $B \\cup (C \\cap A)=\\{$ [ANS] $\\}$", "answer_v2": [ "(2, 4, 8)", "(1, 2, 3, 4, 7, 8)", "(2, 4, 7, 8)", "(1, 2, 4, 7, 8)" ], "answer_type_v2": [ "UOL", "UOL", "UOL", "UOL" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $A=\\{1, 3, 4, 5, 6\\}$, $B=\\{1, 5, 6, 8\\}$, $C=\\{0, 2, 3, 5, 7, 8\\}$. List the elements of the following sets. If there is more than one element write them separated by commas. $A \\cap B=\\{$ [ANS] $\\}$ $A \\cup B=\\{$ [ANS] $\\}$ $(B \\cup C) \\cap A=\\{$ [ANS] $\\}$ $B \\cup (C \\cap A)=\\{$ [ANS] $\\}$", "answer_v3": [ "(1, 5, 6)", "(1, 3, 4, 5, 6, 8)", "(1, 3, 5, 6)", "(1, 3, 5, 6, 8)" ], "answer_type_v3": [ "UOL", "UOL", "UOL", "UOL" ], "options_v3": [ [], [], [], [] ] }, { "id": "Set_theory_and_logic_0009", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "2", "keywords": [ "Probability" ], "problem_v1": "Let U=Universal set $=\\{$ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 $\\}$, $A=\\{1, 2, 4, 5, 7, 8\\}$, \u00a0 and $B=\\{4, 5, 6, 7\\}$. List the elements of the following sets. If there is more than one element write them separated by commas. $\\overline{A}=\\{$ [ANS] $\\}$ $\\overline{A \\cup B}=\\{$ [ANS] $\\}$ $\\overline{A \\cup \\overline{B}}=\\{$ [ANS] $\\}$ $A \\cap \\overline{B}=\\{$ [ANS] $\\}$", "answer_v1": [ "(0, 3, 6, 9)", "(0, 3, 9)", "6", "(1, 2, 8)" ], "answer_type_v1": [ "UOL", "UOL", "NV", "UOL" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let U=Universal set $=\\{$ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 $\\}$, $A=\\{0, 2, 3, 4, 8, 9\\}$, \u00a0 and $B=\\{0, 1, 2, 9\\}$. List the elements of the following sets. If there is more than one element write them separated by commas. $\\overline{A}=\\{$ [ANS] $\\}$ $\\overline{A \\cup B}=\\{$ [ANS] $\\}$ $\\overline{A \\cup \\overline{B}}=\\{$ [ANS] $\\}$ $A \\cap \\overline{B}=\\{$ [ANS] $\\}$", "answer_v2": [ "(1, 5, 6, 7)", "(5, 6, 7)", "1", "(3, 4, 8)" ], "answer_type_v2": [ "UOL", "UOL", "NV", "UOL" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let U=Universal set $=\\{$ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 $\\}$, $A=\\{1, 2, 3, 4, 5, 6\\}$, \u00a0 and $B=\\{2, 3, 6, 9\\}$. List the elements of the following sets. If there is more than one element write them separated by commas. $\\overline{A}=\\{$ [ANS] $\\}$ $\\overline{A \\cup B}=\\{$ [ANS] $\\}$ $\\overline{A \\cup \\overline{B}}=\\{$ [ANS] $\\}$ $A \\cap \\overline{B}=\\{$ [ANS] $\\}$", "answer_v3": [ "(0, 7, 8, 9)", "(0, 7, 8)", "9", "(1, 4, 5)" ], "answer_type_v3": [ "UOL", "UOL", "NV", "UOL" ], "options_v3": [ [], [], [], [] ] }, { "id": "Set_theory_and_logic_0010", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "4", "keywords": [ "proposition" ], "problem_v1": "Determine whether the given equality is true (T) or false (F). We use standard notation for intervals (open, closed, or half-open) in the set of real numbers. For instance, $[a,b)=\\lbrace x \\in {\\mathbb R} \\ | \\ a \\le x < b \\rbrace$, and $(a,\\infty)=\\lbrace x \\in {\\mathbb R} \\ | \\ x > a \\rbrace$. The set of positive integers is denoted by ${\\mathbb N}$. [ANS] 1. $\\bigcap_{n \\in {\\mathbb N}} (\\frac{1}{n},n)=\\emptyset$ [ANS] 2. $\\bigcup_{n \\in {\\mathbb N}} [\\frac{1}{n},n]=[0,\\infty)$ [ANS] 3. $\\bigcup_{n \\in {\\mathbb N}} (\\frac{1}{n},n)=(0,\\infty)$ [ANS] 4. $\\bigcap_{n \\in {\\mathbb N}} [\\frac{1}{n},n]=\\emptyset$ [ANS] 5. $\\bigcup_{n \\in {\\mathbb N}} [\\frac{1}{n},n]=(0,\\infty)$", "answer_v1": [ "T", "F", "T", "F", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Determine whether the given equality is true (T) or false (F). We use standard notation for intervals (open, closed, or half-open) in the set of real numbers. For instance, $[a,b)=\\lbrace x \\in {\\mathbb R} \\ | \\ a \\le x < b \\rbrace$, and $(a,\\infty)=\\lbrace x \\in {\\mathbb R} \\ | \\ x > a \\rbrace$. The set of positive integers is denoted by ${\\mathbb N}$. [ANS] 1. $\\bigcup_{n \\in {\\mathbb N}} (\\frac{1}{n},n)=(0,\\infty)$ [ANS] 2. $\\bigcup_{n \\in {\\mathbb N}} [\\frac{1}{n},n]=[0,\\infty)$ [ANS] 3. $\\bigcup_{n \\in \\{1,3,4\\}} [\\frac{1}{n},n]=[\\frac{1}{4},4]$ [ANS] 4. $\\bigcap_{n \\in {\\mathbb N}} (\\frac{1}{n},n)=\\lbrace 1 \\rbrace$ [ANS] 5. $\\bigcap_{n \\in {\\mathbb N}} [\\frac{1}{n},n]=\\emptyset$", "answer_v2": [ "T", "F", "T", "F", "F" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Determine whether the given equality is true (T) or false (F). We use standard notation for intervals (open, closed, or half-open) in the set of real numbers. For instance, $[a,b)=\\lbrace x \\in {\\mathbb R} \\ | \\ a \\le x < b \\rbrace$, and $(a,\\infty)=\\lbrace x \\in {\\mathbb R} \\ | \\ x > a \\rbrace$. The set of positive integers is denoted by ${\\mathbb N}$. [ANS] 1. $\\bigcap_{n \\in {\\mathbb N}} [\\frac{1}{n},n]=\\emptyset$ [ANS] 2. $\\bigcap_{n \\in {\\mathbb N}} [\\frac{1}{n},n]=\\emptyset$ [ANS] 3. $\\bigcup_{n \\in {\\mathbb N}} [\\frac{1}{n},n]=(0,\\infty)$ [ANS] 4. $\\bigcap_{1\\leq n\\leq 7} [\\frac{1}{n},n]=\\lbrace 1 \\rbrace$ [ANS] 5. $\\bigcap_{n \\in {\\mathbb N}} (\\frac{1}{n},n)=\\emptyset$", "answer_v3": [ "F", "F", "T", "T", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0011", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "2", "keywords": [ "inequalities" ], "problem_v1": "Given the sets\nA=\\{2, 3, 6, 7, 8\\} \\quad \\mbox{and} \\quad B=\\{5, 6, 7, 8\\} find the following sets and write the answer using set notation. Separate multiple elements by commas. Write empty if the set contains no elements.\n(a) $A \\cap B=$ [ANS]\n(b) $A \\cup B=$ [ANS]", "answer_v1": [ "(6,7,8)", "(2,3,5,6,7,8)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Given the sets\nA=\\{2, 4, 5, 10, 11\\} \\quad \\mbox{and} \\quad B=\\{2, 3, 5, 11\\} find the following sets and write the answer using set notation. Separate multiple elements by commas. Write empty if the set contains no elements.\n(a) $A \\cap B=$ [ANS]\n(b) $A \\cup B=$ [ANS]", "answer_v2": [ "(2,5,11)", "(2,3,4,5,10,11)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Given the sets\nA=\\{1, 2, 5, 6, 7\\} \\quad \\mbox{and} \\quad B=\\{2, 6, 7, 10\\} find the following sets and write the answer using set notation. Separate multiple elements by commas. Write empty if the set contains no elements.\n(a) $A \\cap B=$ [ANS]\n(b) $A \\cup B=$ [ANS]", "answer_v3": [ "(2,6,7)", "(1,2,5,6,7,10)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Set_theory_and_logic_0012", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "3", "keywords": [ "set theory", "union", "intersection" ], "problem_v1": "Let $A=\\left(4,91\\right]$ and $B=\\left[38,105\\right)$. Compute: $A \\cap B=$ [ANS]\n$A \\cup B=$ [ANS]\n$(A \\cup B)'=$ [ANS]\n(Note: write your answers using).", "answer_v1": [ "[38,91]", "(4,105)", "(-infinity,4] U [105,infinity)" ], "answer_type_v1": [ "INT", "INT", "INT" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $A=\\left(-\\infty,78\\right]$ and $B=\\left(-\\infty,103\\right)$. Compute: $A \\cap B=$ [ANS]\n$A \\cup B=$ [ANS]\n$A' \\cap B=$ [ANS]\n(Note: write your answers using).", "answer_v2": [ "(-infinity,78]", "(-infinity,103)", "(78,103)" ], "answer_type_v2": [ "INT", "INT", "INT" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $A=\\left(-\\infty,82\\right]$ and $B=\\left(32,\\infty \\right)$. Compute: $A \\cap B=$ [ANS]\n$A \\cup B=$ [ANS]\n$A' \\cap B=$ [ANS]\n(Note: write your answers using).", "answer_v3": [ "(32,82]", "(-infinity,infinity)", "(82,infinity)" ], "answer_type_v3": [ "INT", "INT", "INT" ], "options_v3": [ [], [], [] ] }, { "id": "Set_theory_and_logic_0013", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Boolean operations on sets", "level": "3", "keywords": [ "sets", "intervals", "union", "intersection", "complement", "set difference" ], "problem_v1": "Consider the universal set $\\mathbb{R},$ define the interval $A=[-5,1],$ interval $B=(-1,4),$ and $C$ be the negative real numbers. Complete the following exercises in interval notation.\n1. $A \\cup B=$ [ANS]\n2. $A \\cap B \\cap C=$ [ANS]\n3. $A \\cap (B \\cup C)=$ [ANS]\n4. $A^c\\cup B^c \\cup C^c=$ [ANS]\n5. $(C-A)\\cup(B-C)=$ [ANS]\nNote: You may write the empty set $\\emptyset$ as \"EmptySet\".", "answer_v1": [ "[-5,4)", "(-1,0)", "[-5,1]", "(-infinity,-1] U [0,infinity)", "(-infinity,-5) U [0,4)" ], "answer_type_v1": [ "INT", "INT", "INT", "INT", "INT" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Consider the universal set $\\mathbb{R},$ define the interval $A=[-8,1],$ interval $B=(-1,5),$ and $C$ be the negative real numbers. Complete the following exercises in interval notation.\n1. $A \\cup B=$ [ANS]\n2. $A \\cap B \\cap C=$ [ANS]\n3. $A \\cap (B \\cup C)=$ [ANS]\n4. $A^c\\cup B^c \\cup C^c=$ [ANS]\n5. $(C-A)\\cup(B-C)=$ [ANS]\nNote: You may write the empty set $\\emptyset$ as \"EmptySet\".", "answer_v2": [ "[-8,5)", "(-1,0)", "[-8,1]", "(-infinity,-1] U [0,infinity)", "(-infinity,-8) U [0,5)" ], "answer_type_v2": [ "INT", "INT", "INT", "INT", "INT" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Consider the universal set $\\mathbb{R},$ define the interval $A=[-7,1],$ interval $B=(-1,4),$ and $C$ be the negative real numbers. Complete the following exercises in interval notation.\n1. $A \\cup B=$ [ANS]\n2. $A \\cap B \\cap C=$ [ANS]\n3. $A \\cap (B \\cup C)=$ [ANS]\n4. $A^c\\cup B^c \\cup C^c=$ [ANS]\n5. $(C-A)\\cup(B-C)=$ [ANS]\nNote: You may write the empty set $\\emptyset$ as \"EmptySet\".", "answer_v3": [ "[-7,4)", "(-1,0)", "[-7,1]", "(-infinity,-1] U [0,infinity)", "(-infinity,-7) U [0,4)" ], "answer_type_v3": [ "INT", "INT", "INT", "INT", "INT" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0014", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Membership tables", "level": "2", "keywords": [ "Set", "Intersection", "Union", "Containment", "Complement", "set theory", "subset", "union", "intersection" ], "problem_v1": "Complete the following membership table by filling in the blanks with 1 or 0 as appropriate.\n$\\begin{array}{cccccc}\\hline A & B & B-A & A \\cup B & A \\cap (B-A) & A \\cup (B-A) \\\\\\hline1 & 1 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline1 & 0 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline0 & 1 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline0 & 0 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\nUse the membership table above to answer the following questions. For each part, check the answer that most completely describes the general situation. (1) $A-B$ [ANS] A. $\\subseteq A$ B. $=B-A$ C. $=A$ D. $\\subseteq B$\n(2) $A \\cap (B-A)$ [ANS] A. $=\\emptyset$ B. $=A \\cap B$ C. $=B$ D. $=A$\n(3) $A \\cup (B-A)$ [ANS] A. $=B$ B. $=A \\cap B$ C. $=A \\cup B$ D. $=A$", "answer_v1": [ "0", "1", "0", "1", "0", "1", "0", "1", "1", "1", "0", "1", "0", "0", "0", "0", "A", "A", "C" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Complete the following membership table by filling in the blanks with 1 or 0 as appropriate.\n$\\begin{array}{cccccc}\\hline A & B & B-A & A \\cup B & A \\cap (B-A) & A \\cup (B-A) \\\\\\hline1 & 1 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline1 & 0 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline0 & 1 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline0 & 0 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\nUse the membership table above to answer the following questions. For each part, check the answer that most completely describes the general situation. (1) $A-B$ [ANS] A. $\\subseteq A$ B. $=B-A$ C. $=A$ D. $\\subseteq B$\n(2) $A \\cap (B-A)$ [ANS] A. $=B$ B. $=A \\cap B$ C. $=A$ D. $=\\emptyset$\n(3) $A \\cup (B-A)$ [ANS] A. $=B$ B. $=A \\cap B$ C. $=A$ D. $=A \\cup B$", "answer_v2": [ "0", "1", "0", "1", "0", "1", "0", "1", "1", "1", "0", "1", "0", "0", "0", "0", "A", "D", "D" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Complete the following membership table by filling in the blanks with 1 or 0 as appropriate.\n$\\begin{array}{cccccc}\\hline A & B & B-A & A \\cup B & A \\cap (B-A) & A \\cup (B-A) \\\\\\hline1 & 1 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline1 & 0 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline0 & 1 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline0 & 0 & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\nUse the membership table above to answer the following questions. For each part, check the answer that most completely describes the general situation. (1) $A-B$ [ANS] A. $\\subseteq A$ B. $=A$ C. $=B-A$ D. $\\subseteq B$\n(2) $A \\cap (B-A)$ [ANS] A. $=B$ B. $=\\emptyset$ C. $=A \\cap B$ D. $=A$\n(3) $A \\cup (B-A)$ [ANS] A. $=B$ B. $=A \\cup B$ C. $=A$ D. $=A \\cap B$", "answer_v3": [ "0", "1", "0", "1", "0", "1", "0", "1", "1", "1", "0", "1", "0", "0", "0", "0", "A", "B", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "MCS", "MCS", "MCS" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Set_theory_and_logic_0015", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Products", "level": "2", "keywords": [ "set theory", "cartesian product", "Set", "Intersection", "Union", "Containment", "Complement", "Product" ], "problem_v1": "$A=\\lbrace 1, 3, 5 \\rbrace, B=\\lbrace 2,3 \\rbrace$ Check ALL of the following Cartesian products to which the following elements belong: (b) $(3,1)$ [ANS] A. $A \\times A$ B. $A \\times B$ C. $B \\times B$ D. $B \\times A$\n(c) $(1,1)$ [ANS] A. $B \\times B$ B. $A \\times B$ C. $A \\times A$ D. $B \\times A$\n(d) $(3,3)$ [ANS] A. $B \\times A$ B. $B \\times B$ C. $A \\times B$ D. $A \\times A$\n(a) $(1, 2)$ [ANS] A. $A \\times B$ B. $B \\times B$ C. $B \\times A$ D. $A \\times A$", "answer_v1": [ "AD", "C", "ABCD", "A" ], "answer_type_v1": [ "MCM", "MCS", "MCM", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "$A=\\lbrace 5, 7, 9 \\rbrace, B=\\lbrace 6,7 \\rbrace$ Check ALL of the following Cartesian products to which the following elements belong: (b) $(7,5)$ [ANS] A. $A \\times B$ B. $B \\times B$ C. $A \\times A$ D. $B \\times A$\n(a) $(5, 6)$ [ANS] A. $A \\times B$ B. $B \\times B$ C. $B \\times A$ D. $A \\times A$\n(c) $(5,5)$ [ANS] A. $B \\times B$ B. $A \\times B$ C. $B \\times A$ D. $A \\times A$\n(d) $(7,7)$ [ANS] A. $B \\times B$ B. $A \\times A$ C. $A \\times B$ D. $B \\times A$", "answer_v2": [ "CD", "A", "D", "ABCD" ], "answer_type_v2": [ "MCM", "MCS", "MCS", "MCM" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "$A=\\lbrace 1, 3, 5 \\rbrace, B=\\lbrace 2,3 \\rbrace$ Check ALL of the following Cartesian products to which the following elements belong: (b) $(3,1)$ [ANS] A. $B \\times A$ B. $A \\times B$ C. $A \\times A$ D. $B \\times B$\n(d) $(3,3)$ [ANS] A. $A \\times A$ B. $B \\times A$ C. $B \\times B$ D. $A \\times B$\n(a) $(1, 2)$ [ANS] A. $A \\times B$ B. $B \\times A$ C. $B \\times B$ D. $A \\times A$\n(c) $(1,1)$ [ANS] A. $B \\times B$ B. $A \\times A$ C. $B \\times A$ D. $A \\times B$", "answer_v3": [ "AC", "ABCD", "A", "B" ], "answer_type_v3": [ "MCM", "MCM", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Set_theory_and_logic_0016", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Power sets", "level": "3", "keywords": [ "set theory", "subset", "power set" ], "problem_v1": "Let A be the following set. A=$\\lbrace \\emptyset$, $1$, $\\lbrace1,2 \\rbrace \\rbrace$. Mark each of the following true T or false F. [ANS] 1. $\\lbrace \\emptyset, 1 \\rbrace \\in P(A)$ [ANS] 2. $\\lbrace \\emptyset,1 \\rbrace \\in A \\times A$ [ANS] 3. $\\lbrace 1,2 \\rbrace \\subseteq A$ [ANS] 4. $\\lbrace \\lbrace 1,2 \\rbrace \\rbrace \\subseteq A$ [ANS] 5. $\\lbrace \\emptyset \\rbrace \\in P(A)$", "answer_v1": [ "T", "F", "F", "T", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let A be the following set. A=$\\lbrace \\emptyset$, $1$, $\\lbrace1,2 \\rbrace \\rbrace$. Mark each of the following true T or false F. [ANS] 1. $(1,2) \\in A \\times A$ [ANS] 2. $\\lbrace \\emptyset,1 \\rbrace \\in A \\times A$ [ANS] 3. $\\lbrace \\lbrace 1,2 \\rbrace \\rbrace \\in P(A)$ [ANS] 4. $\\lbrace \\emptyset, 1 \\rbrace \\in P(A)$ [ANS] 5. $\\lbrace \\lbrace 1,2 \\rbrace \\rbrace \\subseteq A$", "answer_v2": [ "F", "F", "T", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let A be the following set. A=$\\lbrace \\emptyset$, $1$, $\\lbrace1,2 \\rbrace \\rbrace$. Mark each of the following true T or false F. [ANS] 1. $(1,2) \\in A \\times A$ [ANS] 2. $\\lbrace \\emptyset,1 \\rbrace \\in A \\times A$ [ANS] 3. $\\lbrace \\lbrace 1,2 \\rbrace \\rbrace \\in P(A)$ [ANS] 4. $\\lbrace \\emptyset \\rbrace \\in P(A)$ [ANS] 5. $\\lbrace 1,2 \\rbrace \\subseteq A$", "answer_v3": [ "F", "F", "T", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0017", "subject": "Set_theory_and_logic", "topic": "Operations on sets", "subtopic": "Cardinality", "level": "5", "keywords": [ "sets", "intersection", "union" ], "problem_v1": "A certain discrete mathematics class consists of $26$ students. Of these, $13$ plan to major in mathematics and $16$ plan to major in computer science. Five students are not planning to major in either subject. How many students are planning to major in both subjects? (Be prepared to explain your reasoning with some sort of diagram.) Number of students majoring in both=[ANS]", "answer_v1": [ "8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A certain discrete mathematics class consists of $30$ students. Of these, $15$ plan to major in mathematics and $13$ plan to major in computer science. Five students are not planning to major in either subject. How many students are planning to major in both subjects? (Be prepared to explain your reasoning with some sort of diagram.) Number of students majoring in both=[ANS]", "answer_v2": [ "3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A certain discrete mathematics class consists of $26$ students. Of these, $13$ plan to major in mathematics and $13$ plan to major in computer science. Five students are not planning to major in either subject. How many students are planning to major in both subjects? (Be prepared to explain your reasoning with some sort of diagram.) Number of students majoring in both=[ANS]", "answer_v3": [ "5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Set_theory_and_logic_0018", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Subset", "level": "2", "keywords": [ "set theory", "subset", "Set", "Intersection", "Union", "Containment", "Complement" ], "problem_v1": "Suppose that $A=\\lbrace 2,4,6 \\rbrace, B=\\lbrace 2,6 \\rbrace, C=\\lbrace 4,6 \\rbrace$ and $D=\\lbrace 4,6,8 \\rbrace$. Determine which of these sets are subsets of which other of these sets. Check ALL correct answers below. [ANS] A. $D \\subseteq C$ B. $B \\subseteq A$ C. $A \\subseteq D$ D. $D \\subseteq A$ E. $C \\subseteq A$ F. $A \\subseteq C$ G. $A \\subseteq B$ H. $B \\subseteq D$ I. $C \\subseteq D$ J. $B \\subseteq C$ K. $D \\subseteq B$", "answer_v1": [ "BEI" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ] ], "problem_v2": "Suppose that $A=\\lbrace 2,4,6 \\rbrace, B=\\lbrace 2,6 \\rbrace, C=\\lbrace 4,6 \\rbrace$ and $D=\\lbrace 4,6,8 \\rbrace$. Determine which of these sets are subsets of which other of these sets. Check ALL correct answers below. [ANS] A. $A \\subseteq C$ B. $B \\subseteq A$ C. $C \\subseteq A$ D. $D \\subseteq A$ E. $A \\subseteq D$ F. $C \\subseteq D$ G. $B \\subseteq D$ H. $D \\subseteq C$ I. $A \\subseteq B$ J. $B \\subseteq C$ K. $D \\subseteq B$", "answer_v2": [ "BCF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ] ], "problem_v3": "Suppose that $A=\\lbrace 2,4,6 \\rbrace, B=\\lbrace 2,6 \\rbrace, C=\\lbrace 4,6 \\rbrace$ and $D=\\lbrace 4,6,8 \\rbrace$. Determine which of these sets are subsets of which other of these sets. Check ALL correct answers below. [ANS] A. $D \\subseteq B$ B. $B \\subseteq A$ C. $A \\subseteq C$ D. $D \\subseteq C$ E. $D \\subseteq A$ F. $B \\subseteq D$ G. $A \\subseteq D$ H. $A \\subseteq B$ I. $C \\subseteq A$ J. $B \\subseteq C$ K. $C \\subseteq D$", "answer_v3": [ "BIK" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ] ] }, { "id": "Set_theory_and_logic_0019", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Subset", "level": "2", "keywords": [], "problem_v1": "Let $T=\\lbrace x,x^2,x^3,x^4,x^5\\rbrace$\nDetermine which of the following sets are subsets of $T$ (Check all that apply) [ANS] A. $\\lbrace\\rbrace$ B. $\\lbrace x^2+x,x^5\\rbrace$ C. $\\lbrace x^5,x^6\\rbrace$ D. $\\lbrace x\\rbrace$ E. $\\lbrace x^4,x^2\\rbrace$", "answer_v1": [ "ADE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Let $T=\\lbrace x,x^2,x^3,x^4,x^5\\rbrace$\nDetermine which of the following sets are subsets of $T$ (Check all that apply) [ANS] A. $\\lbrace\\rbrace$ B. $\\lbrace x^2+x,x^5\\rbrace$ C. $\\lbrace x\\rbrace$ D. $\\lbrace x^5,x^6\\rbrace$ E. $\\lbrace x^4,x^2\\rbrace$", "answer_v2": [ "ACE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Let $T=\\lbrace x,x^2,x^3,x^4,x^5\\rbrace$\nDetermine which of the following sets are subsets of $T$ (Check all that apply) [ANS] A. $\\lbrace x^2+x,x^5\\rbrace$ B. $\\lbrace\\rbrace$ C. $\\lbrace x\\rbrace$ D. $\\lbrace x^5,x^6\\rbrace$ E. $\\lbrace x^4,x^2\\rbrace$", "answer_v3": [ "BCE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0020", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Subset", "level": "2", "keywords": [], "problem_v1": "Let $S=\\lbrace a,b,c,f,g\\rbrace$\nDetermine which of the following sets are subsets of $S$ (Check all that apply) [ANS] A. $\\lbrace a,f,g\\rbrace$ B. $\\lbrace f,g,h\\rbrace$ C. $\\lbrace q,r,s,t\\rbrace$ D. $\\lbrace a,b,d\\rbrace$ E. $\\lbrace a,b,c,f,g\\rbrace$", "answer_v1": [ "AE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Let $S=\\lbrace a,b,c,f,g\\rbrace$\nDetermine which of the following sets are subsets of $S$ (Check all that apply) [ANS] A. $\\lbrace a,b,c,f,g\\rbrace$ B. $\\lbrace f,g,h\\rbrace$ C. $\\lbrace q,r,s,t\\rbrace$ D. $\\lbrace a,f,g\\rbrace$ E. $\\lbrace a,b,d\\rbrace$", "answer_v2": [ "AD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Let $S=\\lbrace a,b,c,f,g\\rbrace$\nDetermine which of the following sets are subsets of $S$ (Check all that apply) [ANS] A. $\\lbrace a,b,d\\rbrace$ B. $\\lbrace a,b,c,f,g\\rbrace$ C. $\\lbrace f,g,h\\rbrace$ D. $\\lbrace q,r,s,t\\rbrace$ E. $\\lbrace a,f,g\\rbrace$", "answer_v3": [ "BE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0021", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Subset", "level": "2", "keywords": [], "problem_v1": "Let $R=\\lbrace\\text{numbers divisible by 2}\\rbrace$\nDetermine which of the following sets are subsets of $R$ [ANS] A. $\\lbrace 0\\rbrace$ B. $\\lbrace 1\\rbrace$ C. $\\lbrace 44,221,90\\rbrace$ D. $\\lbrace integers\\rbrace$ E. $\\lbrace-82,104,16\\rbrace$", "answer_v1": [ "AE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Let $R=\\lbrace\\text{numbers divisible by 2}\\rbrace$\nDetermine which of the following sets are subsets of $R$ [ANS] A. $\\lbrace-82,104,16\\rbrace$ B. $\\lbrace 1\\rbrace$ C. $\\lbrace 44,221,90\\rbrace$ D. $\\lbrace 0\\rbrace$ E. $\\lbrace integers\\rbrace$", "answer_v2": [ "AD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Let $R=\\lbrace\\text{numbers divisible by 2}\\rbrace$\nDetermine which of the following sets are subsets of $R$ [ANS] A. $\\lbrace integers\\rbrace$ B. $\\lbrace-82,104,16\\rbrace$ C. $\\lbrace 1\\rbrace$ D. $\\lbrace 44,221,90\\rbrace$ E. $\\lbrace 0\\rbrace$", "answer_v3": [ "BE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0022", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Subset", "level": "3", "keywords": [ "proposition" ], "problem_v1": "Enter T or F depending on whether the statement is true or false. (You must enter T or F--True and False will not work.) [ANS] 1. $\\{\\emptyset\\}\\subseteq \\{\\emptyset\\}$ [ANS] 2. $\\{\\emptyset\\}\\not\\in \\{\\emptyset,\\{\\emptyset\\}\\}$ [ANS] 3. $\\{\\emptyset\\}\\in \\{\\emptyset\\}$", "answer_v1": [ "T", "F", "F" ], "answer_type_v1": [ "TF", "TF", "TF" ], "options_v1": [ [], [], [] ], "problem_v2": "Enter T or F depending on whether the statement is true or false. (You must enter T or F--True and False will not work.) [ANS] 1. $\\emptyset \\in \\{\\emptyset\\}$ [ANS] 2. $\\emptyset\\subseteq \\{\\emptyset\\}$ [ANS] 3. $\\{\\emptyset\\}\\in \\{\\emptyset\\}$", "answer_v2": [ "T", "T", "F" ], "answer_type_v2": [ "TF", "TF", "TF" ], "options_v2": [ [], [], [] ], "problem_v3": "Enter T or F depending on whether the statement is true or false. (You must enter T or F--True and False will not work.) [ANS] 1. $\\emptyset\\subseteq \\{\\emptyset\\}$ [ANS] 2. $\\emptyset \\not\\subseteq \\emptyset$ [ANS] 3. $\\emptyset \\in \\{\\emptyset\\}$", "answer_v3": [ "T", "F", "T" ], "answer_type_v3": [ "TF", "TF", "TF" ], "options_v3": [ [], [], [] ] }, { "id": "Set_theory_and_logic_0023", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Properties of relations", "level": "4", "keywords": [ "relation", "reflexive", "symmetric", "transitive" ], "problem_v1": "Given the following relations on the set of all integers where $(x,y) \\in R$ if and only if the following is satisfied. (Check ALL correct answers from the following lists):\n(a) $x+y=0$ [ANS] A. transitive B. reflexive C. irreflexive D. antisymmetric E. symmetric\n(b) $x-y$ is an integer [ANS] A. reflexive B. symmetric C. antisymmetric D. transitive E. irreflexive\n(c) $x=2y$ [ANS] A. symmetric B. irreflexive C. antisymmetric D. transitive E. reflexive\n(d) $xy > 1$ [ANS] A. irreflexive B. symmetric C. transitive D. antisymmetric E. reflexive", "answer_v1": [ "E", "ABD", "C", "B" ], "answer_type_v1": [ "MCS", "MCM", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Given the following relations on the set of all integers where $(x,y) \\in R$ if and only if the following is satisfied. (Check ALL correct answers from the following lists):\n(a) $x+y=0$ [ANS] A. antisymmetric B. irreflexive C. symmetric D. transitive E. reflexive\n(b) $x-y$ is an integer [ANS] A. irreflexive B. antisymmetric C. symmetric D. reflexive E. transitive\n(c) $x=2y$ [ANS] A. irreflexive B. reflexive C. transitive D. antisymmetric E. symmetric\n(d) $xy > 1$ [ANS] A. transitive B. irreflexive C. symmetric D. reflexive E. antisymmetric", "answer_v2": [ "C", "CDE", "D", "C" ], "answer_type_v2": [ "MCS", "MCM", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Given the following relations on the set of all integers where $(x,y) \\in R$ if and only if the following is satisfied. (Check ALL correct answers from the following lists):\n(a) $x+y=0$ [ANS] A. reflexive B. transitive C. symmetric D. irreflexive E. antisymmetric\n(b) $x-y$ is an integer [ANS] A. symmetric B. irreflexive C. antisymmetric D. transitive E. reflexive\n(c) $x=2y$ [ANS] A. irreflexive B. antisymmetric C. symmetric D. transitive E. reflexive\n(d) $xy > 1$ [ANS] A. irreflexive B. transitive C. reflexive D. symmetric E. antisymmetric", "answer_v3": [ "C", "ADE", "B", "D" ], "answer_type_v3": [ "MCS", "MCM", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0024", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Properties of relations", "level": "4", "keywords": [ "relation", "reflexive", "symmetric", "transitive" ], "problem_v1": "Define relations $R_1,\\ldots,R_6$ on $\\lbrace 1,2,3,4 \\rbrace$ by $R_1=\\lbrace (2,2),(2,3),(2,4),(3,2),(3,3),(3,4) \\rbrace,$ $R_2=\\lbrace (1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\\rbrace,$ $R_3=\\lbrace (2,4),(4,2) \\rbrace$, $R_4=\\lbrace (1,2),(2,3),(3,4)\\rbrace$, $R_5=\\lbrace (1,1),(2,2),(3,3),(4,4)\\rbrace,$ $R_6=\\lbrace (1,3),(1,4),(2,3),(2,4),(3,1),(3,4) \\rbrace,$ Which of the following statements are correct? Check ALL correct answers below. [ANS] A. $R_2$ is reflexive B. $R_3$ is reflexive C. $R_3$ is transitive D. $R_1$ is reflexive E. $R_5$ is not reflexive F. $R_4$ is symmetric G. $R_3$ is symmetric H. $R_6$ is symmetric I. $R_5$ is transitive J. $R_4$ is antisymmetric K. $R_1$ is not symmetric L. $R_2$ is not transitive M. $R_4$ is transitive", "answer_v1": [ "AGIJK" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M" ] ], "problem_v2": "Define relations $R_1,\\ldots,R_6$ on $\\lbrace 1,2,3,4 \\rbrace$ by $R_1=\\lbrace (2,2),(2,3),(2,4),(3,2),(3,3),(3,4) \\rbrace,$ $R_2=\\lbrace (1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\\rbrace,$ $R_3=\\lbrace (2,4),(4,2) \\rbrace$, $R_4=\\lbrace (1,2),(2,3),(3,4)\\rbrace$, $R_5=\\lbrace (1,1),(2,2),(3,3),(4,4)\\rbrace,$ $R_6=\\lbrace (1,3),(1,4),(2,3),(2,4),(3,1),(3,4) \\rbrace,$ Which of the following statements are correct? Check ALL correct answers below. [ANS] A. $R_5$ is transitive B. $R_3$ is symmetric C. $R_4$ is symmetric D. $R_3$ is transitive E. $R_1$ is reflexive F. $R_4$ is antisymmetric G. $R_6$ is symmetric H. $R_4$ is transitive I. $R_2$ is not transitive J. $R_3$ is reflexive K. $R_1$ is not symmetric L. $R_2$ is reflexive M. $R_5$ is not reflexive", "answer_v2": [ "ABFKL" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M" ] ], "problem_v3": "Define relations $R_1,\\ldots,R_6$ on $\\lbrace 1,2,3,4 \\rbrace$ by $R_1=\\lbrace (2,2),(2,3),(2,4),(3,2),(3,3),(3,4) \\rbrace,$ $R_2=\\lbrace (1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\\rbrace,$ $R_3=\\lbrace (2,4),(4,2) \\rbrace$, $R_4=\\lbrace (1,2),(2,3),(3,4)\\rbrace$, $R_5=\\lbrace (1,1),(2,2),(3,3),(4,4)\\rbrace,$ $R_6=\\lbrace (1,3),(1,4),(2,3),(2,4),(3,1),(3,4) \\rbrace,$ Which of the following statements are correct? Check ALL correct answers below. [ANS] A. $R_5$ is transitive B. $R_5$ is not reflexive C. $R_2$ is reflexive D. $R_3$ is transitive E. $R_4$ is antisymmetric F. $R_3$ is reflexive G. $R_4$ is transitive H. $R_1$ is reflexive I. $R_3$ is symmetric J. $R_4$ is symmetric K. $R_1$ is not symmetric L. $R_2$ is not transitive M. $R_6$ is symmetric", "answer_v3": [ "ACEIK" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M" ] ] }, { "id": "Set_theory_and_logic_0025", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Properties of relations", "level": "4", "keywords": [ "relation", "reflexive", "symmetric", "transitive" ], "problem_v1": "Suppose $R$ and $S$ are relations on a set $A$. Select True or False for each statement below.\n1. If R and S are reflexive relations, then $R \\circ S$ is reflexive. [ANS] 2. If R and S are reflexive relations, then $R-S$ is reflexive. [ANS] 3. If R and S are reflexive relations, then $R \\oplus S$ is reflexive. [ANS]", "answer_v1": [ "TRUE", "FALSE", "FALSE" ], "answer_type_v1": [ "TF", "TF", "TF" ], "options_v1": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v2": "Suppose $R$ and $S$ are relations on a set $A$. Select True or False for each statement below.\n1. If R and S are reflexive relations, then $R \\cup S$ is reflexive. [ANS] 2. If R and S are reflexive relations, then $R \\cap S$ is reflexive. [ANS] 3. If R and S are reflexive relations, then $R \\oplus S$ is reflexive. [ANS]", "answer_v2": [ "TRUE", "TRUE", "FALSE" ], "answer_type_v2": [ "TF", "TF", "TF" ], "options_v2": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v3": "Suppose $R$ and $S$ are relations on a set $A$. Select True or False for each statement below.\n1. If R and S are reflexive relations, then $R \\oplus S$ is reflexive. [ANS] 2. If R and S are reflexive relations, then $R \\circ S$ is reflexive. [ANS] 3. If R and S are reflexive relations, then $R \\cup S$ is reflexive. [ANS]", "answer_v3": [ "FALSE", "TRUE", "TRUE" ], "answer_type_v3": [ "TF", "TF", "TF" ], "options_v3": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ] }, { "id": "Set_theory_and_logic_0026", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Properties of relations", "level": "4", "keywords": [ "relation", "reflexive", "symmetric", "transitive" ], "problem_v1": "Given the following relations on the set of all people. Check ALL correct answers from the following lists:\n(a) $a$ is older than $b$ [ANS] A. transitive B. reflexive C. symmetric D. irreflexive E. antisymmetric\n(b) $a$ and $b$ have a common grandparent [ANS] A. symmetric B. reflexive C. irreflexive D. transitive E. antisymmetric\n(c) $a$ has the same first name as $b$ [ANS] A. symmetric B. irreflexive C. reflexive D. transitive E. antisymmetric\n(d) $a$ and $b$ were born on the same day [ANS] A. antisymmetric B. transitive C. symmetric D. irreflexive E. reflexive", "answer_v1": [ "ADE", "AB", "ACD", "BCE" ], "answer_type_v1": [ "MCM", "MCM", "MCM", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Given the following relations on the set of all people. Check ALL correct answers from the following lists:\n(a) $a$ is older than $b$ [ANS] A. transitive B. reflexive C. irreflexive D. symmetric E. antisymmetric\n(b) $a$ and $b$ have a common grandparent [ANS] A. irreflexive B. antisymmetric C. transitive D. symmetric E. reflexive\n(c) $a$ has the same first name as $b$ [ANS] A. reflexive B. irreflexive C. transitive D. antisymmetric E. symmetric\n(d) $a$ and $b$ were born on the same day [ANS] A. transitive B. antisymmetric C. irreflexive D. reflexive E. symmetric", "answer_v2": [ "ACE", "DE", "ACE", "ADE" ], "answer_type_v2": [ "MCM", "MCM", "MCM", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Given the following relations on the set of all people. Check ALL correct answers from the following lists:\n(a) $a$ is older than $b$ [ANS] A. reflexive B. transitive C. irreflexive D. symmetric E. antisymmetric\n(b) $a$ and $b$ have a common grandparent [ANS] A. reflexive B. irreflexive C. transitive D. antisymmetric E. symmetric\n(c) $a$ has the same first name as $b$ [ANS] A. reflexive B. transitive C. symmetric D. antisymmetric E. irreflexive\n(d) $a$ and $b$ were born on the same day [ANS] A. irreflexive B. reflexive C. transitive D. symmetric E. antisymmetric", "answer_v3": [ "BCE", "AE", "ABC", "BCD" ], "answer_type_v3": [ "MCM", "MCM", "MCM", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0027", "subject": "Set_theory_and_logic", "topic": "Relations between sets", "subtopic": "Properties of relations", "level": "3", "keywords": [ "relation" ], "problem_v1": "Determine which of these relations are transitive. The variables $x$, $y$, $x'$, $y'$ represent integers. [ANS] A. $(x,y) \\sim (x',y')$ if and only if $xy'=x' y$. B. $x \\sim y$ if and only if $x-y$ is a multiple of 10. C. $x \\sim y$ if and only if $x-y$ is negative. D. $(x,y) \\sim (x',y')$ if and only if $x-y=x'-y'$. E. $x \\sim y$ if and only if $x+y$ is odd. F. $x \\sim y$ if and only if $xy$ is negative.", "answer_v1": [ "BCD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Determine which of these relations are transitive. The variables $x$, $y$, $x'$, $y'$ represent integers. [ANS] A. $x \\sim y$ if and only if $xy \\geq 0$. B. $(x,y) \\sim (x',y')$ if and only if $x-y=x'-y'$. C. $x \\sim y$ if and only if $x+y$ is positive. D. $x \\sim y$ if and only if $x+y$ is odd. E. $x \\sim y$ if and only if $x-y$ is positive. F. $x \\sim y$ if and only if $x-y$ is a multiple of 10.", "answer_v2": [ "BEF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Determine which of these relations are transitive. The variables $x$, $y$, $x'$, $y'$ represent integers. [ANS] A. $x \\sim y$ if and only if $x+y$ is positive. B. $(x,y) \\sim (x',y')$ if and only if $xy'=x' y$. C. $x \\sim y$ if and only if $x+y$ is odd. D. $x \\sim y$ if and only if $x-y$ is negative. E. $x \\sim y$ if and only if $x+y$ is even. F. $(x,y) \\sim (x',y')$ if and only if $x-y=x'-y'$.", "answer_v3": [ "DEF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Set_theory_and_logic_0029", "subject": "Set_theory_and_logic", "topic": "Functions", "subtopic": "Injective, surjective, bijective", "level": "4", "keywords": [ "equivalence relations" ], "problem_v1": "Let $X$ be the set $\\lbrace 10, 3, 5 \\rbrace$. For the first three parts of this problem you are asked to define a function $f: X \\rightarrow X$ so that the relation\nu \\sim w \\Leftrightarrow w=f(u) satisfies each of the following conditions.\n(a) $\\sim$ is reflexive\n$\\begin{array}{cccc}\\hline x & 10 & 3 & 5 \\\\\\hline f(x) & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\n(b) $\\sim$ is symmetric\n$\\begin{array}{cccc}\\hline x & 10 & 3 & 5 \\\\\\hline f(x) & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\n(c) $\\sim$ is transitive\n$\\begin{array}{cccc}\\hline x & 10 & 3 & 5 \\\\\\hline f(x) & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$", "answer_v1": [ "(10, 3, 5)", "(10, 3, 5)", "(10, 3, 5)" ], "answer_type_v1": [ "OL", "OL", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $X$ be the set $\\lbrace-17, 18,-14 \\rbrace$. For the first three parts of this problem you are asked to define a function $f: X \\rightarrow X$ so that the relation\nu \\sim w \\Leftrightarrow w=f(u) satisfies each of the following conditions.\n(a) $\\sim$ is reflexive\n$\\begin{array}{cccc}\\hline x &-17 & 18 &-14 \\\\\\hline f(x) & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\n(b) $\\sim$ is symmetric\n$\\begin{array}{cccc}\\hline x &-17 & 18 &-14 \\\\\\hline f(x) & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\n(c) $\\sim$ is transitive\n$\\begin{array}{cccc}\\hline x &-17 & 18 &-14 \\\\\\hline f(x) & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$", "answer_v2": [ "(-17, 18, -14)", "(-17, 18, -14)", "(-17, 18, -14)" ], "answer_type_v2": [ "OL", "OL", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $X$ be the set $\\lbrace-8, 5,-10 \\rbrace$. For the first three parts of this problem you are asked to define a function $f: X \\rightarrow X$ so that the relation\nu \\sim w \\Leftrightarrow w=f(u) satisfies each of the following conditions.\n(a) $\\sim$ is reflexive\n$\\begin{array}{cccc}\\hline x &-8 & 5 &-10 \\\\\\hline f(x) & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\n(b) $\\sim$ is symmetric\n$\\begin{array}{cccc}\\hline x &-8 & 5 &-10 \\\\\\hline f(x) & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\n(c) $\\sim$ is transitive\n$\\begin{array}{cccc}\\hline x &-8 & 5 &-10 \\\\\\hline f(x) & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$", "answer_v3": [ "(-8, 5, -10)", "(-8, 5, -10)", "(-8, 5, -10)" ], "answer_type_v3": [ "OL", "OL", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Set_theory_and_logic_0030", "subject": "Set_theory_and_logic", "topic": "Functions", "subtopic": "Image and inverse image", "level": "4", "keywords": [ "proof\u2019", "\u2018function\u2019", "\u2018sets\u2019", "\u2018image\u2019", "\u2018preimage\u2019" ], "problem_v1": "Let $f(x)=x^{2}$. Find each set. $f ([3, 8))=$ [ANS]\n$f ((-2, 8))=$ [ANS]", "answer_v1": [ "[9,64)", "[0,64)" ], "answer_type_v1": [ "INT", "INT" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)=x^{2}$. Find each set. $f ([4, 5))=$ [ANS]\n$f ((-4, 5))=$ [ANS]", "answer_v2": [ "[16,25)", "[0,25)" ], "answer_type_v2": [ "INT", "INT" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)=x^{2}$. Find each set. $f ([3, 6))=$ [ANS]\n$f ((-4, 6))=$ [ANS]", "answer_v3": [ "[9,36)", "[0,36)" ], "answer_type_v3": [ "INT", "INT" ], "options_v3": [ [], [] ] }, { "id": "Set_theory_and_logic_0031", "subject": "Set_theory_and_logic", "topic": "Functions", "subtopic": "Image and inverse image", "level": "4", "keywords": [ "proof\u2019", "\u2018function\u2019", "\u2018sets\u2019", "\u2018image\u2019", "\u2018preimage\u2019" ], "problem_v1": "Let $f(x)=-4x$. Let $S=[-3, 4)$. What is $f (S)$? [ANS]\nWhat is $f^{-1} (S)$? [ANS]", "answer_v1": [ "(-16,12]", "(-1,0.75]" ], "answer_type_v1": [ "INT", "INT" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)=-7x$. Let $S=[-2, 1)$. What is $f (S)$? [ANS]\nWhat is $f^{-1} (S)$? [ANS]", "answer_v2": [ "(-7,14]", "(-0.142857,0.285714]" ], "answer_type_v2": [ "INT", "INT" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)=-6x$. Let $S=[-3, 2)$. What is $f (S)$? [ANS]\nWhat is $f^{-1} (S)$? [ANS]", "answer_v3": [ "(-12,18]", "(-0.333333,0.5]" ], "answer_type_v3": [ "INT", "INT" ], "options_v3": [ [], [] ] }, { "id": "Set_theory_and_logic_0032", "subject": "Set_theory_and_logic", "topic": "Functions", "subtopic": "Image and inverse image", "level": "4", "keywords": [ "proof\u2019", "function", "\u2018sets\u2019", "\u2018image\u2019", "\u2018preimage\u2019" ], "problem_v1": "Let $f(x)=5x$. Let $S=(0, 6]$. What is $f (S)$? [ANS]\nWhat is $f^{-1} (S)$? [ANS]", "answer_v1": [ "(0,30]", "(0,1.2]" ], "answer_type_v1": [ "INT", "INT" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)=2x$. Let $S=(2, 3]$. What is $f (S)$? [ANS]\nWhat is $f^{-1} (S)$? [ANS]", "answer_v2": [ "(4,6]", "(1,1.5]" ], "answer_type_v2": [ "INT", "INT" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)=3x$. Let $S=(0, 4]$. What is $f (S)$? [ANS]\nWhat is $f^{-1} (S)$? [ANS]", "answer_v3": [ "(0,12]", "(0,1.33333]" ], "answer_type_v3": [ "INT", "INT" ], "options_v3": [ [], [] ] }, { "id": "Set_theory_and_logic_0033", "subject": "Set_theory_and_logic", "topic": "Functions", "subtopic": "Image and inverse image", "level": "4", "keywords": [ "proof\u2019", "\u2018function\u2019", "\u2018sets\u2019", "\u2018image\u2019", "\u2018preimage\u2019" ], "problem_v1": "[A capital U denotes \u201cunion\u201d.] Let $f(x)=2|x|$. Let $S=(5, 7]$. What is $f (S)$? [ANS]\nWhat is $f^{-1} (S)$? [ANS]", "answer_v1": [ "(2*5,2*7]", "[-0.5*7,-0.5*5) U (0.5*5,0.5*7]" ], "answer_type_v1": [ "INT", "INT" ], "options_v1": [ [], [] ], "problem_v2": "[A capital U denotes \u201cunion\u201d.] Let $f(x)=2|x|$. Let $S=(1, 4]$. What is $f (S)$? [ANS]\nWhat is $f^{-1} (S)$? [ANS]", "answer_v2": [ "(2*1,2*4]", "[-0.5*4,-0.5*1) U (0.5*1,0.5*4]" ], "answer_type_v2": [ "INT", "INT" ], "options_v2": [ [], [] ], "problem_v3": "[A capital U denotes \u201cunion\u201d.] Let $f(x)=2|x|$. Let $S=(2, 4]$. What is $f (S)$? [ANS]\nWhat is $f^{-1} (S)$? [ANS]", "answer_v3": [ "(2*2,2*4]", "[-0.5*4,-0.5*2) U (0.5*2,0.5*4]" ], "answer_type_v3": [ "INT", "INT" ], "options_v3": [ [], [] ] }, { "id": "Set_theory_and_logic_0035", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Translation", "level": "5", "keywords": [ "logic", "deduction" ], "problem_v1": "Suppose this is true: All widgets are gadgets.\nWhich is the correct conditional form of the sentence? [ANS] A. If it's a widget, then it's a gadget B. If it's a gadget, then it's a widget\nWhat can be deduced from that and this additional fact? It's a gadget [ANS] A. It's a widget B. It is not a gadget C. It is not a widget D. It's a gadget E. Nothing\nWhat can be deduced from that and this additional fact? It's not a widget [ANS] A. It's a widget B. It is not a gadget C. It is not a widget D. It's a gadget E. Nothing\nWhat can be deduced from that and this additional fact? It's not a gadget [ANS] A. It is not a widget B. It is not a gadget C. It's a gadget D. It's a widget E. Nothing", "answer_v1": [ "A", "E", "E", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Suppose this is true: All widgets are gadgets.\nWhich is the correct conditional form of the sentence? [ANS] A. If it's a widget, then it's a gadget B. If it's a gadget, then it's a widget\nWhat can be deduced from that and this additional fact? It's a gadget [ANS] A. It is not a gadget B. It's a gadget C. It is not a widget D. It's a widget E. Nothing\nWhat can be deduced from that and this additional fact? It's not a widget [ANS] A. It is not a gadget B. It's a gadget C. It is not a widget D. It's a widget E. Nothing\nWhat can be deduced from that and this additional fact? It's not a gadget [ANS] A. It's a gadget B. It is not a gadget C. It's a widget D. It is not a widget E. Nothing", "answer_v2": [ "A", "E", "E", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Suppose this is true: All widgets are gadgets.\nWhich is the correct conditional form of the sentence? [ANS] A. If it's a gadget, then it's a widget B. If it's a widget, then it's a gadget\nWhat can be deduced from that and this additional fact? It's a gadget [ANS] A. It is not a gadget B. It's a widget C. It is not a widget D. It's a gadget E. Nothing\nWhat can be deduced from that and this additional fact? It's not a widget [ANS] A. It is not a gadget B. It's a widget C. It is not a widget D. It's a gadget E. Nothing\nWhat can be deduced from that and this additional fact? It's not a gadget [ANS] A. It's a gadget B. It is not a widget C. It is not a gadget D. It's a widget E. Nothing", "answer_v3": [ "B", "E", "E", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0036", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Translation", "level": "2", "keywords": [ "Logic", "Inclusive", "Exlusive" ], "problem_v1": "For each of the following sentences, determine whether an \"inclusive or\" or an \"exclusive or\" is usually what is meant by the sentence. Enter \"I\" for the inclusive case and \"E\" for the exclusive case. [ANS] 1. Lunch includes soup or salad. [ANS] 2. To enter the country you need a passport or a voter registration card. [ANS] 3. Publish or perish. [ANS]", "answer_v1": [ "E", "I", "E", "I" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "I", "E" ], [ "I", "E" ], [ "I", "E" ], [ "I", "E" ] ], "problem_v2": "For each of the following sentences, determine whether an \"inclusive or\" or an \"exclusive or\" is usually what is meant by the sentence. Enter \"I\" for the inclusive case and \"E\" for the exclusive case. [ANS] 1. To enter the country you need a passport or a voter registration card. [ANS] 2. Publish or perish. [ANS] 3. Lunch includes soup or salad. [ANS]", "answer_v2": [ "I", "E", "E", "I" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "I", "E" ], [ "I", "E" ], [ "I", "E" ], [ "I", "E" ] ], "problem_v3": "For each of the following sentences, determine whether an \"inclusive or\" or an \"exclusive or\" is usually what is meant by the sentence. Enter \"I\" for the inclusive case and \"E\" for the exclusive case. [ANS] 1. To enter the country you need a passport or a voter registration card. [ANS] 2. Experience with C++or Java is required. [ANS] 3. Lunch includes soup or salad. [ANS]", "answer_v3": [ "I", "I", "E", "E" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "I", "E" ], [ "I", "E" ], [ "I", "E" ], [ "I", "E" ] ] }, { "id": "Set_theory_and_logic_0037", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Translation", "level": "2", "keywords": [ "Logic", "Proposition" ], "problem_v1": "Enter \"T\" for each true proposition, \"F\" for each false proposition and \"N\" for each statement which is not a proposition. [ANS] 1. x+1=5 if x=1. [ANS] 2. What time is it? [ANS] 3. 5+7=10. [ANS] 4. All ants are insects. [ANS] 5. This statement is false. [ANS] 6. Do not pass go. [ANS] 7. 2+3=5. [ANS] 8. x+y=y+x for every pair of real numbers x and y.", "answer_v1": [ "F", "N", "F", "T", "N", "N", "T", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Enter \"T\" for each true proposition, \"F\" for each false proposition and \"N\" for each statement which is not a proposition. [ANS] 1. x+1=5 if x=1. [ANS] 2. What time is it? [ANS] 3. This statement is false. [ANS] 4. 2+3=5. [ANS] 5. All insects are ants. [ANS] 6. All ants are insects. [ANS] 7. Do not pass go. [ANS] 8. 5+7=10.", "answer_v2": [ "F", "N", "N", "T", "F", "T", "N", "F" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Enter \"T\" for each true proposition, \"F\" for each false proposition and \"N\" for each statement which is not a proposition. [ANS] 1. What time is it? [ANS] 2. This statement is false. [ANS] 3. 5+7=10. [ANS] 4. 2+3=5. [ANS] 5. Do not pass go. [ANS] 6. All insects are ants. [ANS] 7. x+y=y+x for every pair of real numbers x and y. [ANS] 8. x+1=5 if x=1.", "answer_v3": [ "N", "N", "F", "T", "N", "F", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0038", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Translation", "level": "2", "keywords": [ "discrete mathematics", "logic" ], "problem_v1": "Enter T or F depending on whether the statement is a proposition or not. [ANS] 1. Load the packages in the car. [ANS] 2. Androids dream of electric sheep. [ANS] 3. 1024 is the smallest four-digit number that is a prime. [ANS] 4. The college of engineering. [ANS] 5. My name is Enigo Montoya. [ANS] 6. What a car! [ANS] 7. If x=3 then xx=6. [ANS] 8. E=mcc [ANS] 9. The integer 1 is the smallest positive integer. [ANS] 10. What color is your parachute?", "answer_v1": [ "F", "T", "T", "F", "T", "F", "T", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Enter T or F depending on whether the statement is a proposition or not. [ANS] 1. 8 is a prime number. [ANS] 2. The job market is very bad these days. [ANS] 3. What is the Matrix? [ANS] 4. The integer 1 is the smallest positive integer. [ANS] 5. All continuous functions are differentiable. [ANS] 6. I am French. [ANS] 7. All differentiable functions are continuous. [ANS] 8. Do androids dream of electric sheep? [ANS] 9. 1024. [ANS] 10. 1024 is the smallest four-digit number that is a perfect square.", "answer_v2": [ "T", "T", "F", "T", "T", "T", "T", "F", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Enter T or F depending on whether the statement is a proposition or not. [ANS] 1. The college of engineering. [ANS] 2. She is a mathematics major. [ANS] 3. What color is your parachute? [ANS] 4. Load the packages in the car. [ANS] 5. Four score and seven years ago. [ANS] 6. I am American. [ANS] 7. 128=2. [ANS] 8. Look out for that car! [ANS] 9. Brett Favre was the best quarterback this year. [ANS] 10. Do androids dream of electric sheep?", "answer_v3": [ "F", "T", "F", "F", "F", "T", "T", "F", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0039", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Operations on propositions", "level": "2", "keywords": [ "Logic", "Boolean", "Operation" ], "problem_v1": "What is the value of $x$ after each of the following statements is encountered in a computer program, if $x=7$ before the statement is reached?\n(a) if $x < 6$ then $x:=x+1$ $x=$ [ANS]\n(b) if $(1+1=3)$ OR $(6 > x)$ then $x:=x+1$ $x=$ [ANS]\n(c) if $(2+3=5)$ AND $(3+4=7)$ then $x:=x+1$ $x=$ [ANS]\n(d) if $(1+1=2)$ XOR $(x=6)$ then $x:=x+1$ $x=$ [ANS]", "answer_v1": [ "7", "7", "8", "8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "What is the value of $x$ after each of the following statements is encountered in a computer program, if $x=1$ before the statement is reached?\n(a) if $x < 9$ then $x:=x+1$ $x=$ [ANS]\n(b) if $(1+1=3)$ OR $(9 > x)$ then $x:=x+1$ $x=$ [ANS]\n(c) if $(2+3=5)$ AND $(3+4=7)$ then $x:=x+1$ $x=$ [ANS]\n(d) if $(1+1=2)$ XOR $(x=2)$ then $x:=x+1$ $x=$ [ANS]", "answer_v2": [ "2", "2", "2", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "What is the value of $x$ after each of the following statements is encountered in a computer program, if $x=3$ before the statement is reached?\n(a) if $x < 6$ then $x:=x+1$ $x=$ [ANS]\n(b) if $(1+1=3)$ OR $(6 > x)$ then $x:=x+1$ $x=$ [ANS]\n(c) if $(2+3=5)$ AND $(3+4=7)$ then $x:=x+1$ $x=$ [ANS]\n(d) if $(1+1=2)$ XOR $(x=3)$ then $x:=x+1$ $x=$ [ANS]", "answer_v3": [ "4", "4", "4", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Set_theory_and_logic_0040", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Operations on propositions", "level": "2", "keywords": [ "discrete mathematics", "logic", "literal" ], "problem_v1": "Enter T or F depending on whether the formula is a literal or not. [ANS] 1. $p \\wedge q$ [ANS] 2. $p \\vee q$ [ANS] 3. $\\neg p$\nEnter T or F depending on whether the formula is a clause or not. [ANS] 1. p [ANS] 2. $p \\vee q \\vee \\neg r \\vee \\neg s$ [ANS] 3. $p \\rightarrow q$ [ANS] 4. $(p \\wedge q) \\vee (\\neg p \\wedge r)$\nEnter T or F depending on whether the formula is a Horn clause or not. [ANS] 1. $p \\vee q \\vee \\neg r \\vee \\neg s$ [ANS] 2. $\\neg p \\vee \\neg q \\vee \\neg r \\vee s$ [ANS] 3. p [ANS] 4. $p \\vee q \\vee \\neg r$\nHow to represent $\\neg p$ in a logic program? [ANS] A. $\\rightarrow p$ B. $\\leftarrow p$ C. impossible D. $p$ E. $p \\leftarrow$\nHow to represent $p$ in a logic program? [ANS] A. $p \\leftarrow$ B. $\\rightarrow p$ C. impossible D. $p$ E. $\\leftarrow p$\nHow to represent $\\neg p \\vee \\neg q$ in a logic program? [ANS] A. $p, q \\leftarrow$ B. $\\leftarrow p \\wedge q$ C. $\\rightarrow p,q$ D. $p \\vee q$ E. impossible F. $\\leftarrow p, q$\nHow to represent $\\neg p \\vee \\neg q \\vee \\neg r \\vee s$ in a logic program? [ANS] A. $\\neg (p \\wedge q \\wedge r) \\rightarrow s$ B. $s \\rightarrow p,q,r$ C. $p, q, s \\leftarrow r$ D. $s \\leftarrow p, q, r$ E. $s \\leftarrow p \\wedge q \\wedge r$ F. impossible\nHow to represent $\\neg p \\vee \\neg q \\vee r \\vee s$ in a logic program? [ANS] A. $\\neg (p \\wedge q \\wedge r) \\rightarrow s$ B. $s \\leftarrow p \\wedge q \\wedge r$ C. $s \\leftarrow p, q, r$ D. impossible E. $s \\rightarrow p,q,r$ F. $p, q, s \\leftarrow r$", "answer_v1": [ "F", "F", "T", "T", "T", "F", "F", "F", "T", "T", "F", "B", "A", "F", "D", "D" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Enter T or F depending on whether the formula is a literal or not. [ANS] 1. $p \\wedge q$ [ANS] 2. p [ANS] 3. $\\neg p$\nEnter T or F depending on whether the formula is a clause or not. [ANS] 1. $(p \\wedge q) \\vee (\\neg p \\wedge r)$ [ANS] 2. p [ANS] 3. $p \\vee q \\vee \\neg r \\vee \\neg s$ [ANS] 4. $p \\rightarrow q$\nEnter T or F depending on whether the formula is a Horn clause or not. [ANS] 1. p [ANS] 2. $\\neg p \\vee \\neg q \\vee \\neg r \\vee \\neg s$ [ANS] 3. $(p \\vee q) \\wedge (\\neg p \\vee r)$ [ANS] 4. $\\neg (p \\vee q)$\nHow to represent $\\neg p$ in a logic program? [ANS] A. impossible B. $\\rightarrow p$ C. $\\leftarrow p$ D. $p \\leftarrow$ E. $p$\nHow to represent $p$ in a logic program? [ANS] A. $\\rightarrow p$ B. impossible C. $p \\leftarrow$ D. $\\leftarrow p$ E. $p$\nHow to represent $\\neg p \\vee \\neg q$ in a logic program? [ANS] A. $\\leftarrow p, q$ B. $\\leftarrow p \\wedge q$ C. $\\rightarrow p,q$ D. $p \\vee q$ E. $p, q \\leftarrow$ F. impossible\nHow to represent $\\neg p \\vee \\neg q \\vee \\neg r \\vee s$ in a logic program? [ANS] A. $\\neg (p \\wedge q \\wedge r) \\rightarrow s$ B. $s \\rightarrow p,q,r$ C. $s \\leftarrow p, q, r$ D. impossible E. $p, q, s \\leftarrow r$ F. $s \\leftarrow p \\wedge q \\wedge r$\nHow to represent $\\neg p \\vee \\neg q \\vee r \\vee s$ in a logic program? [ANS] A. $s \\leftarrow p \\wedge q \\wedge r$ B. $s \\leftarrow p, q, r$ C. $p, q, s \\leftarrow r$ D. impossible E. $s \\rightarrow p,q,r$ F. $\\neg (p \\wedge q \\wedge r) \\rightarrow s$", "answer_v2": [ "F", "T", "T", "F", "T", "T", "F", "T", "T", "F", "F", "C", "C", "A", "C", "D" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Enter T or F depending on whether the formula is a literal or not. [ANS] 1. p [ANS] 2. $p \\wedge q$ [ANS] 3. $\\neg p$\nEnter T or F depending on whether the formula is a clause or not. [ANS] 1. $\\neg p$ [ANS] 2. $p \\rightarrow q$ [ANS] 3. $(p \\vee q)$ [ANS] 4. $\\neg (p \\vee q)$\nEnter T or F depending on whether the formula is a Horn clause or not. [ANS] 1. $\\neg p \\vee \\neg q$ [ANS] 2. $\\neg (p \\vee q)$ [ANS] 3. $\\neg p$ [ANS] 4. $p \\vee q \\vee \\neg r \\vee \\neg s$\nHow to represent $\\neg p$ in a logic program? [ANS] A. $\\rightarrow p$ B. impossible C. $p \\leftarrow$ D. $\\leftarrow p$ E. $p$\nHow to represent $p$ in a logic program? [ANS] A. $p$ B. $\\rightarrow p$ C. impossible D. $p \\leftarrow$ E. $\\leftarrow p$\nHow to represent $\\neg p \\vee \\neg q$ in a logic program? [ANS] A. $\\rightarrow p,q$ B. $p \\vee q$ C. $\\leftarrow p \\wedge q$ D. impossible E. $p, q \\leftarrow$ F. $\\leftarrow p, q$\nHow to represent $\\neg p \\vee \\neg q \\vee \\neg r \\vee s$ in a logic program? [ANS] A. $s \\leftarrow p, q, r$ B. $s \\rightarrow p,q,r$ C. impossible D. $p, q, s \\leftarrow r$ E. $\\neg (p \\wedge q \\wedge r) \\rightarrow s$ F. $s \\leftarrow p \\wedge q \\wedge r$\nHow to represent $\\neg p \\vee \\neg q \\vee r \\vee s$ in a logic program? [ANS] A. $s \\rightarrow p,q,r$ B. $\\neg (p \\wedge q \\wedge r) \\rightarrow s$ C. $p, q, s \\leftarrow r$ D. $s \\leftarrow p, q, r$ E. $s \\leftarrow p \\wedge q \\wedge r$ F. impossible", "answer_v3": [ "T", "F", "T", "T", "F", "T", "F", "T", "F", "T", "F", "D", "D", "F", "A", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Set_theory_and_logic_0041", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Operations on propositions", "level": "2", "keywords": [ "discrete mathematics", "logic", "ambiguous" ], "problem_v1": "Enter T or F depending on whether the formula is ambiguous (T) or not (F). [ANS] 1. $p \\wedge q \\vee (q \\wedge r)$ [ANS] 2. $(p \\wedge q) \\wedge r$ [ANS] 3. $p \\rightarrow q$ [ANS] 4. $p \\rightarrow q$ [ANS] 5. $p \\wedge q \\wedge r$\nEnter T or F depending on whether the formula is a conjunctive normal form or not. [ANS] 1. p [ANS] 2. $(p \\vee q) \\wedge (\\neg p \\vee r \\vee q) \\wedge (t \\vee w)$ [ANS] 3. $\\neg (p \\vee q)$ [ANS] 4. $\\neg (p \\wedge q)$ [ANS] 5. $(p \\wedge q) \\vee (\\neg p \\wedge r)$\nEnter T or F depending on whether the formula is a disjunctive normal form or not. [ANS] 1. $p \\rightarrow q$ [ANS] 2. $(p \\wedge q) \\vee (\\neg p \\wedge r \\wedge q) \\vee (t \\wedge w)$ [ANS] 3. $(p \\wedge q) \\vee (\\neg p \\wedge r)$ [ANS] 4. p [ANS] 5. $(p \\wedge q) \\vee (\\neg p \\wedge r)$\nWhat formula is a disjunctive normal form of $\\neg(b \\vee (a \\wedge c))$? [ANS] A. $(\\neg b \\wedge \\neg a) \\vee (b \\wedge \\neg c)$ B. $(\\neg b \\wedge \\neg a) \\vee (\\neg b \\wedge \\neg c)$ C. $(\\neg b \\wedge a) \\vee (\\neg b \\wedge \\neg c)$ D. $(\\neg b \\wedge \\neg c) \\vee (b \\wedge \\neg c)$ E. $(\\neg b \\wedge \\neg a) \\vee (\\neg a \\wedge \\neg c)$ F. $(\\neg b \\wedge \\neg a) \\vee (\\neg b \\wedge c)$ G. $(b \\wedge a) \\vee (\\neg b \\wedge \\neg c)$\n$\\begin{array}{cc}\\hline a & Idempotent Law \\\\\\hlineb & Double Negation \\\\\\hlinec & De Morgan's Law \\\\\\hlined & Commutative Properties \\\\\\hlinee & Associative Properties \\\\\\hlinef & Distributive Properties \\\\\\hlineg & Equivalence of Contrapositive \\\\\\hlineh & Definition of Implication \\\\\\hlinei & Definition of Equivalence \\\\\\hlinej & Identity Laws \\((p \\vee F \\equiv p \\wedge T \\equiv p) \\) \\\\\\hlinek & Tautology \\((p \\vee \\neg p \\equiv T) \\) \\\\\\hlinel & Contradiction \\((p \\wedge \\neg p \\equiv F) \\) \\\\\\hline\\end{array}$\nProvide the justifications for the following transformation in disjunctive normal form at each step, using the equivalences listed above. We start with a formula in conjunctive normal form. $(a \\vee b) \\wedge (\\neg a \\vee d)$ $\\equiv$ $((a \\vee b) \\wedge \\neg a) \\vee ((a \\vee b) \\wedge d)$ by [ANS] $\\equiv$ $(a \\wedge \\neg a) \\vee (b \\wedge \\neg a) \\vee (a \\wedge d) \\vee (b \\wedge d)$ by [ANS] $\\equiv$ $(b \\wedge \\neg a) \\vee (a \\wedge d) \\vee (b \\wedge d)$ by [ANS] (Conjunctive normal form)", "answer_v1": [ "T", "F", "F", "F", "F", "T", "T", "F", "F", "F", "F", "T", "T", "T", "T", "B", "f", "f", "l" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS", "OE", "OE", "OE" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E", "F", "G" ], [], [], [] ], "problem_v2": "Enter T or F depending on whether the formula is ambiguous (T) or not (F). [ANS] 1. $p \\rightarrow q$ [ANS] 2. $(p \\wedge q) \\wedge r$ [ANS] 3. $\\neg p$ [ANS] 4. $p \\wedge q \\rightarrow s$ [ANS] 5. $p \\vee q \\vee r \\vee s \\vee w$\nEnter T or F depending on whether the formula is a conjunctive normal form or not. [ANS] 1. $(p \\vee q) \\wedge r$ [ANS] 2. $\\neg (p \\vee q)$ [ANS] 3. $(p \\wedge q) \\vee (\\neg p \\wedge r)$ [ANS] 4. $\\neg (p \\wedge q)$ [ANS] 5. p\nEnter T or F depending on whether the formula is a disjunctive normal form or not. [ANS] 1. $(p \\vee q)$ [ANS] 2. $(p \\wedge q) \\vee (\\neg p \\wedge r \\wedge q) \\vee (t \\wedge w)$ [ANS] 3. p [ANS] 4. $r \\leftrightarrow q$ [ANS] 5. $(p \\vee q) \\wedge (\\neg p \\vee r \\vee q) \\wedge (t \\vee w)$\nWhat formula is a disjunctive normal form of $\\neg(b \\vee (a \\wedge c))$? [ANS] A. $(\\neg b \\wedge \\neg a) \\vee (\\neg a \\wedge \\neg c)$ B. $(\\neg b \\wedge \\neg a) \\vee (\\neg b \\wedge c)$ C. $(\\neg b \\wedge a) \\vee (\\neg b \\wedge \\neg c)$ D. $(\\neg b \\wedge \\neg a) \\vee (b \\wedge \\neg c)$ E. $(\\neg b \\wedge \\neg a) \\vee (\\neg b \\wedge \\neg c)$ F. $(\\neg b \\wedge \\neg c) \\vee (b \\wedge \\neg c)$ G. $(b \\wedge a) \\vee (\\neg b \\wedge \\neg c)$\n$\\begin{array}{cc}\\hline a & Idempotent Law \\\\\\hlineb & Double Negation \\\\\\hlinec & De Morgan's Law \\\\\\hlined & Commutative Properties \\\\\\hlinee & Associative Properties \\\\\\hlinef & Distributive Properties \\\\\\hlineg & Equivalence of Contrapositive \\\\\\hlineh & Definition of Implication \\\\\\hlinei & Definition of Equivalence \\\\\\hlinej & Identity Laws \\((p \\vee F \\equiv p \\wedge T \\equiv p) \\) \\\\\\hlinek & Tautology \\((p \\vee \\neg p \\equiv T) \\) \\\\\\hlinel & Contradiction \\((p \\wedge \\neg p \\equiv F) \\) \\\\\\hline\\end{array}$\nProvide the justifications for the following transformation in disjunctive normal form at each step, using the equivalences listed above. We start with a formula in conjunctive normal form. $(a \\vee b) \\wedge (\\neg a \\vee d)$ $\\equiv$ $((a \\vee b) \\wedge \\neg a) \\vee ((a \\vee b) \\wedge d)$ by [ANS] $\\equiv$ $(a \\wedge \\neg a) \\vee (b \\wedge \\neg a) \\vee (a \\wedge d) \\vee (b \\wedge d)$ by [ANS] $\\equiv$ $(b \\wedge \\neg a) \\vee (a \\wedge d) \\vee (b \\wedge d)$ by [ANS] (Conjunctive normal form)", "answer_v2": [ "F", "F", "F", "T", "F", "T", "F", "F", "F", "T", "T", "T", "T", "F", "F", "E", "f", "f", "l" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS", "OE", "OE", "OE" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E", "F", "G" ], [], [], [] ], "problem_v3": "Enter T or F depending on whether the formula is ambiguous (T) or not (F). [ANS] 1. $(p \\wedge q) \\wedge r$ [ANS] 2. $p \\vee q \\vee r \\vee s \\vee w$ [ANS] 3. $p \\wedge q$ [ANS] 4. $p \\wedge q \\wedge r$ [ANS] 5. $p \\wedge q \\vee r$\nEnter T or F depending on whether the formula is a conjunctive normal form or not. [ANS] 1. $(p \\vee q)$ [ANS] 2. $\\neg p$ [ANS] 3. $(p \\vee q) \\wedge r$ [ANS] 4. $(p \\wedge q) \\vee (\\neg p \\wedge r)$ [ANS] 5. $p \\wedge q$\nEnter T or F depending on whether the formula is a disjunctive normal form or not. [ANS] 1. $\\neg p$ [ANS] 2. $(p \\vee q) \\wedge (\\neg p \\vee r \\vee q) \\wedge (t \\vee w)$ [ANS] 3. $\\neg (p \\vee q)$ [ANS] 4. $(p \\wedge q) \\vee (\\neg p \\wedge r)$ [ANS] 5. $(p \\wedge q) \\vee r$\nWhat formula is a disjunctive normal form of $\\neg(b \\vee (a \\wedge c))$? [ANS] A. $(\\neg b \\wedge a) \\vee (\\neg b \\wedge \\neg c)$ B. $(\\neg b \\wedge \\neg a) \\vee (b \\wedge \\neg c)$ C. $(\\neg b \\wedge \\neg c) \\vee (b \\wedge \\neg c)$ D. $(\\neg b \\wedge \\neg a) \\vee (\\neg b \\wedge \\neg c)$ E. $(\\neg b \\wedge \\neg a) \\vee (\\neg a \\wedge \\neg c)$ F. $(\\neg b \\wedge \\neg a) \\vee (\\neg b \\wedge c)$ G. $(b \\wedge a) \\vee (\\neg b \\wedge \\neg c)$\n$\\begin{array}{cc}\\hline a & Idempotent Law \\\\\\hlineb & Double Negation \\\\\\hlinec & De Morgan's Law \\\\\\hlined & Commutative Properties \\\\\\hlinee & Associative Properties \\\\\\hlinef & Distributive Properties \\\\\\hlineg & Equivalence of Contrapositive \\\\\\hlineh & Definition of Implication \\\\\\hlinei & Definition of Equivalence \\\\\\hlinej & Identity Laws \\((p \\vee F \\equiv p \\wedge T \\equiv p) \\) \\\\\\hlinek & Tautology \\((p \\vee \\neg p \\equiv T) \\) \\\\\\hlinel & Contradiction \\((p \\wedge \\neg p \\equiv F) \\) \\\\\\hline\\end{array}$\nProvide the justifications for the following transformation in disjunctive normal form at each step, using the equivalences listed above. We start with a formula in conjunctive normal form. $(a \\vee b) \\wedge (\\neg a \\vee d)$ $\\equiv$ $((a \\vee b) \\wedge \\neg a) \\vee ((a \\vee b) \\wedge d)$ by [ANS] $\\equiv$ $(a \\wedge \\neg a) \\vee (b \\wedge \\neg a) \\vee (a \\wedge d) \\vee (b \\wedge d)$ by [ANS] $\\equiv$ $(b \\wedge \\neg a) \\vee (a \\wedge d) \\vee (b \\wedge d)$ by [ANS] (Conjunctive normal form)", "answer_v3": [ "F", "F", "F", "F", "T", "T", "T", "T", "F", "T", "T", "F", "F", "T", "T", "D", "f", "f", "l" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS", "OE", "OE", "OE" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E", "F", "G" ], [], [], [] ] }, { "id": "Set_theory_and_logic_0042", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Operations on propositions", "level": "2", "keywords": [ "discrete mathematics", "logic", "negation" ], "problem_v1": "Which proposition is a necessary condition for the following statement to be true: \"I don't vote and I feel enfranchised.\" [ANS] A. I feel enfranchised B. I vote in the election and I feel enfranchised. C. I vote in the election D. I vote in the election if and only if I feel enfranchised. E. I feel disenfranchised\nWhich proposition is a necessary condition for the following statement to be true: \"I am hungry and I do not eat an apple.\" [ANS] A. I am hungry if and only if I eat an apple. B. I am hungry C. I eat an apple D. I am hungry and I eat an apple. E. I am not hungry\nWhich proposition is a sufficient condition for the following statement to be true: \"If I am on time for work then I catch the 8:05 bus.\" [ANS] A. I am on time for work or I catch the 8:05 bus. B. I miss the 8:05 bus C. I am on time for work D. I am on time for work and I miss the 8:05 bus. E. I am late for work F. If I catch the 8:05 bus then I am on time for work. G. I am late for work or I miss the 8:05 bus.\nRewrite the following as an equivalent if then statement: \"'n is prime' is a necessary condition for 'n is odd or n is 2'.\" [ANS] A. If n is prime then n is odd or n is 2. B. If n is prime then n is even but not 2. C. If n is composite then n is odd or n is 2. D. If n is odd or n is 2 then n is composite. E. If n is even but not 2 then n is composite. F. If n is composite then n is even but not 2. G. If n is even but not 2 then n is prime. H. If n is odd or n is 2 then n is prime.\nRewrite the following as an equivalent if then statement: \"'I raise my grades' is a necessary condition for 'I form a study group'.\" [ANS] A. If I form a study group then I raise my grades. B. If I lower my grades then I form a study group. C. If I work alone then I raise my grades. D. If I raise my grades then I form a study group. E. If I form a study group then I lower my grades. F. If I raise my grades then I work alone. G. If I lower my grades then I work alone. H. If I work alone then I lower my grades.\nRewrite the following as an equivalent if then statement: \"'I feel tired' is a sufficient condition for 'I exercise'.\" [ANS] A. If I don't exercise then I feel tired. B. If I feel tired then I exercise. C. If I exercise then I feel tired. D. If I feel tired then I don't exercise. E. If I feel envigorated then I exercise. F. If I don't exercise then I feel envigorated. G. If I exercise then I feel envigorated. H. If I feel envigorated then I don't exercise.", "answer_v1": [ "A", "B", "E", "H", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v2": "Which proposition is a necessary condition for the following statement to be true: \"I don't go to Paris and I visit the Eiffel Tower.\" [ANS] A. I go to Paris B. I visit the Eiffel Tower C. I do not visit the Eiffel Tower. D. I go to Paris if and only if I visit the Eiffel Tower. E. I go to Paris and I visit the Eiffel Tower.\nWhich proposition is a necessary condition for the following statement to be true: \"I am on time for work and I miss the 8:05 bus.\" [ANS] A. I am on time for work B. I am on time for work and I catch the 8:05 bus. C. I catch the 8:05 bus D. I am late for work E. I am on time for work if and only if I catch the 8:05 bus.\nWhich proposition is a sufficient condition for the following statement to be true: \"If n is divisible by 6 then n is divisible by both 2 and 3.\" [ANS] A. n is not divisible by 6 or n is not divisible by both 2 and 3. B. n is not divisible by both 2 and 3 C. n is divisible by 6 and n is not divisible by both 2 and 3. D. n is divisible by 6 E. n is not divisible by 6 F. If n is divisible by both 2 and 3 then n is divisible by 6. G. n is divisible by 6 or n is divisible by both 2 and 3. Rewrite the following as an equivalent if then statement: \"'I go to Paris' is a necessary condition for 'I visit the Eiffel Tower'.\" [ANS] A. If I don't go to Paris then I do not visit the Eiffel Tower. B. If I do not visit the Eiffel Tower then I don't go to Paris. C. If I go to Paris then I visit the Eiffel Tower. D. If I visit the Eiffel Tower then I don't go to Paris. E. If I don't go to Paris then I visit the Eiffel Tower. F. If I do not visit the Eiffel Tower then I go to Paris. G. If I visit the Eiffel Tower then I go to Paris. H. If I go to Paris then I do not visit the Eiffel Tower.\nRewrite the following as an equivalent if then statement: \"'I visit the Eiffel Tower' is a necessary condition for 'I go to Paris'.\" [ANS] A. If I visit the Eiffel Tower then I go to Paris. B. If I don't go to Paris then I do not visit the Eiffel Tower. C. If I do not visit the Eiffel Tower then I go to Paris. D. If I don't go to Paris then I visit the Eiffel Tower. E. If I go to Paris then I do not visit the Eiffel Tower. F. If I do not visit the Eiffel Tower then I don't go to Paris. G. If I go to Paris then I visit the Eiffel Tower. H. If I visit the Eiffel Tower then I don't go to Paris.\nRewrite the following as an equivalent if then statement: \"'I visit the Eiffel Tower' is a sufficient condition for 'I go to Paris'.\" [ANS] A. If I go to Paris then I visit the Eiffel Tower. B. If I visit the Eiffel Tower then I don't go to Paris. C. If I visit the Eiffel Tower then I go to Paris. D. If I don't go to Paris then I visit the Eiffel Tower. E. If I don't go to Paris then I do not visit the Eiffel Tower. F. If I do not visit the Eiffel Tower then I go to Paris. G. If I go to Paris then I do not visit the Eiffel Tower. H. If I do not visit the Eiffel Tower then I don't go to Paris.", "answer_v2": [ "B", "A", "E", "G", "G", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v3": "Which proposition is a necessary condition for the following statement to be true: \"P is not a square and P is a rectangle.\" [ANS] A. P is a square B. P is not a rectangle C. P is a rectangle D. P is a square if and only if P is a rectangle. E. P is a square and P is a rectangle.\nWhich proposition is a necessary condition for the following statement to be true: \"I am hungry and I do not eat an apple.\" [ANS] A. I am hungry if and only if I eat an apple. B. I am not hungry C. I am hungry D. I am hungry and I eat an apple. E. I eat an apple\nWhich proposition is a sufficient condition for the following statement to be true: \"If this triangle has two 45 degree angles then it is a right triangle.\" [ANS] A. this triangle has two 45 degree angles or it is a right triangle. B. this triangle has two 45 degree angles C. this triangle does not have two 45 degree angles or it is not a right triangle. D. this triangle does not have two 45 degree angles E. it is not a right triangle F. this triangle has two 45 degree angles and it is not a right triangle. G. If it is a right triangle then this triangle has two 45 degree angles.\nRewrite the following as an equivalent if then statement: \"'this triangle has two 45 degree angles' is a necessary condition for 'it is a right triangle'.\" [ANS] A. If it is a right triangle then this triangle does not have two 45 degree angles. B. If this triangle has two 45 degree angles then it is not a right triangle. C. If it is not a right triangle then this triangle does not have two 45 degree angles. D. If it is not a right triangle then this triangle has two 45 degree angles. E. If this triangle does not have two 45 degree angles then it is not a right triangle. F. If this triangle does not have two 45 degree angles then it is a right triangle. G. If this triangle has two 45 degree angles then it is a right triangle. H. If it is a right triangle then this triangle has two 45 degree angles.\nRewrite the following as an equivalent if then statement: \"'I catch the 8:05 bus' is a necessary condition for 'I am on time for work'.\" [ANS] A. If I miss the 8:05 bus then I am late for work. B. If I miss the 8:05 bus then I am on time for work. C. If I am on time for work then I miss the 8:05 bus. D. If I am late for work then I catch the 8:05 bus. E. If I catch the 8:05 bus then I am late for work. F. If I catch the 8:05 bus then I am on time for work. G. If I am on time for work then I catch the 8:05 bus. H. If I am late for work then I miss the 8:05 bus.\nRewrite the following as an equivalent if then statement: \"'the decimal expansion of r is repeating' is a sufficient condition for 'r is rational'.\" [ANS] A. If the decimal expansion of r is repeating then r is rational. B. If r is irrational then the decimal expansion of r is repeating. C. If r is rational then the decimal expansion of r does not repeat. D. If the decimal expansion of r does not repeat then r is rational. E. If r is rational then the decimal expansion of r is repeating. F. If the decimal expansion of r is repeating then r is irrational. G. If the decimal expansion of r does not repeat then r is irrational. H. If r is irrational then the decimal expansion of r does not repeat.", "answer_v3": [ "C", "C", "D", "H", "G", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ] ] }, { "id": "Set_theory_and_logic_0043", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Operations on propositions", "level": "2", "keywords": [ "discrete mathematics", "logic", "negation" ], "problem_v1": "What is the negation of the following: \"If I vote in the election then I feel enfranchised.\" [ANS] A. I vote in the election or I feel disenfranchised. B. I vote in the election or I feel enfranchised. C. If I don't vote then I feel disenfranchised. D. If I don't vote then I feel enfranchised. E. If I feel disenfranchised then I don't vote. F. I vote in the election and I feel enfranchised. G. I don't vote or I feel enfranchised. H. I vote in the election and I feel disenfranchised. I. I don't vote and I feel enfranchised. J. If I feel enfranchised then I vote in the election. K. If I vote in the election then I feel enfranchised.\nWhat is the negation of the following: \"If I am hungry then I eat an apple.\" [ANS] A. If I don't eat an apple then I'm not hungry. B. I am not hungry and I don't eat an apple. C. If I am not hungry then I eat an apple. D. If I am not hungry then I don't eat an apple. E. I am hungry or I don't eat an apple. F. I am hungry or I eat an apple. G. If I eat an apple then I am hungry. H. I am hungry and I don't eat an apple. I. I am hungry and I eat an apple. J. If I am hungry then I eat an apple. K. I am not hungry or I eat an apple.\nWhat is the negation of the following: \"I go to Paris and I visit the Eiffel Tower.\" [ANS] A. I go to Paris and I visit the Eiffel Tower. B. I go to Paris or I don't visit the Eiffel Tower. C. I don't go to Paris and I don't visit the Eiffel Tower. D. I don't go to Paris or I visit the Eiffel Tower. E. I go to Paris or I visit the Eiffel Tower. F. I don't go to Paris or I don't visit the Eiffel Tower. G. I don't go to Paris and I visit the Eiffel Tower. H. I go to Paris and I don't visit the Eiffel Tower.\nWhat is the negation of the following: \"I go to Paris or I visit the Eiffel Tower.\" [ANS] A. I don't go to Paris and I don't visit the Eiffel Tower. B. I don't go to Paris or I visit the Eiffel Tower. C. I go to Paris and I visit the Eiffel Tower. D. I don't go to Paris or I don't visit the Eiffel Tower. E. I don't go to Paris and I visit the Eiffel Tower. F. I go to Paris or I visit the Eiffel Tower. G. I go to Paris or I don't visit the Eiffel Tower. H. I go to Paris and I don't visit the Eiffel Tower.\nWhat is the converse of the following: \"If I exercise then I feel tired.\" [ANS] A. If I feel envigorated then I don't exercise. B. If I exercise then I feel envigorated. C. If I don't exercise then I feel envigorated. D. If I feel tired then I don't exercise. E. If I feel tired then I exercise. F. If I exercise then I feel tired.\nWhat is the inverse of the following: \"If I am hungry then I eat an apple.\" [ANS] A. If I'm not hungry then I don't eat an apple. B. If I am hungry then I eat an apple. C. If I don't eat an apple then I'm not hungry. D. If I eat an apple then I am hungry. E. If I eat an apple then I am not hungry. F. If I'm hungry then I eat an apple.\nWhat is the contrapositive of the following: \"If I am on time for work then I catch the 8:05 bus.\" [ANS] A. If I catch the 8:05 bus then I am on time for work. B. If I am on time for work then I catch the 8:05 bus. C. If I am on time for work then I miss the 8:05 bus. D. If I catch the 8:05 bus then I am late for work. E. If I am late for work then I miss the 8:05 bus. F. If I miss the 8:05 bus then I am late for work.", "answer_v1": [ "H", "H", "F", "A", "E", "A", "F" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "What is the negation of the following: \"If I go to Paris then I visit the Eiffel Tower.\" [ANS] A. I go to Paris and I visit the Eiffel Tower. B. If I don't go to Paris then I don't visit the Eiffel Tower. C. I go to Paris or I don't visit the Eiffel Tower. D. If I don't go to Paris then I visit the Eiffel Tower. E. I don't go to Paris and I don't visit the Eiffel Tower. F. I go to Paris and I don't visit the Eiffel Tower. G. If I don't visit the Eiffel Tower then I don't go to Paris. H. If I go to Paris then I visit the Eiffel Tower. I. I don't go to Paris or I visit the Eiffel Tower. J. I go to Paris or I visit the Eiffel Tower. K. If I visit the Eiffel Tower then I go to Paris.\nWhat is the negation of the following: \"If I am on time for work then I catch the 8:05 bus.\" [ANS] A. If I miss the 8:05 bus then I am late for work. B. If I am on time for work then I catch the 8:05 bus. C. I am on time for work and I miss the 8:05 bus. D. I am late for work and I catch the 8:05 bus. E. If I catch the 8:05 bus then I am on time for work. F. I am late for work or I catch the 8:05 bus. G. If I am late for work then I catch the 8:05 bus. H. If I am late for work then I miss the 8:05 bus. I. I am on time for work or I miss the 8:05 bus. J. I am on time for work or I catch the 8:05 bus. K. I am on time for work and I catch the 8:05 bus.\nWhat is the negation of the following: \"It rains and I take an umbrella.\" [ANS] A. It rains or I take an umbrella. B. It rains and I don't take an umbrella. C. It does not rain and I don't take an umbrella. D. It does not rain or I do not take an umbrella. E. It rains or I don't take an umbrella. F. It does not rain and I take an umbrella. G. It rains and I take an umbrella. H. It does not rain or I take an umbrella.\nWhat is the negation of the following: \"It rains or I take an umbrella.\" [ANS] A. It rains and I don't take an umbrella. B. It rains or I take an umbrella. C. It does not rain or I do not take an umbrella. D. It does not rain or I take an umbrella. E. It does not rain and I take an umbrella. F. It rains or I don't take an umbrella. G. It does not rain and I don't take an umbrella. H. It rains and I take an umbrella.\nWhat is the converse of the following: \"If r is rational then the decimal expansion of r is repeating.\" [ANS] A. If r is rational then the decimal expansion of r is repeating. B. If r is rational then the decimal expansion of r does not repeat. C. If the decimal expansion of r is repeating then r is irrational. D. If r is irrational then the decimal expansion of r does not repeat. E. If the decimal expansion of r is repeating then r is rational. F. If the decimal expansion of r does not repeat then r is irrational.\nWhat is the inverse of the following: \"If I exercise then I feel tired.\" [ANS] A. If I feel tired then I exercise. B. If I feel tired then I don't exercise. C. If I feel envigorated then I don't exercise. D. If I don't exercise then I feel envigorated. E. If I exercise then I feel tired. F. If I exercise then I feel envigorated.\nWhat is the contrapositive of the following: \"If P is a square then P is a rectangle.\" [ANS] A. If P is a square then P is not a rectangle. B. If P is not a rectangle then P is not a square. C. If P is not a square then P is not a rectangle. D. If P is a rectangle then P is not a square. E. If P is a square then P is a rectangle. F. If P is a rectangle then P is a square.", "answer_v2": [ "F", "C", "D", "G", "E", "D", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "What is the negation of the following: \"If P is a square then P is a rectangle.\" [ANS] A. P is not a square or P is a rectangle. B. P is a square and P is a rectangle. C. If P is a square then P is a rectangle. D. P is a square or P is a rectangle. E. P is a square or P is not a rectangle. F. If P is not a rectangle then P is not a square. G. If P is not a square then P is not a rectangle. H. If P is not a square then P is a rectangle. I. If P is a rectangle then P is a square. J. P is a square and P is not a rectangle. K. P is not a square and P is a rectangle.\nWhat is the negation of the following: \"If I am hungry then I eat an apple.\" [ANS] A. If I eat an apple then I am hungry. B. If I don't eat an apple then I'm not hungry. C. If I am not hungry then I eat an apple. D. If I am not hungry then I don't eat an apple. E. I am not hungry or I eat an apple. F. I am hungry or I eat an apple. G. I am hungry or I don't eat an apple. H. I am not hungry and I don't eat an apple. I. If I am hungry then I eat an apple. J. I am hungry and I eat an apple. K. I am hungry and I don't eat an apple.\nWhat is the negation of the following statement: \"I vote in the election and I feel enfranchised.\" [ANS] A. I vote in the election or I feel enfranchised. B. I don't vote or I feel enfranchised. C. I don't vote or I feel disenfranchised. D. I vote in the election and I feel enfranchised. E. I vote in the election and I feel disenfranchised. F. I don't vote and I feel enfranchised. G. I don't vote and I feel disenfranchised. H. I vote in the election or I feel disenfranchised.\nWhat is the negation of the following: \"I am hungry or I eat an apple.\" [ANS] A. I am not hungry and I eat an apple. B. I am hungry or I eat an apple. C. I am not hungry and I don't eat an apple. D. I am not hungry or I don't eat an apple. E. I am hungry or I don't eat an apple. F. I am hungry and I eat an apple. G. I am not hungry or I eat an apple. H. I am hungry and I don't eat an apple.\nWhat is the converse of the following: \"If P is a square then P is a rectangle.\" [ANS] A. If P is not a square then P is not a rectangle. B. If P is a square then P is a rectangle. C. If P is a rectangle then P is not a square. D. If P is not a rectangle then P is not a square. E. If P is a rectangle then P is a square. F. If P is a square then P is not a rectangle.\nWhat is the inverse of the following: \"If I am hungry then I eat an apple.\" [ANS] A. If I eat an apple then I am not hungry. B. If I don't eat an apple then I'm not hungry. C. If I am hungry then I eat an apple. D. If I eat an apple then I am hungry. E. If I'm not hungry then I don't eat an apple. F. If I'm hungry then I eat an apple.\nWhat is the contrapositive of the following: \"If I am on time for work then I catch the 8:05 bus.\" [ANS] A. If I am on time for work then I miss the 8:05 bus. B. If I am on time for work then I catch the 8:05 bus. C. If I catch the 8:05 bus then I am on time for work. D. If I am late for work then I miss the 8:05 bus. E. If I miss the 8:05 bus then I am late for work. F. If I catch the 8:05 bus then I am late for work.", "answer_v3": [ "J", "K", "C", "C", "E", "E", "E" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F", "G", "H" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Set_theory_and_logic_0044", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Operations on propositions", "level": "2", "keywords": [ "discrete mathematics", "logic", "nor", "nand" ], "problem_v1": "Consider the following truth table. $\\begin{array}{cccc}\\hline p & q & r & f(p,q,r) \\\\\\hline0 & 0 & 0 & 1 \\\\\\hline0 & 0 & 1 & 0 \\\\\\hline0 & 1 & 0 & 0 \\\\\\hline0 & 1 & 1 & 1 \\\\\\hline1 & 0 & 0 & 0 \\\\\\hline1 & 0 & 1 & 0 \\\\\\hline1 & 1 & 0 & 0 \\\\\\hline1 & 1 & 1 & 1 \\\\\\hline\\end{array}$\nGive a disjunctive normal form to represent $f(p,q,r)$. [ANS] A. $(\\neg p \\wedge \\neg q \\wedge \\neg r) \\vee (\\neg p \\wedge q \\wedge r) \\vee (p \\wedge q \\wedge r)$ B. $(\\neg p \\wedge q \\wedge r) \\vee (p \\wedge q \\wedge r)$ C. $(\\neg p \\wedge \\neg q \\wedge r) \\vee (\\neg p \\wedge q \\wedge \\neg r) \\vee (p \\wedge \\neg q \\wedge \\neg r) \\vee (p \\wedge \\neg q \\wedge r) \\vee (p \\wedge q \\wedge \\neg r)$ D. $(\\neg p \\wedge \\neg q \\wedge \\neg r) \\vee (\\neg p \\wedge q \\wedge r)$\nGive a conjunctive normal form to represent $f(p,q,r)$. [ANS] A. $(p \\vee q \\vee \\neg r) \\wedge (p \\vee \\neg q \\vee r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (\\neg p \\vee q \\vee \\neg r) \\wedge (\\neg p \\vee \\neg q \\vee r)$ B. $(p \\vee \\neg q \\vee r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (\\neg p \\vee q \\vee \\neg r) \\wedge (\\neg p \\vee \\neg q \\vee r)$ C. $(p \\vee q \\vee \\neg r) \\wedge (p \\vee \\neg q \\vee r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (\\neg p \\vee q \\vee \\neg r)$ D. $(\\neg p \\vee \\neg q \\vee \\neg r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (p \\vee q \\vee r)$\nExpress $a \\wedge b$ using only a NOR. [ANS] A. NOR(NOR(a,a),NOR(b,b)) B. NOR(a,b) C. NOR(NOR(b,b),NOR(b,a)) D. NOR(NOR(a,b),NOR(a,b))\nExpress $a \\vee b$ using only a NAND. [ANS] A. NAND(NAND(b,b),NAND(b,a)) B. NAND(NAND(b,a),NAND(a,a)) C. NAND(a,b) D. NAND(NAND(a,a),NAND(b,b))", "answer_v1": [ "A", "A", "A", "D" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the following truth table. $\\begin{array}{cccc}\\hline p & q & r & f(p,q,r) \\\\\\hline0 & 0 & 0 & 1 \\\\\\hline0 & 0 & 1 & 0 \\\\\\hline0 & 1 & 0 & 0 \\\\\\hline0 & 1 & 1 & 1 \\\\\\hline1 & 0 & 0 & 0 \\\\\\hline1 & 0 & 1 & 0 \\\\\\hline1 & 1 & 0 & 0 \\\\\\hline1 & 1 & 1 & 1 \\\\\\hline\\end{array}$\nGive a disjunctive normal form to represent $f(p,q,r)$. [ANS] A. $(\\neg p \\wedge \\neg q \\wedge \\neg r) \\vee (\\neg p \\wedge q \\wedge r) \\vee (p \\wedge q \\wedge r)$ B. $(\\neg p \\wedge q \\wedge r) \\vee (p \\wedge q \\wedge r)$ C. $(\\neg p \\wedge \\neg q \\wedge r) \\vee (\\neg p \\wedge q \\wedge \\neg r) \\vee (p \\wedge \\neg q \\wedge \\neg r) \\vee (p \\wedge \\neg q \\wedge r) \\vee (p \\wedge q \\wedge \\neg r)$ D. $(\\neg p \\wedge \\neg q \\wedge \\neg r) \\vee (\\neg p \\wedge q \\wedge r)$\nGive a conjunctive normal form to represent $f(p,q,r)$. [ANS] A. $(p \\vee q \\vee \\neg r) \\wedge (p \\vee \\neg q \\vee r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (\\neg p \\vee q \\vee \\neg r)$ B. $(p \\vee \\neg q \\vee r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (\\neg p \\vee q \\vee \\neg r) \\wedge (\\neg p \\vee \\neg q \\vee r)$ C. $(\\neg p \\vee \\neg q \\vee \\neg r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (p \\vee q \\vee r)$ D. $(p \\vee q \\vee \\neg r) \\wedge (p \\vee \\neg q \\vee r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (\\neg p \\vee q \\vee \\neg r) \\wedge (\\neg p \\vee \\neg q \\vee r)$\nExpress $a \\vee b$ using only a NOR. [ANS] A. NOR(NOR(a,a),NOR(b,b)) B. NOR(a,b) C. NOR(NOR(a,b),NOR(b,b)) D. NOR(NOR(a,b),NOR(a,b))\nExpress $a \\wedge b$ using only a NAND. [ANS] A. NAND(NAND(a,a),NAND(b,b)) B. NAND(NAND(a,b),NAND(a,b)) C. NAND(a,b) D. NAND(NAND(b,a),NAND(b,b))", "answer_v2": [ "A", "D", "D", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the following truth table. $\\begin{array}{cccc}\\hline p & q & r & f(p,q,r) \\\\\\hline0 & 0 & 0 & 1 \\\\\\hline0 & 0 & 1 & 0 \\\\\\hline0 & 1 & 0 & 0 \\\\\\hline0 & 1 & 1 & 1 \\\\\\hline1 & 0 & 0 & 0 \\\\\\hline1 & 0 & 1 & 0 \\\\\\hline1 & 1 & 0 & 0 \\\\\\hline1 & 1 & 1 & 1 \\\\\\hline\\end{array}$\nGive a disjunctive normal form to represent $f(p,q,r)$. [ANS] A. $(\\neg p \\wedge \\neg q \\wedge \\neg r) \\vee (\\neg p \\wedge q \\wedge r) \\vee (p \\wedge q \\wedge r)$ B. $(\\neg p \\wedge \\neg q \\wedge r) \\vee (\\neg p \\wedge q \\wedge \\neg r) \\vee (p \\wedge \\neg q \\wedge \\neg r) \\vee (p \\wedge \\neg q \\wedge r) \\vee (p \\wedge q \\wedge \\neg r)$ C. $(\\neg p \\wedge q \\wedge r) \\vee (p \\wedge q \\wedge r)$ D. $(\\neg p \\wedge \\neg q \\wedge \\neg r) \\vee (\\neg p \\wedge q \\wedge r)$\nGive a conjunctive normal form to represent $f(p,q,r)$. [ANS] A. $(p \\vee q \\vee \\neg r) \\wedge (p \\vee \\neg q \\vee r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (\\neg p \\vee q \\vee \\neg r)$ B. $(p \\vee q \\vee \\neg r) \\wedge (p \\vee \\neg q \\vee r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (\\neg p \\vee q \\vee \\neg r) \\wedge (\\neg p \\vee \\neg q \\vee r)$ C. $(p \\vee \\neg q \\vee r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (\\neg p \\vee q \\vee \\neg r) \\wedge (\\neg p \\vee \\neg q \\vee r)$ D. $(\\neg p \\vee \\neg q \\vee \\neg r) \\wedge (\\neg p \\vee q \\vee r) \\wedge (p \\vee q \\vee r)$\nExpress $a \\vee b$ using only a NOR. [ANS] A. NOR(NOR(a,b),NOR(a,b)) B. NOR(NOR(a,b),NOR(b,b)) C. NOR(NOR(a,a),NOR(b,b)) D. NOR(a,b) Express $a \\vee b$ using only a NAND. [ANS] A. NAND(NAND(b,a),NAND(a,a)) B. NAND(NAND(a,a),NAND(b,b)) C. NAND(a,b) D. NAND(NAND(b,b),NAND(b,a))", "answer_v3": [ "A", "B", "A", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Set_theory_and_logic_0045", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Truth tables", "level": "2", "keywords": [ "truth table" ], "problem_v1": "Complete the truth table and determine whether or not the following statement is a tautology, a contradiction, or neither.\n$(p\\wedge (p\\rightarrow q))\\rightarrow q$\n$\\begin{array}{ccccc}\\hline p & q & p\\rightarrow q & p\\wedge (p\\rightarrow q) & (p\\wedge (p\\rightarrow q))\\rightarrow q \\\\\\hlineT & T & [ANS] & [ANS] & [ANS] \\\\\\hlineT & F & [ANS] & [ANS] & [ANS] \\\\\\hlineF & T & [ANS] & [ANS] & [ANS] \\\\\\hlineF & F & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\nThe statement is a [ANS] A. Contradiction, because the statement is always false B. Contradiction, because the statement is always true C. Tautology, because the statement is always false D. Tautology, because the statement is always true E. Neither", "answer_v1": [ "T", "T", "T", "F", "F", "T", "T", "F", "T", "T", "F", "T", "D" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Complete the truth table and determine whether or not the following statement is a tautology, a contradiction, or neither.\n$(p\\wedge (p\\rightarrow q))\\rightarrow q$\n$\\begin{array}{ccccc}\\hline p & q & p\\rightarrow q & p\\wedge (p\\rightarrow q) & (p\\wedge (p\\rightarrow q))\\rightarrow q \\\\\\hlineT & T & [ANS] & [ANS] & [ANS] \\\\\\hlineT & F & [ANS] & [ANS] & [ANS] \\\\\\hlineF & T & [ANS] & [ANS] & [ANS] \\\\\\hlineF & F & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\nThe statement is a [ANS] A. Tautology, because the statement is always false B. Tautology, because the statement is always true C. Contradiction, because the statement is always false D. Contradiction, because the statement is always true E. Neither", "answer_v2": [ "T", "T", "T", "F", "F", "T", "T", "F", "T", "T", "F", "T", "B" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Complete the truth table and determine whether or not the following statement is a tautology, a contradiction, or neither.\n$(p\\wedge (p\\rightarrow q))\\rightarrow q$\n$\\begin{array}{ccccc}\\hline p & q & p\\rightarrow q & p\\wedge (p\\rightarrow q) & (p\\wedge (p\\rightarrow q))\\rightarrow q \\\\\\hlineT & T & [ANS] & [ANS] & [ANS] \\\\\\hlineT & F & [ANS] & [ANS] & [ANS] \\\\\\hlineF & T & [ANS] & [ANS] & [ANS] \\\\\\hlineF & F & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\nThe statement is a [ANS] A. Contradiction, because the statement is always true B. Contradiction, because the statement is always false C. Tautology, because the statement is always true D. Tautology, because the statement is always false E. Neither", "answer_v3": [ "T", "T", "T", "F", "F", "T", "T", "F", "T", "T", "F", "T", "C" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0046", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Truth tables", "level": "2", "keywords": [ "logic", "predicate" ], "problem_v1": "Complete the following truth table by filling in the blanks with T or F as appropriate.\n$\\begin{array}{cccccccc}\\hline p & q & r & p \\rightarrow q & p \\rightarrow r & [p \\rightarrow q] \\vee [p \\rightarrow r] & q \\vee r & p \\rightarrow [q \\vee r] \\\\\\hlineT & T & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineT & T & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineT & F & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineT & F & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & T & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & T & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & F & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & F & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\n\" $[p \\rightarrow q] \\vee [p \\rightarrow r]$ \" and \" $p \\rightarrow [q \\vee r]$ \" are [ANS] A. not logically comparable B. not logically equivalent C. logically equivalent", "answer_v1": [ "T", "T", "T", "T", "T", "T", "F", "T", "T", "T", "F", "T", "T", "T", "T", "F", "F", "F", "F", "F", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "F", "T", "C" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C" ] ], "problem_v2": "Complete the following truth table by filling in the blanks with T or F as appropriate.\n$\\begin{array}{cccccccc}\\hline p & q & r & p \\rightarrow q & p \\rightarrow r & [p \\rightarrow q] \\vee [p \\rightarrow r] & q \\vee r & p \\rightarrow [q \\vee r] \\\\\\hlineT & T & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineT & T & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineT & F & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineT & F & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & T & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & T & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & F & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & F & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\n\" $[p \\rightarrow q] \\vee [p \\rightarrow r]$ \" and \" $p \\rightarrow [q \\vee r]$ \" are [ANS] A. logically equivalent B. not logically equivalent C. not logically comparable", "answer_v2": [ "T", "T", "T", "T", "T", "T", "F", "T", "T", "T", "F", "T", "T", "T", "T", "F", "F", "F", "F", "F", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "F", "T", "A" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C" ] ], "problem_v3": "Complete the following truth table by filling in the blanks with T or F as appropriate.\n$\\begin{array}{cccccccc}\\hline p & q & r & p \\rightarrow q & p \\rightarrow r & [p \\rightarrow q] \\vee [p \\rightarrow r] & q \\vee r & p \\rightarrow [q \\vee r] \\\\\\hlineT & T & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineT & T & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineT & F & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineT & F & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & T & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & T & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & F & T & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hlineF & F & F & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\\\hline\\end{array}$\n\" $[p \\rightarrow q] \\vee [p \\rightarrow r]$ \" and \" $p \\rightarrow [q \\vee r]$ \" are [ANS] A. not logically comparable B. logically equivalent C. not logically equivalent", "answer_v3": [ "T", "T", "T", "T", "T", "T", "F", "T", "T", "T", "F", "T", "T", "T", "T", "F", "F", "F", "F", "F", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "T", "F", "T", "B" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "MCS" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [ "A", "B", "C" ] ] }, { "id": "Set_theory_and_logic_0047", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Rules of inference", "level": "5", "keywords": [], "problem_v1": "Suppose this is true: If $x \\leq 6$ then $y > 11$.\nWhat can be deduced from that and this additional fact? $y=12$ [ANS] A. $x \\leq 6$ B. $y \\leq 11$ C. $y > 11$ D. $x > 6$ E. Nothing\nWhat can be deduced from that and this additional fact? $y=9$ [ANS] A. $x > 6$ B. $y > 11$ C. $y \\leq 11$ D. $x \\leq 6$ E. Nothing\nWhat can be deduced from that and this additional fact? $x=5$ [ANS] A. $y \\leq 11$ B. $x > 6$ C. $y > 11$ D. $x \\leq 6$ E. Nothing\nWhat can be deduced from that and this additional fact? $x=8$ [ANS] A. $x \\leq 6$ B. $y \\leq 11$ C. $y > 11$ D. $x > 6$ E. Nothing\nWhat can be deduced from that and this additional fact? $x^{2}=4$ [ANS] A. $y \\leq 11$ B. $x > 6$ C. $y > 11$ D. $x \\leq 6$ E. Nothing", "answer_v1": [ "E", "A", "C", "E", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Suppose this is true: If $x \\leq 3$ then $y > 13$.\nWhat can be deduced from that and this additional fact? $y=14$ [ANS] A. $x \\leq 3$ B. $y \\leq 13$ C. $x > 3$ D. $y > 13$ E. Nothing\nWhat can be deduced from that and this additional fact? $y=11$ [ANS] A. $y \\leq 13$ B. $y > 13$ C. $x \\leq 3$ D. $x > 3$ E. Nothing\nWhat can be deduced from that and this additional fact? $x=2$ [ANS] A. $y \\leq 13$ B. $x > 3$ C. $x \\leq 3$ D. $y > 13$ E. Nothing\nWhat can be deduced from that and this additional fact? $x=5$ [ANS] A. $x \\leq 3$ B. $y \\leq 13$ C. $x > 3$ D. $y > 13$ E. Nothing\nWhat can be deduced from that and this additional fact? $x^{2}=4$ [ANS] A. $y \\leq 13$ B. $x > 3$ C. $x \\leq 3$ D. $y > 13$ E. Nothing", "answer_v2": [ "E", "D", "D", "E", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Suppose this is true: If $x \\leq 4$ then $y > 12$.\nWhat can be deduced from that and this additional fact? $y=13$ [ANS] A. $y > 12$ B. $x \\leq 4$ C. $y \\leq 12$ D. $x > 4$ E. Nothing\nWhat can be deduced from that and this additional fact? $y=10$ [ANS] A. $x > 4$ B. $x \\leq 4$ C. $y \\leq 12$ D. $y > 12$ E. Nothing\nWhat can be deduced from that and this additional fact? $x=3$ [ANS] A. $y \\leq 12$ B. $y > 12$ C. $x \\leq 4$ D. $x > 4$ E. Nothing\nWhat can be deduced from that and this additional fact? $x=6$ [ANS] A. $y > 12$ B. $x \\leq 4$ C. $y \\leq 12$ D. $x > 4$ E. Nothing\nWhat can be deduced from that and this additional fact? $x^{2}=4$ [ANS] A. $y \\leq 12$ B. $y > 12$ C. $x \\leq 4$ D. $x > 4$ E. Nothing", "answer_v3": [ "E", "A", "B", "E", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0048", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Rules of inference", "level": "5", "keywords": [ "logic", "deduction" ], "problem_v1": "Suppose this is true: If $|x| > 6$ then $y \\leq-1$.\nWhat can be deduced from that and this additional fact? $x=-7$ [ANS] A. $y >-1$ B. $|x| > 6$ C. $y \\leq-1$ D. $|x| \\leq 6$ E. Nothing\nWhat can be deduced from that and this additional fact? $y=-1$ [ANS] A. $|x| \\leq 6$ B. $y \\leq-1$ C. $y >-1$ D. $|x| > 6$ E. Nothing\nWhat can be deduced from that and this additional fact? $x=4$ [ANS] A. $|x| \\leq 6$ B. $y \\leq-1$ C. $y >-1$ D. $|x| > 6$ E. Nothing\nWhat can be deduced from that and this additional fact? $x^{2}=49$ [ANS] A. $y >-1$ B. $|x| > 6$ C. $y \\leq-1$ D. $|x| \\leq 6$ E. Nothing", "answer_v1": [ "C", "E", "E", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Suppose this is true: If $|x| > 3$ then $y \\leq 1$.\nWhat can be deduced from that and this additional fact? $x=-4$ [ANS] A. $y > 1$ B. $|x| > 3$ C. $|x| \\leq 3$ D. $y \\leq 1$ E. Nothing\nWhat can be deduced from that and this additional fact? $y=1$ [ANS] A. $|x| \\leq 3$ B. $y \\leq 1$ C. $|x| > 3$ D. $y > 1$ E. Nothing\nWhat can be deduced from that and this additional fact? $x=1$ [ANS] A. $|x| \\leq 3$ B. $y \\leq 1$ C. $|x| > 3$ D. $y > 1$ E. Nothing\nWhat can be deduced from that and this additional fact? $x^{2}=16$ [ANS] A. $y > 1$ B. $|x| > 3$ C. $|x| \\leq 3$ D. $y \\leq 1$ E. Nothing", "answer_v2": [ "D", "E", "E", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Suppose this is true: If $|x| > 4$ then $y \\leq-1$.\nWhat can be deduced from that and this additional fact? $x=-5$ [ANS] A. $y >-1$ B. $y \\leq-1$ C. $|x| \\leq 4$ D. $|x| > 4$ E. Nothing\nWhat can be deduced from that and this additional fact? $y=-1$ [ANS] A. $y >-1$ B. $|x| \\leq 4$ C. $y \\leq-1$ D. $|x| > 4$ E. Nothing\nWhat can be deduced from that and this additional fact? $x=2$ [ANS] A. $y >-1$ B. $|x| \\leq 4$ C. $y \\leq-1$ D. $|x| > 4$ E. Nothing\nWhat can be deduced from that and this additional fact? $x^{2}=25$ [ANS] A. $y >-1$ B. $y \\leq-1$ C. $|x| \\leq 4$ D. $|x| > 4$ E. Nothing", "answer_v3": [ "B", "E", "E", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Set_theory_and_logic_0049", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Rules of inference", "level": "5", "keywords": [ "logic", "deduction" ], "problem_v1": "Suppose the following statement is true Statement: $x > 7 \\Rightarrow z < 11$. In each of the following check every answer that is correct. (There may be more than one.) What can be deduced from the statement and this additional fact: $x > 8$? [ANS] A. $z < 10$ B. $z \\geq 11$ C. $x \\leq 7$ D. $x > 7$ E. Nothing F. None of the above\nWhat can be deduced from the statement and this additional fact: $z=10$? [ANS] A. Nothing B. $x \\leq 7$ C. $z < 11$ D. $x > 7$ E. $z \\geq 11$ F. None of the above\nWhat can be deduced from the statement and this additional fact: $z^2=144$? [ANS] A. $z < 11$ B. $z \\geq 11$ C. $x > 7$ D. Nothing E. $x \\leq 7$ F. None of the above\nWhat can be deduced from the statement and this additional fact? $z > 12$? [ANS] A. $x < 8$ B. Nothing C. $z < 11$ D. $z \\geq 11$ E. $x > 7$ F. None of the above", "answer_v1": [ "D", "C", "F", "AD" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Suppose the following statement is true Statement: $x > 2 \\Rightarrow z < 16$. In each of the following check every answer that is correct. (There may be more than one.) What can be deduced from the statement and this additional fact: $x > 3$? [ANS] A. Nothing B. $x \\leq 2$ C. $z < 15$ D. $z \\geq 16$ E. $x > 2$ F. None of the above\nWhat can be deduced from the statement and this additional fact: $z=15$? [ANS] A. $z \\geq 16$ B. $x \\leq 2$ C. $x > 2$ D. Nothing E. $z < 16$ F. None of the above\nWhat can be deduced from the statement and this additional fact: $z^2=289$? [ANS] A. Nothing B. $z \\geq 16$ C. $x > 2$ D. $z < 16$ E. $x \\leq 2$ F. None of the above\nWhat can be deduced from the statement and this additional fact? $z > 17$? [ANS] A. $z \\geq 16$ B. $x < 3$ C. Nothing D. $x > 2$ E. $z < 16$ F. None of the above", "answer_v2": [ "E", "E", "F", "AB" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Suppose the following statement is true Statement: $x > 4 \\Rightarrow z < 11$. In each of the following check every answer that is correct. (There may be more than one.) What can be deduced from the statement and this additional fact: $x > 5$? [ANS] A. $x \\leq 4$ B. Nothing C. $z \\geq 11$ D. $z < 10$ E. $x > 4$ F. None of the above\nWhat can be deduced from the statement and this additional fact: $z=10$? [ANS] A. Nothing B. $x > 4$ C. $z < 11$ D. $z \\geq 11$ E. $x \\leq 4$ F. None of the above\nWhat can be deduced from the statement and this additional fact: $z^2=144$? [ANS] A. Nothing B. $z < 11$ C. $z \\geq 11$ D. $x \\leq 4$ E. $x > 4$ F. None of the above\nWhat can be deduced from the statement and this additional fact? $z > 12$? [ANS] A. $z \\geq 11$ B. $x < 5$ C. $x > 4$ D. Nothing E. $z < 11$ F. None of the above", "answer_v3": [ "E", "C", "F", "AB" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Set_theory_and_logic_0050", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Rules of inference", "level": "5", "keywords": [ "proof\u2019", "\u2018logic\u2019", "\u2018equivalence\u2019" ], "problem_v1": "Select all the sentences that are logically equivalent to $x > 7 \\Rightarrow x^{2} > k$. [ANS] A. $x \\leq 7 \\:$ or $\\: x^{2} > k$ B. $x^{2} \\leq k \\Rightarrow x \\leq 7$ C. $x \\leq 6 \\Rightarrow x^{2} \\leq k$ D. $x > 7 \\:$ and $\\: x^{2} > k$ E. $x^{2} > k \\Rightarrow x > 7$ F. None of the above", "answer_v1": [ "AB" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Select all the sentences that are logically equivalent to $x > 4 \\Rightarrow x^{2} > k$. [ANS] A. $x \\leq 6 \\Rightarrow x^{2} \\leq k$ B. $x^{2} > k \\Rightarrow x > 4$ C. $x > 4 \\:$ and $\\: x^{2} > k$ D. $x \\leq 4 \\:$ or $\\: x^{2} > k$ E. $x^{2} \\leq k \\Rightarrow x \\leq 4$ F. None of the above", "answer_v2": [ "DE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Select all the sentences that are logically equivalent to $x > 5 \\Rightarrow x^{2} > k$. [ANS] A. $x^{2} \\leq k \\Rightarrow x \\leq 5$ B. $x \\leq 6 \\Rightarrow x^{2} \\leq k$ C. $x > 5 \\:$ and $\\: x^{2} > k$ D. $x^{2} > k \\Rightarrow x > 5$ E. $x \\leq 5 \\:$ or $\\: x^{2} > k$ F. None of the above", "answer_v3": [ "AE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Set_theory_and_logic_0051", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Rules of inference", "level": "5", "keywords": [ "proof\u2019", "\u2018logic\u2019", "\u2018equivalence\u2019" ], "problem_v1": "Select all the sentences that are logically equivalent to $x \\in S \\Rightarrow x \\in T$. [ANS] A. $x \\not\\in T \\Rightarrow x \\notin S$ B. $x \\in T \\Rightarrow x \\in S$ C. $x \\in S$ or $x \\not\\in T$ D. $x \\in T$ or $x \\not\\in S$ E. $y \\in S \\Rightarrow y \\in T$ F. None of the above", "answer_v1": [ "ADE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Select all the sentences that are logically equivalent to $x \\in S \\Rightarrow x \\in T$. [ANS] A. $x \\not\\in T \\Rightarrow x \\notin S$ B. $x \\in T \\Rightarrow x \\in S$ C. $x \\in T$ or $x \\not\\in S$ D. $x \\in S$ or $x \\not\\in T$ E. $y \\in S \\Rightarrow y \\in T$ F. None of the above", "answer_v2": [ "ACE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Select all the sentences that are logically equivalent to $x \\in S \\Rightarrow x \\in T$. [ANS] A. $x \\in T \\Rightarrow x \\in S$ B. $x \\not\\in T \\Rightarrow x \\notin S$ C. $x \\in T$ or $x \\not\\in S$ D. $x \\in S$ or $x \\not\\in T$ E. $y \\in S \\Rightarrow y \\in T$ F. None of the above", "answer_v3": [ "BCE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Set_theory_and_logic_0052", "subject": "Set_theory_and_logic", "topic": "Propositional logic", "subtopic": "Rules of inference", "level": "2", "keywords": [ "Logic", "Reasoning" ], "problem_v1": "Which rule of inference is used in each of the following arguments? Check the correct answers. 1. Colleen is a cat. Colleen is gray. Therefore Colleen is a gray cat. [ANS] A. Conjuction. B. Modus ponens. C. Modus tollens. D. Hypothetical syllogism. E. Addition. F. Simplification. G. Disjunctive syllogism.\n2. If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, I will understand the material. [ANS] A. Disjunctive syllogism. B. Addition. C. Modus ponens. D. Simplification. E. Conjuction. F. Hypothetical syllogism. G. Modus tollens.\n3. Steve will work at a computer company this summer. Therefore, this summer Steve will work at a computer company or be a beach bum. [ANS] A. Modus tollens. B. Addition. C. Conjuction. D. Hypothetical syllogism. E. Disjunctive syllogism. F. Modus ponens. G. Simplication.\n4. If Pete had hit gold then Pete would be rich. Pete is not rich. Therefore, he did not hit gold. [ANS] A. Simplification. B. Conjuction. C. Modus ponens. D. Hypothetical syllogism. E. Disjunctive syllogism. F. Addition. G. Modus tollens.", "answer_v1": [ "A", "F", "B", "G" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Which rule of inference is used in each of the following arguments? Check the correct answers. 1. If it is rainy, then the pool will be closed. It is rainy. Therfore, the pool is closed. [ANS] A. Disjunctive syllogism. B. Addition. C. Simplification. D. Modus ponens. E. Conjunction. F. Modus tollens. G. Hypothetical syllogism.\n2. Linda is an excellent swimmer. If Linda is an excellent swimmer, then she can work as a lifeguard. Therefore, Linda can work as a lifeguard. [ANS] A. Modus tollens. B. Modus ponens. C. Disjunctive syllogism. D. Simplification. E. Hypothetical syllogism. F. Conjuction. G. Addition.\n3. It is either hotter than 100 degrees today or the pollution outside is dangerous. It is less than 100 degrees outside today. Therefore, the pollution is dangerous. [ANS] A. Hypothetical syllogism. B. Modus tollens. C. Disjunctive syllogism. D. Addition. E. Conjuction. F. Modus ponens. G. Simplification.\n4. If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, I will understand the material. [ANS] A. Conjuction. B. Disjunctive syllogism. C. Modus tollens. D. Addition. E. Hypothetical syllogism. F. Modus ponens. G. Simplification.", "answer_v2": [ "D", "B", "C", "E" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Which rule of inference is used in each of the following arguments? Check the correct answers. 1. Kangaroos live in Australia and are marsupials. Therefore, kangaroos are marsupials. [ANS] A. Hypothetical syllogism. B. Simplification. C. Modus tollens. D. Conjuction. E. Addition. F. Disjunctive syllogism. G. Modus ponens.\n2. Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major. [ANS] A. Addition. B. Modus ponens. C. Hypothetical syllogism. D. Conjunction. E. Disjunctive syllogism. F. Modus tollens. G. Simplification.\n3. If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn. [ANS] A. Addition. B. Hypothetical syllogism. C. Modus ponens. D. Modus tollens. E. Simplification. F. Conjunction. G. Disjunctive syllogism.\n4. If it is rainy, then the pool will be closed. It is rainy. Therfore, the pool is closed. [ANS] A. Simplification. B. Addition. C. Hypothetical syllogism. D. Disjunctive syllogism. E. Conjunction. F. Modus tollens. G. Modus ponens.", "answer_v3": [ "B", "A", "B", "G" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Set_theory_and_logic_0053", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Predicates", "level": "2", "keywords": [ "predicate" ], "problem_v1": "Let $P(x)$ be the predicate $3-x > x$. Determine whether the following expressions are true or false. $P(5)$ [ANS] A. True B. False C. Not enough information\n$P(-3)$ [ANS] A. True B. False C. Not enough information\n$P(y)$ [ANS] A. False B. True C. Not enough information\n$P(0)$ [ANS] A. True B. False C. Not enough information", "answer_v1": [ "B", "A", "C", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Let $P(x)$ be the predicate $3-x > x$. Determine whether the following expressions are true or false. $P(5)$ [ANS] A. False B. True C. Not enough information\n$P(-3)$ [ANS] A. True B. False C. Not enough information\n$P(y)$ [ANS] A. True B. False C. Not enough information\n$P(0)$ [ANS] A. False B. True C. Not enough information", "answer_v2": [ "A", "A", "C", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Let $P(x)$ be the predicate $3-x > x$. Determine whether the following expressions are true or false. $P(5)$ [ANS] A. False B. True C. Not enough information\n$P(-3)$ [ANS] A. True B. False C. Not enough information\n$P(y)$ [ANS] A. True B. False C. Not enough information\n$P(0)$ [ANS] A. True B. False C. Not enough information", "answer_v3": [ "A", "A", "C", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Set_theory_and_logic_0054", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Predicates", "level": "2", "keywords": [ "predicate" ], "problem_v1": "Consider the following statement: $\\text{Not everyone is good at sports.}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{is good at sports.}$ Write the statement as a universal statement [ANS] A. $\\sim (\\forall x\\in X, P(x))$ B. $\\forall x\\in X, \\sim P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\sim \\exists x\\in X: \\sim P(x)$\nWrite the statement as an existential statement [ANS] A. $\\exists x\\in X: \\sim P(x)$ B. $\\exists x\\in X: P(x)$ C. $\\sim (\\exists x\\in X: P(x))$ D. $\\sim\\exists x\\in X: P(x)$\nWrite the existential statement in English [ANS] A. There exists someone who is good at sports B. There does not exist someone who is good at sports C. There exists someone who is not good at sports D. There does not exist someone who is not good at sports", "answer_v1": [ "A", "A", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the following statement: $\\text{Not everyone is good at sports.}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{is good at sports.}$ Write the statement as a universal statement [ANS] A. $\\sim (\\forall x\\in X, P(x))$ B. $\\forall x\\in X, \\sim P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\sim \\exists x\\in X: \\sim P(x)$\nWrite the statement as an existential statement [ANS] A. $\\sim (\\exists x\\in X: P(x))$ B. $\\exists x\\in X: P(x)$ C. $\\sim\\exists x\\in X: P(x)$ D. $\\exists x\\in X: \\sim P(x)$\nWrite the existential statement in English [ANS] A. There exists someone who is good at sports B. There does not exist someone who is good at sports C. There does not exist someone who is not good at sports D. There exists someone who is not good at sports", "answer_v2": [ "A", "D", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the following statement: $\\text{Not everyone is good at sports.}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{is good at sports.}$ Write the statement as a universal statement [ANS] A. $\\sim (\\forall x\\in X, P(x))$ B. $\\forall x\\in X, \\sim P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\sim \\exists x\\in X: \\sim P(x)$\nWrite the statement as an existential statement [ANS] A. $\\sim (\\exists x\\in X: P(x))$ B. $\\exists x\\in X: \\sim P(x)$ C. $\\exists x\\in X: P(x)$ D. $\\sim\\exists x\\in X: P(x)$\nWrite the existential statement in English [ANS] A. There exists someone who is good at sports B. There exists someone who is not good at sports C. There does not exist someone who is not good at sports D. There does not exist someone who is good at sports", "answer_v3": [ "A", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Set_theory_and_logic_0055", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Predicates", "level": "2", "keywords": [ "predicate" ], "problem_v1": "Let $T(x)$ be the predicate $x\\text{mod} 8\\text {is prime}$. Determine whether the following expressions are true or false. $P(22)$ [ANS] A. True B. False C. Not enough information\n$P(75)$ [ANS] A. True B. False C. Not enough information\n$P(56)$ [ANS] A. False B. True C. Not enough information\n$P(34)$ [ANS] A. True B. False C. Not enough information", "answer_v1": [ "B", "A", "A", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "Let $T(x)$ be the predicate $x\\text{mod} 8\\text {is prime}$. Determine whether the following expressions are true or false. $P(22)$ [ANS] A. False B. True C. Not enough information\n$P(75)$ [ANS] A. True B. False C. Not enough information\n$P(56)$ [ANS] A. True B. False C. Not enough information\n$P(34)$ [ANS] A. False B. True C. Not enough information", "answer_v2": [ "A", "A", "B", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "Let $T(x)$ be the predicate $x\\text{mod} 8\\text {is prime}$. Determine whether the following expressions are true or false. $P(22)$ [ANS] A. False B. True C. Not enough information\n$P(75)$ [ANS] A. True B. False C. Not enough information\n$P(56)$ [ANS] A. False B. True C. Not enough information\n$P(34)$ [ANS] A. True B. False C. Not enough information", "answer_v3": [ "A", "A", "A", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Set_theory_and_logic_0056", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "3", "keywords": [ "proof\u2019", "\u2018logic\u2019", "\u2018definition\u2019", "\u2018reading\u2019" ], "problem_v1": "Use a comma to separate correct answers. Here is a definition: $n^{*}=n/2$ if $n$ is even, and $n^{*}=3n$ if $n$ is odd. Solve for $n$: $n^{*}=(7^{*})(8^{*})$ $n=$ [ANS]", "answer_v1": [ "(168, 28)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Use a comma to separate correct answers. Here is a definition: $n^{*}=n/2$ if $n$ is even, and $n^{*}=3n$ if $n$ is odd. Solve for $n$: $n^{*}=(3^{*})(10^{*})$ $n=$ [ANS]", "answer_v2": [ "(90, 15)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Use a comma to separate correct answers. Here is a definition: $n^{*}=n/2$ if $n$ is even, and $n^{*}=3n$ if $n$ is odd. Solve for $n$: $n^{*}=(3^{*})(8^{*})$ $n=$ [ANS]", "answer_v3": [ "(72, 12)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Set_theory_and_logic_0057", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "3", "keywords": [ "proof\u2019", "\u2018logic\u2019", "\u2018definition\u2019", "\u2018reading\u2019" ], "problem_v1": "Here is a definition: $f(a, b)=2a-b$. Solve for $x$: $f(6, x)=f(8, 6). \\hspace{1em} x=$ [ANS]", "answer_v1": [ "2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Here is a definition: $f(a, b)=2a-b$. Solve for $x$: $f(2, x)=f(10, 3). \\hspace{1em} x=$ [ANS]", "answer_v2": [ "-13" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Here is a definition: $f(a, b)=2a-b$. Solve for $x$: $f(3, x)=f(8, 4). \\hspace{1em} x=$ [ANS]", "answer_v3": [ "-6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Set_theory_and_logic_0058", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "3", "keywords": [ "proof\u2019", "\u2018logic\u2019", "\u2018definition\u2019", "\u2018reading\u2019" ], "problem_v1": "Here is a definition: $x \\: \\Box \\: y=2x+3y$. Read it and use it to: Find $4 \\: \\Box \\: 7 \\:=$ [ANS]\nFind $c \\:\\Box \\: x \\:=$ [ANS]\nSolve for $x$ if $(x+1) \\: \\Box \\: x=23$ $x=$ [ANS]", "answer_v1": [ "29", "2*c+3*x", "21/5" ], "answer_type_v1": [ "NV", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Here is a definition: $x \\: \\Box \\: y=2x+3y$. Read it and use it to: Find $1 \\: \\Box \\: 9 \\:=$ [ANS]\nFind $c \\:\\Box \\: x \\:=$ [ANS]\nSolve for $x$ if $(x+1) \\: \\Box \\: x=13$ $x=$ [ANS]", "answer_v2": [ "29", "2*c+3*x", "11/5" ], "answer_type_v2": [ "NV", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Here is a definition: $x \\: \\Box \\: y=2x+3y$. Read it and use it to: Find $2 \\: \\Box \\: 8 \\:=$ [ANS]\nFind $c \\:\\Box \\: x \\:=$ [ANS]\nSolve for $x$ if $(x+1) \\: \\Box \\: x=15$ $x=$ [ANS]", "answer_v3": [ "28", "2*c+3*x", "13/5" ], "answer_type_v3": [ "NV", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Set_theory_and_logic_0059", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "5", "keywords": [ "proof\u2019", "\u2018grammar\u2019", "\u2018expression\u2019", "open sentence" ], "problem_v1": "Select all the sentences that are open sentences. [ANS] A. $S \\subset T$ B. $x=4$ C. $5 \\in (4, 7)$ D. $x \\in \\lbrace 4, 5 \\rbrace$ E. $\\lbrace 4, 5 \\rbrace \\subset [4, 5]$ F. None of the above", "answer_v1": [ "ABD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Select all the sentences that are open sentences. [ANS] A. $\\lbrace 1, 2 \\rbrace \\subset [1, 2]$ B. $2 \\in (1, 4)$ C. $x=1$ D. $S \\subset T$ E. $x \\in \\lbrace 1, 2 \\rbrace$ F. None of the above", "answer_v2": [ "CDE" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Select all the sentences that are open sentences. [ANS] A. $x=2$ B. $\\lbrace 2, 3 \\rbrace \\subset [2, 3]$ C. $3 \\in (2, 5)$ D. $x \\in \\lbrace 2, 3 \\rbrace$ E. $S \\subset T$ F. None of the above", "answer_v3": [ "ADE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Set_theory_and_logic_0060", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "5", "keywords": [ "proof\u2019", "\u2018grammar\u2019", "\u2018expression\u2019", "open sentence" ], "problem_v1": "Select all the sentences that are open sentences. [ANS] A. $x-a=b \\:$ iff $\\: x=b+a$ B. $4+1=5$ C. $x/2=c$ D. $x > 4$ E. $x/2=c \\:$ iff $\\: x=2c$ F. $x=a$ G. None of the above", "answer_v1": [ "CDF" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Select all the sentences that are open sentences. [ANS] A. $x > 1$ B. $x/2=c$ C. $x/2=c \\:$ iff $\\: x=2c$ D. $x=a$ E. $x-a=b \\:$ iff $\\: x=b+a$ F. $1+1=2$ G. None of the above", "answer_v2": [ "ABD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Select all the sentences that are open sentences. [ANS] A. $x-a=b \\:$ iff $\\: x=b+a$ B. $x/2=c \\:$ iff $\\: x=2c$ C. $x > 2$ D. $x=a$ E. $x/2=c$ F. $2+1=3$ G. None of the above", "answer_v3": [ "CDE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Set_theory_and_logic_0061", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "", "keywords": [ "predicate" ], "problem_v1": "Consider the following statement: $\\text{Some people are allergic to cats.}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{is allergic to cats.}$ Write the statement in quantified form [ANS] A. $\\exists x\\in X: P(x)$ B. $\\exists x\\in X: \\sim P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\forall x\\in X, P(x)$\nNegate the quantified statement [ANS] A. $\\forall x\\in X, \\sim P(x)$ B. $\\forall x\\in X, P(x)$ C. $\\exists x\\in X, \\sim P(x)$ D. $\\exists x\\in X: P(x)$\nTranslate the negated statement into English [ANS] A. Everyone is allergic to cats B. There exists someone that is allergic to cats C. Nobody is allergic to cats D. There exists someone that is not allergic to cats", "answer_v1": [ "A", "A", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the following statement: $\\text{Some people are allergic to cats.}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{is allergic to cats.}$ Write the statement in quantified form [ANS] A. $\\exists x\\in X: P(x)$ B. $\\exists x\\in X: \\sim P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\forall x\\in X, P(x)$\nNegate the quantified statement [ANS] A. $\\exists x\\in X, \\sim P(x)$ B. $\\forall x\\in X, P(x)$ C. $\\exists x\\in X: P(x)$ D. $\\forall x\\in X, \\sim P(x)$\nTranslate the negated statement into English [ANS] A. Everyone is allergic to cats B. There exists someone that is allergic to cats C. There exists someone that is not allergic to cats D. Nobody is allergic to cats", "answer_v2": [ "A", "D", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the following statement: $\\text{Some people are allergic to cats.}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{is allergic to cats.}$ Write the statement in quantified form [ANS] A. $\\exists x\\in X: P(x)$ B. $\\forall x\\in X, \\sim P(x)$ C. $\\exists x\\in X: \\sim P(x)$ D. $\\forall x\\in X, P(x)$\nNegate the quantified statement [ANS] A. $\\exists x\\in X, \\sim P(x)$ B. $\\forall x\\in X, \\sim P(x)$ C. $\\forall x\\in X, P(x)$ D. $\\exists x\\in X: P(x)$\nTranslate the negated statement into English [ANS] A. Everyone is allergic to cats B. Nobody is allergic to cats C. There exists someone that is not allergic to cats D. There exists someone that is allergic to cats", "answer_v3": [ "A", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Set_theory_and_logic_0062", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "", "keywords": [ "predicate" ], "problem_v1": "Consider the following statement: $\\text{Everyone likes playing games.}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{likes playing games.}$ Write the statement in quantified form [ANS] A. $\\forall x\\in X, P(x)$ B. $\\exists x\\in X: \\sim P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\exists x\\in X: P(x)$\nNegate the quantified statement [ANS] A. $\\exists x\\in X: \\sim P(x)$ B. $\\forall x\\in X, P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\exists x\\in X: P(x)$", "answer_v1": [ "A", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the following statement: $\\text{Everyone likes playing games.}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{likes playing games.}$ Write the statement in quantified form [ANS] A. $\\forall x\\in X, P(x)$ B. $\\exists x\\in X: \\sim P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\exists x\\in X: P(x)$\nNegate the quantified statement [ANS] A. $\\forall x\\in X, \\sim P(x)$ B. $\\forall x\\in X, P(x)$ C. $\\exists x\\in X: P(x)$ D. $\\exists x\\in X: \\sim P(x)$", "answer_v2": [ "A", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the following statement: $\\text{Everyone likes playing games.}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{likes playing games.}$ Write the statement in quantified form [ANS] A. $\\forall x\\in X, P(x)$ B. $\\forall x\\in X, \\sim P(x)$ C. $\\exists x\\in X: \\sim P(x)$ D. $\\exists x\\in X: P(x)$\nNegate the quantified statement [ANS] A. $\\forall x\\in X, \\sim P(x)$ B. $\\exists x\\in X: \\sim P(x)$ C. $\\forall x\\in X, P(x)$ D. $\\exists x\\in X: P(x)$", "answer_v3": [ "A", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Set_theory_and_logic_0063", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "2", "keywords": [ "predicate" ], "problem_v1": "Consider the following statement: $\\text{Nobody likes the smell of a skunk}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{likes the smell of a skunk.}$ Write the statement as an existential statement [ANS] A. $\\sim\\exists x\\in X: P(x)$ B. $\\exists x\\in X: \\sim P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\exists x\\in X: P(x)$\nWrite the statement as a universal statement [ANS] A. $\\forall x\\in X, \\sim P(x)$ B. $\\sim\\forall x\\in X, \\sim P(x)$ C. $\\forall x\\in X, P(x)$ D. $\\sim\\exists x\\in X: P(x)$", "answer_v1": [ "A", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the following statement: $\\text{Nobody likes the smell of a skunk}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{likes the smell of a skunk.}$ Write the statement as an existential statement [ANS] A. $\\sim\\exists x\\in X: P(x)$ B. $\\exists x\\in X: \\sim P(x)$ C. $\\forall x\\in X, \\sim P(x)$ D. $\\exists x\\in X: P(x)$\nWrite the statement as a universal statement [ANS] A. $\\forall x\\in X, P(x)$ B. $\\sim\\forall x\\in X, \\sim P(x)$ C. $\\sim\\exists x\\in X: P(x)$ D. $\\forall x\\in X, \\sim P(x)$", "answer_v2": [ "A", "D" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the following statement: $\\text{Nobody likes the smell of a skunk}$ For the following questions, Let $X=\\lbrace people\\rbrace$ $P(x)$ be the predicate $x\\text{likes the smell of a skunk.}$ Write the statement as an existential statement [ANS] A. $\\sim\\exists x\\in X: P(x)$ B. $\\forall x\\in X, \\sim P(x)$ C. $\\exists x\\in X: \\sim P(x)$ D. $\\exists x\\in X: P(x)$\nWrite the statement as a universal statement [ANS] A. $\\forall x\\in X, P(x)$ B. $\\forall x\\in X, \\sim P(x)$ C. $\\sim\\forall x\\in X, \\sim P(x)$ D. $\\sim\\exists x\\in X: P(x)$", "answer_v3": [ "A", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Set_theory_and_logic_0064", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "2", "keywords": [ "logic", "predicate" ], "problem_v1": "Let I(x) be the statement \"x has an Internet connection\", let C(x,y) be the statement \"x and y have chatted over the internet\". Express each of the following statements in terms of I(x) and C(x,y), quantifiers, and logical connectives. Let the universe of discourse for the variables x and y consist of all students in your class. Put the appropriate letter next to the corresponding symbolic form. [ANS] 1. $\\exists x \\exists y (y \\not=x \\wedge \\lnot C(x,y)))$ [ANS] 2. $C(Jan,Sharon)$ [ANS] 3. $\\exists x \\lnot I(x)$ [ANS] 4. $\\exists x \\exists y (y \\not=x \\wedge \\forall z \\lnot (C(x,z)\\wedge C(y,z))$ [ANS] 5. $\\forall x (I(x)) \\rightarrow \\exists y (x \\not=y \\wedge C(x,y))$ [ANS] 6. $\\lnot (C(Rachel,Chelsea)$ [ANS] 7. $\\forall x \\lnot C(x,Bob)$ [ANS] 8. $\\exists x (I(x) \\wedge \\forall y (I(y) \\rightarrow y=x))$\na) Rachel has not chatted over the internet with Chelsea. b) Jan and Sharon have chatted over the internet. c) No one in the class has chatted with Bob. d) Someone in your class does not have internet connection. e) There are two students in your class who have not chatted over the internet. f) Exactly one student in your class has an internet connection. g) Everyone in your class with an internet connection has chatted over the internet with at least one other student in your class. h) There are at least two students in your class who have not chatted with the same person in your class.", "answer_v1": [ "E", "B", "D", "h", "g", "a", "c", "f" ], "answer_type_v1": [ "OE", "OE", "OE", "OE", "OE", "OE", "OE", "OE" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let I(x) be the statement \"x has an Internet connection\", let C(x,y) be the statement \"x and y have chatted over the internet\". Express each of the following statements in terms of I(x) and C(x,y), quantifiers, and logical connectives. Let the universe of discourse for the variables x and y consist of all students in your class. Put the appropriate letter next to the corresponding symbolic form. [ANS] 1. $\\forall x (I(x)) \\rightarrow \\exists y (x \\not=y \\wedge C(x,y))$ [ANS] 2. $C(Jan,Sharon)$ [ANS] 3. $\\forall x \\lnot C(x,Bob)$ [ANS] 4. $\\exists x (I(x) \\wedge \\forall y (I(y) \\rightarrow y=x))$ [ANS] 5. $\\exists x \\exists y (y \\not=x \\wedge \\lnot C(x,y)))$ [ANS] 6. $\\exists x \\exists y (y \\not=x \\wedge \\forall z \\lnot (C(x,z)\\wedge C(y,z))$ [ANS] 7. $\\lnot (C(Rachel,Chelsea)$ [ANS] 8. $\\exists x \\lnot I(x)$\na) Rachel has not chatted over the internet with Chelsea. b) Jan and Sharon have chatted over the internet. c) No one in the class has chatted with Bob. d) Someone in your class does not have internet connection. e) There are two students in your class who have not chatted over the internet. f) Exactly one student in your class has an internet connection. g) Everyone in your class with an internet connection has chatted over the internet with at least one other student in your class. h) There are at least two students in your class who have not chatted with the same person in your class.", "answer_v2": [ "G", "B", "C", "f", "e", "h", "a", "d" ], "answer_type_v2": [ "OE", "OE", "OE", "OE", "OE", "OE", "OE", "OE" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let I(x) be the statement \"x has an Internet connection\", let C(x,y) be the statement \"x and y have chatted over the internet\". Express each of the following statements in terms of I(x) and C(x,y), quantifiers, and logical connectives. Let the universe of discourse for the variables x and y consist of all students in your class. Put the appropriate letter next to the corresponding symbolic form. [ANS] 1. $\\lnot (C(Rachel,Chelsea)$ [ANS] 2. $C(Jan,Sharon)$ [ANS] 3. $\\exists x \\exists y (y \\not=x \\wedge \\forall z \\lnot (C(x,z)\\wedge C(y,z))$ [ANS] 4. $\\exists x \\lnot I(x)$ [ANS] 5. $\\forall x \\lnot C(x,Bob)$ [ANS] 6. $\\exists x \\exists y (y \\not=x \\wedge \\lnot C(x,y)))$ [ANS] 7. $\\forall x (I(x)) \\rightarrow \\exists y (x \\not=y \\wedge C(x,y))$ [ANS] 8. $\\exists x (I(x) \\wedge \\forall y (I(y) \\rightarrow y=x))$\na) Rachel has not chatted over the internet with Chelsea. b) Jan and Sharon have chatted over the internet. c) No one in the class has chatted with Bob. d) Someone in your class does not have internet connection. e) There are two students in your class who have not chatted over the internet. f) Exactly one student in your class has an internet connection. g) Everyone in your class with an internet connection has chatted over the internet with at least one other student in your class. h) There are at least two students in your class who have not chatted with the same person in your class.", "answer_v3": [ "A", "B", "H", "d", "c", "e", "g", "f" ], "answer_type_v3": [ "OE", "OE", "OE", "OE", "OE", "OE", "OE", "OE" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0065", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "2", "keywords": [ "logic", "predicate" ], "problem_v1": "Let $C(x)$ be the statement \" $x$ has a cat\", let $D(x)$ be the statement \" $x$ has a dog\" and let $F(x)$ be the statement \" $x$ has a ferret\". Express each of the following statements in terms of $C(x)$, $D(x)$, and $F(x)$, quantifiers, and logical connectives. Let the universe of discourse consist of all students in your class. Put the appropriate letter next to the corresponding symbolic form. [ANS] 1. $\\exists x (C(x)) \\wedge (\\exists x D(x)) \\wedge (\\exists x F(x))$ [ANS] 2. $\\exists x (C(x) \\wedge F(x) \\wedge \\lnot D(x))$ [ANS] 3. $\\lnot \\exists x (C(x) \\wedge D(x) \\wedge F(x))$ [ANS] 4. $\\forall x (C(x) \\vee D(x) \\vee F(x))$ [ANS] 5. $\\exists x (C(x) \\wedge D(x) \\wedge F(x))$\na) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret but not a dog. d) No student in this class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has one of these animals.", "answer_v1": [ "E", "C", "D", "b", "a" ], "answer_type_v1": [ "OE", "OE", "OE", "OE", "OE" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let $C(x)$ be the statement \" $x$ has a cat\", let $D(x)$ be the statement \" $x$ has a dog\" and let $F(x)$ be the statement \" $x$ has a ferret\". Express each of the following statements in terms of $C(x)$, $D(x)$, and $F(x)$, quantifiers, and logical connectives. Let the universe of discourse consist of all students in your class. Put the appropriate letter next to the corresponding symbolic form. [ANS] 1. $\\forall x (C(x) \\vee D(x) \\vee F(x))$ [ANS] 2. $\\exists x (C(x) \\wedge F(x) \\wedge \\lnot D(x))$ [ANS] 3. $\\exists x (C(x) \\wedge D(x) \\wedge F(x))$ [ANS] 4. $\\exists x (C(x)) \\wedge (\\exists x D(x)) \\wedge (\\exists x F(x))$ [ANS] 5. $\\lnot \\exists x (C(x) \\wedge D(x) \\wedge F(x))$\na) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret but not a dog. d) No student in this class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has one of these animals.", "answer_v2": [ "B", "C", "A", "e", "d" ], "answer_type_v2": [ "OE", "OE", "OE", "OE", "OE" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let $C(x)$ be the statement \" $x$ has a cat\", let $D(x)$ be the statement \" $x$ has a dog\" and let $F(x)$ be the statement \" $x$ has a ferret\". Express each of the following statements in terms of $C(x)$, $D(x)$, and $F(x)$, quantifiers, and logical connectives. Let the universe of discourse consist of all students in your class. Put the appropriate letter next to the corresponding symbolic form. [ANS] 1. $\\lnot \\exists x (C(x) \\wedge D(x) \\wedge F(x))$ [ANS] 2. $\\exists x (C(x) \\wedge F(x) \\wedge \\lnot D(x))$ [ANS] 3. $\\exists x (C(x) \\wedge D(x) \\wedge F(x))$ [ANS] 4. $\\forall x (C(x) \\vee D(x) \\vee F(x))$ [ANS] 5. $\\exists x (C(x)) \\wedge (\\exists x D(x)) \\wedge (\\exists x F(x))$\na) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret but not a dog. d) No student in this class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has one of these animals.", "answer_v3": [ "D", "C", "A", "b", "e" ], "answer_type_v3": [ "OE", "OE", "OE", "OE", "OE" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0066", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Translation", "level": "2", "keywords": [ "logic", "predicate" ], "problem_v1": "Let P(x) be the statement \"x is a duck\", let Q(x) be the statement \"x is one of my poultry\", let R(x) be the statement \"x is an officer\", and let S(x) be the statement \"x is willing to waltz\". Express each of the following statements in terms of P(x), Q(x), R(x) and S(x), quantifiers, and logical connectives. Let the universe of discourse consist of all living creatures. Put the appropriate letter next to the corresponding symbolic form. [ANS] 1. $\\forall x (Q(x) \\rightarrow \\lnot R(x))$ [ANS] 2. $\\forall x (R(x) \\rightarrow S(x))$ [ANS] 3. $\\forall x (Q(x) \\rightarrow P(x))$ [ANS] 4. $\\forall x (P(x) \\rightarrow \\lnot S(x))$ [ANS] 5. $\\exists x (P(x) \\wedge \\lnot S(x))$\na) Some ducks are not willing to waltz. b) No ducks are willing to waltz. c) No officers ever decline to waltz. d) All my poultry are ducks. e) My poultry are not officers.", "answer_v1": [ "E", "C", "D", "b", "a" ], "answer_type_v1": [ "OE", "OE", "OE", "OE", "OE" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let P(x) be the statement \"x is a duck\", let Q(x) be the statement \"x is one of my poultry\", let R(x) be the statement \"x is an officer\", and let S(x) be the statement \"x is willing to waltz\". Express each of the following statements in terms of P(x), Q(x), R(x) and S(x), quantifiers, and logical connectives. Let the universe of discourse consist of all living creatures. Put the appropriate letter next to the corresponding symbolic form. [ANS] 1. $\\forall x (P(x) \\rightarrow \\lnot S(x))$ [ANS] 2. $\\forall x (R(x) \\rightarrow S(x))$ [ANS] 3. $\\exists x (P(x) \\wedge \\lnot S(x))$ [ANS] 4. $\\forall x (Q(x) \\rightarrow \\lnot R(x))$ [ANS] 5. $\\forall x (Q(x) \\rightarrow P(x))$\na) Some ducks are not willing to waltz. b) No ducks are willing to waltz. c) No officers ever decline to waltz. d) All my poultry are ducks. e) My poultry are not officers.", "answer_v2": [ "B", "C", "A", "e", "d" ], "answer_type_v2": [ "OE", "OE", "OE", "OE", "OE" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let P(x) be the statement \"x is a duck\", let Q(x) be the statement \"x is one of my poultry\", let R(x) be the statement \"x is an officer\", and let S(x) be the statement \"x is willing to waltz\". Express each of the following statements in terms of P(x), Q(x), R(x) and S(x), quantifiers, and logical connectives. Let the universe of discourse consist of all living creatures. Put the appropriate letter next to the corresponding symbolic form. [ANS] 1. $\\forall x (Q(x) \\rightarrow P(x))$ [ANS] 2. $\\forall x (R(x) \\rightarrow S(x))$ [ANS] 3. $\\exists x (P(x) \\wedge \\lnot S(x))$ [ANS] 4. $\\forall x (P(x) \\rightarrow \\lnot S(x))$ [ANS] 5. $\\forall x (Q(x) \\rightarrow \\lnot R(x))$\na) Some ducks are not willing to waltz. b) No ducks are willing to waltz. c) No officers ever decline to waltz. d) All my poultry are ducks. e) My poultry are not officers.", "answer_v3": [ "D", "C", "A", "b", "e" ], "answer_type_v3": [ "OE", "OE", "OE", "OE", "OE" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0067", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Semantics of quantifiers", "level": "2", "keywords": [ "Logic", "Quantifiers" ], "problem_v1": "Determine the truth value of the following statements if the universe of discourse of each variable is the set of real numbers. [ANS] 1. $\\forall x \\exists y (x+y=1)$ [ANS] 2. $\\forall x \\exists y (x=y^2)$ [ANS] 3. $\\forall x \\forall y \\exists z (z=\\frac{x+y}{2})$ [ANS] 4. $\\forall x \\exists y ((x+y=2) \\land (2x-y=1))$ [ANS] 5. $\\forall x \\neq 0 \\exists y (xy=1)$ [ANS] 6. $\\exists x \\exists y (x+y \\neq y+x)$ [ANS] 7. $\\exists x \\exists y ((x+2y=2) \\land (2x+4y=5))$ [ANS] 8. $\\exists x \\forall y \\neq 0 (xy=1)$ [ANS] 9. $\\exists x \\forall y (xy=0)$ [ANS] 10. $\\exists x (x^2=2)$ [ANS] 11. $\\forall x \\exists y (x^2=y)$ [ANS] 12. $\\exists x (x^2=-1)$", "answer_v1": [ "T", "F", "T", "F", "T", "F", "F", "F", "T", "T", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Determine the truth value of the following statements if the universe of discourse of each variable is the set of real numbers. [ANS] 1. $\\exists x \\exists y ((x+2y=2) \\land (2x+4y=5))$ [ANS] 2. $\\exists x (x^2=-1)$ [ANS] 3. $\\forall x \\exists y (x^2=y)$ [ANS] 4. $\\forall x \\forall y \\exists z (z=\\frac{x+y}{2})$ [ANS] 5. $\\forall x \\exists y (x=y^2)$ [ANS] 6. $\\exists x \\forall y \\neq 0 (xy=1)$ [ANS] 7. $\\exists x (x^2=2)$ [ANS] 8. $\\exists x \\exists y (x+y \\neq y+x)$ [ANS] 9. $\\forall x \\exists y (x+y=1)$ [ANS] 10. $\\forall x \\neq 0 \\exists y (xy=1)$ [ANS] 11. $\\forall x \\exists y ((x+y=2) \\land (2x-y=1))$ [ANS] 12. $\\exists x \\forall y (xy=0)$", "answer_v2": [ "F", "F", "T", "T", "F", "F", "T", "F", "T", "T", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Determine the truth value of the following statements if the universe of discourse of each variable is the set of real numbers. [ANS] 1. $\\forall x \\exists y ((x+y=2) \\land (2x-y=1))$ [ANS] 2. $\\forall x \\exists y (x+y=1)$ [ANS] 3. $\\forall x \\neq 0 \\exists y (xy=1)$ [ANS] 4. $\\exists x (x^2=-1)$ [ANS] 5. $\\exists x \\forall y (xy=0)$ [ANS] 6. $\\forall x \\exists y (x=y^2)$ [ANS] 7. $\\exists x \\exists y (x+y \\neq y+x)$ [ANS] 8. $\\forall x \\forall y \\exists z (z=\\frac{x+y}{2})$ [ANS] 9. $\\exists x (x^2=2)$ [ANS] 10. $\\forall x \\exists y (x^2=y)$ [ANS] 11. $\\exists x \\forall y \\neq 0 (xy=1)$ [ANS] 12. $\\exists x \\exists y ((x+2y=2) \\land (2x+4y=5))$", "answer_v3": [ "F", "T", "T", "F", "T", "F", "F", "T", "T", "T", "F", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0068", "subject": "Set_theory_and_logic", "topic": "First order logic", "subtopic": "Semantics of quantifiers", "level": "2", "keywords": [ "logic", "predicate" ], "problem_v1": "Let $Q(x,y)$ be the statement \" $x+y=x-y$ \". If the universe of discourse for both variables consists of all integers, what are the truth values? [ANS] 1. $\\forall y \\exists x\\ Q(x,y)$ [ANS] 2. $\\forall y\\ Q(1,y)$ [ANS] 3. $\\forall x \\exists y\\ Q(x,y)$ [ANS] 4. $\\forall x \\exists y\\ (Q(x,y) \\land (2x-y=1))$ [ANS] 5. $\\exists x\\ Q(x,2)$ [ANS] 6. $Q(2,0)$ [ANS] 7. $\\forall x \\exists y\\ (x=y^2)$ [ANS] 8. $\\exists y \\forall x\\ Q(x,y)$", "answer_v1": [ "F", "F", "T", "F", "F", "T", "F", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let $Q(x,y)$ be the statement \" $x+y=x-y$ \". If the universe of discourse for both variables consists of all integers, what are the truth values? [ANS] 1. $\\exists x \\forall y\\ Q(x,y)$ [ANS] 2. $\\forall y\\ Q(1,y)$ [ANS] 3. $\\exists x\\ Q(x,2)$ [ANS] 4. $\\exists x \\exists y\\ Q(x,y)$ [ANS] 5. $\\forall x \\exists y\\ (Q(x,y) \\land (2x-y=1))$ [ANS] 6. $\\forall x \\forall y\\ Q(x,y)$ [ANS] 7. $Q(1,1)$ [ANS] 8. $\\forall x \\exists y\\ Q(x,y)$", "answer_v2": [ "F", "F", "F", "T", "F", "F", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let $Q(x,y)$ be the statement \" $x+y=x-y$ \". If the universe of discourse for both variables consists of all integers, what are the truth values? [ANS] 1. $Q(2,0)$ [ANS] 2. $\\forall y\\ Q(1,y)$ [ANS] 3. $\\exists x\\ Q(x,2)$ [ANS] 4. $\\forall x \\exists y\\ Q(x,y)$ [ANS] 5. $Q(1,1)$ [ANS] 6. $\\forall y \\exists x\\ Q(x,y)$ [ANS] 7. $\\forall x \\exists y\\ (x=y^2)$ [ANS] 8. $\\exists x \\forall y\\ Q(x,y)$", "answer_v3": [ "T", "F", "F", "T", "F", "F", "F", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Set_theory_and_logic_0069", "subject": "Set_theory_and_logic", "topic": "Pattern matching", "subtopic": "Numeric", "level": "4", "keywords": [], "problem_v1": "Find the next item in each list:\n6, 11, 16, 21, 26, [ANS]\n5, 11, 17, 23, 29, [ANS]\n61, 55, 49, 43, 37, [ANS]", "answer_v1": [ "31", "35", "31" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the next item in each list:\n2, 9, 16, 23, 30, [ANS]\n3, 7, 11, 15, 19, [ANS]\n57, 51, 45, 39, 33, [ANS]", "answer_v2": [ "37", "23", "27" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the next item in each list:\n2, 7, 12, 17, 22, [ANS]\n3, 9, 15, 21, 27, [ANS]\n63, 57, 51, 45, 39, [ANS]", "answer_v3": [ "27", "33", "33" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Set_theory_and_logic_0070", "subject": "Set_theory_and_logic", "topic": "Pattern matching", "subtopic": "Numeric", "level": "4", "keywords": [ "prealgebra", "common core" ], "problem_v1": "Find the next three numbers in the pattern:\n$1,-2,7,-8,13,-14,$ [ANS]\n(Separate your answers by commas.)", "answer_v1": [ "(19, -20, 25)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the next three numbers in the pattern:\n$3,2,5,0,7,-2,$ [ANS]\n(Separate your answers by commas.)", "answer_v2": [ "(9, -4, 11)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the next three numbers in the pattern:\n$1,0,3,-2,5,-4,$ [ANS]\n(Separate your answers by commas.)", "answer_v3": [ "(7, -6, 9)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] } ]