Mathematical Reasoning — 176 Discrete Mathematics MCQs
176 practice MCQs based on the supplied Discrete Mathematics course outline. Each question has four options and an expandable answer with an explanation. This is an original practice bank, not a copy of the recommended textbooks.
Sets & Subsets
Q1. If A has 3 distinct elements, how many subsets does A have?
- A. 8
- B. 4
- C. 9
- D. 7
Show correct answer and explanation
Correct answer: A. 8
Each element is either included or excluded, giving 2^3 = 8 subsets.
Q2. How many proper subsets does a 3-element set have?
- A. 8
- B. 3
- C. 4
- D. 7
Show correct answer and explanation
Correct answer: D. 7
There are 2^3 subsets; exclude the set itself.
Q3. What is the cardinality of the power set of a set with 2 elements?
- A. 3
- B. 5
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: C. 4
The power set contains all 2^2 subsets.
Q4. If A has 4 distinct elements, how many subsets does A have?
- A. 15
- B. 17
- C. 16
- D. 8
Show correct answer and explanation
Correct answer: C. 16
Each element is either included or excluded, giving 2^4 = 16 subsets.
Q5. How many proper subsets does a 4-element set have?
- A. 8
- B. 15
- C. 16
- D. 4
Show correct answer and explanation
Correct answer: B. 15
There are 2^4 subsets; exclude the set itself.
Q6. What is the cardinality of the power set of a set with 3 elements?
- A. 8
- B. 3
- C. 6
- D. 7
Show correct answer and explanation
Correct answer: A. 8
The power set contains all 2^3 subsets.
Q7. If A has 5 distinct elements, how many subsets does A have?
- A. 32
- B. 16
- C. 25
- D. 31
Show correct answer and explanation
Correct answer: A. 32
Each element is either included or excluded, giving 2^5 = 32 subsets.
Q8. How many proper subsets does a 5-element set have?
- A. 32
- B. 5
- C. 16
- D. 31
Show correct answer and explanation
Correct answer: D. 31
There are 2^5 subsets; exclude the set itself.
Q9. What is the cardinality of the power set of a set with 4 elements?
- A. 8
- B. 15
- C. 16
- D. 4
Show correct answer and explanation
Correct answer: C. 16
The power set contains all 2^4 subsets.
Q10. If A has 6 distinct elements, how many subsets does A have?
- A. 36
- B. 63
- C. 64
- D. 32
Show correct answer and explanation
Correct answer: C. 64
Each element is either included or excluded, giving 2^6 = 64 subsets.
Q11. How many proper subsets does a 6-element set have?
- A. 32
- B. 63
- C. 64
- D. 6
Show correct answer and explanation
Correct answer: B. 63
There are 2^6 subsets; exclude the set itself.
Q12. What is the cardinality of the power set of a set with 5 elements?
- A. 32
- B. 5
- C. 10
- D. 31
Show correct answer and explanation
Correct answer: A. 32
The power set contains all 2^5 subsets.
Q13. If A has 7 distinct elements, how many subsets does A have?
- A. 128
- B. 64
- C. 49
- D. 127
Show correct answer and explanation
Correct answer: A. 128
Each element is either included or excluded, giving 2^7 = 128 subsets.
Q14. How many proper subsets does a 7-element set have?
- A. 128
- B. 7
- C. 64
- D. 127
Show correct answer and explanation
Correct answer: D. 127
There are 2^7 subsets; exclude the set itself.
Q15. What is the cardinality of the power set of a set with 6 elements?
- A. 12
- B. 63
- C. 64
- D. 6
Show correct answer and explanation
Correct answer: C. 64
The power set contains all 2^6 subsets.
Q16. If A has 8 distinct elements, how many subsets does A have?
- A. 64
- B. 255
- C. 256
- D. 128
Show correct answer and explanation
Correct answer: C. 256
Each element is either included or excluded, giving 2^8 = 256 subsets.
Q17. How many proper subsets does a 8-element set have?
- A. 128
- B. 255
- C. 256
- D. 8
Show correct answer and explanation
Correct answer: B. 255
There are 2^8 subsets; exclude the set itself.
Q18. What is the cardinality of the power set of a set with 7 elements?
- A. 128
- B. 7
- C. 14
- D. 127
Show correct answer and explanation
Correct answer: A. 128
The power set contains all 2^7 subsets.
Q19. If A has 9 distinct elements, how many subsets does A have?
- A. 512
- B. 256
- C. 81
- D. 511
Show correct answer and explanation
Correct answer: A. 512
Each element is either included or excluded, giving 2^9 = 512 subsets.
Q20. How many proper subsets does a 9-element set have?
- A. 512
- B. 9
- C. 256
- D. 511
Show correct answer and explanation
Correct answer: D. 511
There are 2^9 subsets; exclude the set itself.
Q21. What is the cardinality of the power set of a set with 8 elements?
- A. 16
- B. 255
- C. 256
- D. 8
Show correct answer and explanation
Correct answer: C. 256
The power set contains all 2^8 subsets.
Q22. If A has 10 distinct elements, how many subsets does A have?
- A. 100
- B. 1023
- C. 1024
- D. 512
Show correct answer and explanation
Correct answer: C. 1024
Each element is either included or excluded, giving 2^10 = 1024 subsets.
Q23. How many proper subsets does a 10-element set have?
- A. 512
- B. 1023
- C. 1024
- D. 10
Show correct answer and explanation
Correct answer: B. 1023
There are 2^10 subsets; exclude the set itself.
Q24. What is the cardinality of the power set of a set with 9 elements?
- A. 512
- B. 9
- C. 18
- D. 511
Show correct answer and explanation
Correct answer: A. 512
The power set contains all 2^9 subsets.
Algebra of Sets
Q25. If |A|=10, |B|=8, and |A∩B|=3, find |A∪B|.
- A. 3
- B. 15
- C. 18
- D. 21
Show correct answer and explanation
Correct answer: B. 15
Inclusion–exclusion: |A∪B| = 10+8−3 = 15.
Q26. In a universal set of 24 elements, A has 11 elements. Find |Aᶜ|.
- A. 13
- B. 11
- C. 24
- D. 35
Show correct answer and explanation
Correct answer: A. 13
The complement contains 24−11 = 13 elements.
Q27. If |A|=9, |B|=7, and |A∩B|=2, how many elements belong to A only?
- A. 9
- B. 5
- C. 11
- D. 7
Show correct answer and explanation
Correct answer: D. 7
A only means |A\B| = |A|−|A∩B| = 7.
Q28. Which expression is equal to (A∪B)ᶜ (De Morgan’s law), for arbitrary sets A and B?
- A. A∩B
- B. A∪B
- C. Aᶜ∩Bᶜ
- D. Aᶜ∪Bᶜ
Show correct answer and explanation
Correct answer: C. Aᶜ∩Bᶜ
Complement of a union is the intersection of the complements.
Q29. Which identity holds for any set A and any universe U?
- A. A∩U = ∅
- B. A∪Aᶜ = U
- C. A∩Aᶜ = U
- D. A∪∅ = U
Show correct answer and explanation
Correct answer: B. A∪Aᶜ = U
Every element of U lies either in A or in its complement.
Q30. If |A|=11, |B|=9, and |A∩B|=4, find |A∪B|.
- A. 20
- B. 24
- C. 4
- D. 16
Show correct answer and explanation
Correct answer: D. 16
Inclusion–exclusion: |A∪B| = 11+9−4 = 16.
Q31. In a universal set of 25 elements, A has 12 elements. Find |Aᶜ|.
- A. 25
- B. 37
- C. 13
- D. 12
Show correct answer and explanation
Correct answer: C. 13
The complement contains 25−12 = 13 elements.
Q32. If |A|=10, |B|=8, and |A∩B|=2, how many elements belong to A only?
- A. 12
- B. 8
- C. 10
- D. 6
Show correct answer and explanation
Correct answer: B. 8
A only means |A\B| = |A|−|A∩B| = 8.
Q33. If A={1,2,3,4} and B={3,4,5}, what is A\B?
- A. {1,2}
- B. {3,4}
- C. {5}
- D. {1,2,5}
Show correct answer and explanation
Correct answer: A. {1,2}
A\B contains elements of A not in B.
Q34. If A={1,2,3} and B={3,4}, what is the symmetric difference A△B?
- A. {3}
- B. {1,2,4}
- C. {1,2,3,4}
- D. {1,2}
Show correct answer and explanation
Correct answer: B. {1,2,4}
Symmetric difference contains elements belonging to exactly one of the sets.
Q35. If |A|=12, |B|=10, and |A∩B|=5, find |A∪B|.
- A. 5
- B. 17
- C. 22
- D. 27
Show correct answer and explanation
Correct answer: B. 17
Inclusion–exclusion: |A∪B| = 12+10−5 = 17.
Q36. In a universal set of 26 elements, A has 13 elements. Find |Aᶜ|.
- A. 13
- B. 26
- C. 39
- D. 14
Show correct answer and explanation
Correct answer: A. 13
The complement contains 26−13 = 13 elements.
Q37. If |A|=11, |B|=9, and |A∩B|=2, how many elements belong to A only?
- A. 11
- B. 7
- C. 13
- D. 9
Show correct answer and explanation
Correct answer: D. 9
A only means |A\B| = |A|−|A∩B| = 9.
Q38. If |A|=3 and |B|=4, how many ordered pairs are in A×B?
- A. 7
- B. 12
- C. 1
- D. 24
Show correct answer and explanation
Correct answer: B. 12
The Cartesian product contains |A||B|=3×4=12 ordered pairs.
Q39. If A⊆B and B⊆A, which conclusion follows?
- A. A∩B=∅
- B. A=B
- C. A∪B=∅
- D. A≠B
Show correct answer and explanation
Correct answer: B. A=B
Mutual inclusion is the defining criterion for equality of sets.
Q40. If |A|=13, |B|=11, and |A∩B|=6, find |A∪B|.
- A. 24
- B. 30
- C. 6
- D. 18
Show correct answer and explanation
Correct answer: D. 18
Inclusion–exclusion: |A∪B| = 13+11−6 = 18.
Q41. In a universal set of 27 elements, A has 14 elements. Find |Aᶜ|.
- A. 27
- B. 41
- C. 13
- D. 14
Show correct answer and explanation
Correct answer: C. 13
The complement contains 27−14 = 13 elements.
Q42. If |A|=12, |B|=10, and |A∩B|=2, how many elements belong to A only?
- A. 14
- B. 10
- C. 12
- D. 8
Show correct answer and explanation
Correct answer: B. 10
A only means |A\B| = |A|−|A∩B| = 10.
Q43. For arbitrary sets A and B, what is A∩(A∪B)?
- A. B
- B. A∪B
- C. A
- D. ∅
Show correct answer and explanation
Correct answer: C. A
The absorption law gives A∩(A∪B)=A.
Q44. For arbitrary sets A and B, what is A∪(A∩B)?
- A. A
- B. B
- C. A∩B
- D. U
Show correct answer and explanation
Correct answer: A. A
The absorption law gives A∪(A∩B)=A.
Q45. If |A|=14, |B|=12, and |A∩B|=7, find |A∪B|.
- A. 7
- B. 19
- C. 26
- D. 33
Show correct answer and explanation
Correct answer: B. 19
Inclusion–exclusion: |A∪B| = 14+12−7 = 19.
Q46. In a universal set of 28 elements, A has 15 elements. Find |Aᶜ|.
- A. 13
- B. 15
- C. 28
- D. 43
Show correct answer and explanation
Correct answer: A. 13
The complement contains 28−15 = 13 elements.
Q47. If |A|=13, |B|=11, and |A∩B|=2, how many elements belong to A only?
- A. 13
- B. 9
- C. 15
- D. 11
Show correct answer and explanation
Correct answer: D. 11
A only means |A\B| = |A|−|A∩B| = 11.
Q48. Which identity is the distributive law of intersection over union?
- A. A∩(B∪C)=(A∩B)∪(A∩C)
- B. A∩(B∪C)=A∪B∪C
- C. A∩(B∪C)=A∩B∩C
- D. A∩(B∪C)=A∪(B∩C)
Show correct answer and explanation
Correct answer: A. A∩(B∪C)=(A∩B)∪(A∩C)
Intersection distributes over union.
Q49. Which expression equals (A∩B)ᶜ for sets in a fixed universe?
- A. Aᶜ∩Bᶜ
- B. A∪B
- C. Aᶜ∪Bᶜ
- D. A∩B
Show correct answer and explanation
Correct answer: C. Aᶜ∪Bᶜ
By De Morgan’s second law, the complement of an intersection is the union of complements.
Q50. If |A|=15, |B|=13, and |A∩B|=8, find |A∪B|.
- A. 28
- B. 36
- C. 8
- D. 20
Show correct answer and explanation
Correct answer: D. 20
Inclusion–exclusion: |A∪B| = 15+13−8 = 20.
Q51. In a universal set of 29 elements, A has 16 elements. Find |Aᶜ|.
- A. 29
- B. 45
- C. 13
- D. 16
Show correct answer and explanation
Correct answer: C. 13
The complement contains 29−16 = 13 elements.
Q52. If |A|=14, |B|=12, and |A∩B|=2, how many elements belong to A only?
- A. 16
- B. 12
- C. 14
- D. 10
Show correct answer and explanation
Correct answer: B. 12
A only means |A\B| = |A|−|A∩B| = 12.
Q53. If A and B are disjoint, |A|=6 and |B|=5, find |A∪B|.
- A. 30
- B. 6
- C. 11
- D. 1
Show correct answer and explanation
Correct answer: C. 11
Disjoint sets have empty intersection, so |A∪B|=6+5=11.
Q54. If A={a,b,c}, how many subsets contain a?
- A. 2
- B. 4
- C. 6
- D. 8
Show correct answer and explanation
Correct answer: B. 4
Fix a as included, then independently choose whether to include b and c: 2²=4.
Q55. If |A|=16, |B|=14, and |A∩B|=9, find |A∪B|.
- A. 9
- B. 21
- C. 30
- D. 39
Show correct answer and explanation
Correct answer: B. 21
Inclusion–exclusion: |A∪B| = 16+14−9 = 21.
Q56. In a universal set of 30 elements, A has 17 elements. Find |Aᶜ|.
- A. 13
- B. 17
- C. 30
- D. 47
Show correct answer and explanation
Correct answer: A. 13
The complement contains 30−17 = 13 elements.
Q57. If |A|=15, |B|=13, and |A∩B|=2, how many elements belong to A only?
- A. 15
- B. 11
- C. 17
- D. 13
Show correct answer and explanation
Correct answer: D. 13
A only means |A\B| = |A|−|A∩B| = 13.
Q58. For any set A, which statement about A∩∅ is true?
- A. A∩∅=A
- B. A∩∅=U
- C. A∩∅=∅
- D. A∩∅=Aᶜ
Show correct answer and explanation
Correct answer: C. A∩∅=∅
No element belongs to the empty set, so its intersection with any set is empty.
Q59. If |A∪B|=18, |A|=12 and |B|=10, find |A∩B|.
- A. 4
- B. 8
- C. 22
- D. 6
Show correct answer and explanation
Correct answer: A. 4
Rearrange inclusion-exclusion: |A∩B|=12+10−18=4.
Q60. If |A|=17, |B|=15, and |A∩B|=10, find |A∪B|.
- A. 32
- B. 42
- C. 10
- D. 22
Show correct answer and explanation
Correct answer: D. 22
Inclusion–exclusion: |A∪B| = 17+15−10 = 22.
Q61. In a universal set of 31 elements, A has 18 elements. Find |Aᶜ|.
- A. 31
- B. 49
- C. 13
- D. 18
Show correct answer and explanation
Correct answer: C. 13
The complement contains 31−18 = 13 elements.
Q62. If |A|=16, |B|=14, and |A∩B|=2, how many elements belong to A only?
- A. 18
- B. 14
- C. 16
- D. 12
Show correct answer and explanation
Correct answer: B. 14
A only means |A\B| = |A|−|A∩B| = 14.
Q63. If A={2,4,6} and B={1,2,3}, what is A∩B?
- A. {1,3}
- B. {2}
- C. {2,4,6}
- D. ∅
Show correct answer and explanation
Correct answer: B. {2}
Only the element 2 is common to both sets.
Q64. If A has five elements, how many 2-element subsets does it have?
- A. 5
- B. 20
- C. 10
- D. 25
Show correct answer and explanation
Correct answer: C. 10
Choose any two of five elements: C(5,2)=10.
Propositional Logic
Q65. Let p be true and q be true. What is p ∧ q?
- A. True
- B. False
- C. Both true and false
- D. Undetermined
Show correct answer and explanation
Correct answer: A. True
A conjunction is true only when both component propositions are true.
Q66. Let p be true and q be true. What is p ∨ q (inclusive OR)?
- A. False
- B. Neither
- C. Cannot be determined
- D. True
Show correct answer and explanation
Correct answer: D. True
Inclusive OR is false only if both propositions are false.
Q67. Which is the contrapositive of "If an integer n is divisible by 2, then n has property P"?
- A. If n does not have property P, then n is not divisible by 2.
- B. If n has property P, then n is divisible by 2.
- C. If n is not divisible by 2, then n does not have property P.
- D. n is divisible by 2 and has property P.
Show correct answer and explanation
Correct answer: A. If n does not have property P, then n is not divisible by 2.
The contrapositive of p→q is ¬q→¬p.
Q68. Let p be false and q be false. What is p ∧ q?
- A. Both true and false
- B. Undetermined
- C. False
- D. True
Show correct answer and explanation
Correct answer: C. False
A conjunction is true only when both component propositions are true.
Q69. Let p be false and q be false. What is p ∨ q (inclusive OR)?
- A. Cannot be determined
- B. False
- C. True
- D. Neither
Show correct answer and explanation
Correct answer: B. False
Inclusive OR is false only if both propositions are false.
Q70. Which is the contrapositive of "If an integer n is divisible by 3, then n has property P"?
- A. If n is not divisible by 3, then n does not have property P.
- B. n is divisible by 3 and has property P.
- C. If n does not have property P, then n is not divisible by 3.
- D. If n has property P, then n is divisible by 3.
Show correct answer and explanation
Correct answer: C. If n does not have property P, then n is not divisible by 3.
The contrapositive of p→q is ¬q→¬p.
Q71. Let p be true and q be false. What is p ∧ q?
- A. False
- B. True
- C. Both true and false
- D. Undetermined
Show correct answer and explanation
Correct answer: A. False
A conjunction is true only when both component propositions are true.
Q72. Let p be true and q be false. What is p ∨ q (inclusive OR)?
- A. False
- B. Neither
- C. Cannot be determined
- D. True
Show correct answer and explanation
Correct answer: D. True
Inclusive OR is false only if both propositions are false.
Q73. Which is the contrapositive of "If an integer n is divisible by 4, then n has property P"?
- A. If n does not have property P, then n is not divisible by 4.
- B. If n has property P, then n is divisible by 4.
- C. If n is not divisible by 4, then n does not have property P.
- D. n is divisible by 4 and has property P.
Show correct answer and explanation
Correct answer: A. If n does not have property P, then n is not divisible by 4.
The contrapositive of p→q is ¬q→¬p.
Q74. Let p be false and q be true. What is p ∧ q?
- A. Both true and false
- B. Undetermined
- C. False
- D. True
Show correct answer and explanation
Correct answer: C. False
A conjunction is true only when both component propositions are true.
Q75. Let p be false and q be true. What is p ∨ q (inclusive OR)?
- A. Cannot be determined
- B. True
- C. False
- D. Neither
Show correct answer and explanation
Correct answer: B. True
Inclusive OR is false only if both propositions are false.
Q76. Which is the contrapositive of "If an integer n is divisible by 5, then n has property P"?
- A. If n is not divisible by 5, then n does not have property P.
- B. n is divisible by 5 and has property P.
- C. If n does not have property P, then n is not divisible by 5.
- D. If n has property P, then n is divisible by 5.
Show correct answer and explanation
Correct answer: C. If n does not have property P, then n is not divisible by 5.
The contrapositive of p→q is ¬q→¬p.
Q77. If p is true and q is false, what is the truth value of p ⊕ q (exclusive OR)?
- A. True
- B. False
- C. Undefined
- D. Both
Show correct answer and explanation
Correct answer: A. True
Exclusive OR is true when exactly one proposition is true.
Q78. If p is true and q is true, what is ¬(p∧q)?
- A. True
- B. False
- C. Undetermined
- D. p
Show correct answer and explanation
Correct answer: B. False
Both propositions are true, so their conjunction is true and its negation is false.
Q79. Which is the contrapositive of "If an integer n is divisible by 6, then n has property P"?
- A. If n does not have property P, then n is not divisible by 6.
- B. If n has property P, then n is divisible by 6.
- C. If n is not divisible by 6, then n does not have property P.
- D. n is divisible by 6 and has property P.
Show correct answer and explanation
Correct answer: A. If n does not have property P, then n is not divisible by 6.
The contrapositive of p→q is ¬q→¬p.
Q80. If p is false and q is true, what is ¬p∧q?
- A. False
- B. p
- C. True
- D. Undetermined
Show correct answer and explanation
Correct answer: C. True
Since p is false, ¬p is true, and q is true.
Q81. If p is false and q is false, what is ¬(p∨q)?
- A. False
- B. True
- C. Undetermined
- D. p∨q
Show correct answer and explanation
Correct answer: B. True
The disjunction is false, so its negation is true.
Q82. Which is the contrapositive of "If an integer n is divisible by 7, then n has property P"?
- A. If n is not divisible by 7, then n does not have property P.
- B. n is divisible by 7 and has property P.
- C. If n does not have property P, then n is not divisible by 7.
- D. If n has property P, then n is divisible by 7.
Show correct answer and explanation
Correct answer: C. If n does not have property P, then n is not divisible by 7.
The contrapositive of p→q is ¬q→¬p.
Q83. For p=true and q=false, evaluate ¬p∨q.
- A. True
- B. False
- C. Neither
- D. Undefined
Show correct answer and explanation
Correct answer: B. False
Both ¬p and q are false, making their disjunction false.
Q84. For p=false and q=true, evaluate p∧¬q.
- A. True
- B. False
- C. Both
- D. Undefined
Show correct answer and explanation
Correct answer: B. False
Both p and ¬q are false, so their conjunction is false.
Q85. Which is the contrapositive of "If an integer n is divisible by 8, then n has property P"?
- A. If n does not have property P, then n is not divisible by 8.
- B. If n has property P, then n is divisible by 8.
- C. If n is not divisible by 8, then n does not have property P.
- D. n is divisible by 8 and has property P.
Show correct answer and explanation
Correct answer: A. If n does not have property P, then n is not divisible by 8.
The contrapositive of p→q is ¬q→¬p.
Q86. For p=true and q=true, evaluate (p∨q)∧¬q.
- A. True
- B. False
- C. Undefined
- D. Neither
Show correct answer and explanation
Correct answer: B. False
The disjunction is true but ¬q is false, so the conjunction is false.
Q87. For p=false and q=false, evaluate ¬p∨¬q.
- A. False
- B. Cannot tell
- C. True
- D. Neither
Show correct answer and explanation
Correct answer: C. True
Both negations are true, hence their disjunction is true.
Q88. Which is the contrapositive of "If an integer n is divisible by 9, then n has property P"?
- A. If n is not divisible by 9, then n does not have property P.
- B. n is divisible by 9 and has property P.
- C. If n does not have property P, then n is not divisible by 9.
- D. If n has property P, then n is divisible by 9.
Show correct answer and explanation
Correct answer: C. If n does not have property P, then n is not divisible by 9.
The contrapositive of p→q is ¬q→¬p.
Conditional Statements
Q89. If p is true and q is true, what is the truth value of p → q?
- A. Undefined
- B. Both
- C. True
- D. False
Show correct answer and explanation
Correct answer: C. True
A conditional is false only when its antecedent is true and its consequent false.
Q90. If p is false and q is false, what is the truth value of p → q?
- A. True
- B. False
- C. Undefined
- D. Both
Show correct answer and explanation
Correct answer: A. True
A conditional is false only when its antecedent is true and its consequent false.
Q91. If p is true and q is false, what is the truth value of p → q?
- A. Undefined
- B. Both
- C. False
- D. True
Show correct answer and explanation
Correct answer: C. False
A conditional is false only when its antecedent is true and its consequent false.
Q92. If p is false and q is true, what is the truth value of p → q?
- A. True
- B. False
- C. Undefined
- D. Both
Show correct answer and explanation
Correct answer: A. True
A conditional is false only when its antecedent is true and its consequent false.
Q93. For p=true and q=false, what is the truth value of q→p?
- A. False
- B. True
- C. Undefined
- D. Both
Show correct answer and explanation
Correct answer: B. True
A conditional with a false antecedent is true.
Q94. For p=false and q=true, what is p→¬q?
- A. False
- B. True
- C. Undetermined
- D. Neither
Show correct answer and explanation
Correct answer: B. True
The antecedent p is false, so the implication is true.
Q95. For p=true and q=true, what is ¬p→q?
- A. False
- B. True
- C. Both
- D. Undetermined
Show correct answer and explanation
Correct answer: B. True
The antecedent ¬p is false, hence the conditional is true.
Q96. For p=false and q=false, evaluate ¬p→q.
- A. True
- B. False
- C. Cannot tell
- D. Neither
Show correct answer and explanation
Correct answer: B. False
Here ¬p is true and q is false, so the implication is false.
Biconditional Statements
Q97. If p is true and q is true, evaluate p ↔ q.
- A. Always true
- B. True
- C. False
- D. Not a proposition
Show correct answer and explanation
Correct answer: B. True
A biconditional is true exactly when the two truth values match.
Q98. If p is false and q is false, evaluate p ↔ q.
- A. False
- B. Not a proposition
- C. Always true
- D. True
Show correct answer and explanation
Correct answer: D. True
A biconditional is true exactly when the two truth values match.
Q99. If p is true and q is false, evaluate p ↔ q.
- A. Always true
- B. False
- C. True
- D. Not a proposition
Show correct answer and explanation
Correct answer: B. False
A biconditional is true exactly when the two truth values match.
Q100. If p is false and q is true, evaluate p ↔ q.
- A. True
- B. Not a proposition
- C. Always true
- D. False
Show correct answer and explanation
Correct answer: D. False
A biconditional is true exactly when the two truth values match.
Q101. For p=true and q=false, what is p↔¬q?
- A. True
- B. False
- C. Neither
- D. Undefined
Show correct answer and explanation
Correct answer: A. True
Both p and ¬q are true, making the biconditional true.
Q102. For p=false and q=true, what is ¬p↔q?
- A. False
- B. True
- C. Cannot tell
- D. Neither
Show correct answer and explanation
Correct answer: B. True
Both ¬p and q are true, so the biconditional is true.
Q103. For p=true and q=true, evaluate p↔¬q.
- A. True
- B. Undefined
- C. False
- D. Neither
Show correct answer and explanation
Correct answer: C. False
p is true but ¬q is false, so the biconditional is false.
Q104. For p=false and q=false, evaluate ¬p↔q.
- A. True
- B. False
- C. Cannot tell
- D. Neither
Show correct answer and explanation
Correct answer: B. False
¬p is true while q is false, so the biconditional is false.
Tautologies & Contradictions
Q105. For a proposition p, which expression is always true?
- A. p ∧ ¬p
- B. p ↔ ¬p
- C. ¬(p ∨ ¬p)
- D. p ∨ ¬p
Show correct answer and explanation
Correct answer: D. p ∨ ¬p
By the law of excluded middle, p∨¬p is a tautology.
Q106. For a proposition p about a number, which expression is always true?
- A. ¬(p ∨ ¬p)
- B. p ∨ ¬p
- C. p ∧ ¬p
- D. p ↔ ¬p
Show correct answer and explanation
Correct answer: B. p ∨ ¬p
By the law of excluded middle, p∨¬p is a tautology.
Q107. Which proposition is logically equivalent to ¬(p∧q)?
- A. ¬p∧¬q
- B. ¬p∨¬q
- C. p∨q
- D. p∧q
Show correct answer and explanation
Correct answer: B. ¬p∨¬q
De Morgan’s law states ¬(p∧q) ≡ ¬p∨¬q.
Q108. Which expression is logically equivalent to p→q?
- A. p∧q
- B. p∨¬q
- C. ¬p∨q
- D. ¬p∧q
Show correct answer and explanation
Correct answer: C. ¬p∨q
A material implication is equivalent to ¬p∨q.
Q109. Which proposition is a contradiction for every p?
- A. p∨¬p
- B. p→p
- C. p∧¬p
- D. ¬(p∧¬p)
Show correct answer and explanation
Correct answer: C. p∧¬p
A proposition cannot be true and false simultaneously.
Q110. Which expression is equivalent to ¬¬p in classical logic?
- A. p
- B. ¬p
- C. p∧¬p
- D. False
Show correct answer and explanation
Correct answer: A. p
The double-negation law states ¬¬p ≡ p.
Q111. Which expression is equivalent to p∧(p∨q)?
- A. q
- B. p
- C. p∧q
- D. p∨q
Show correct answer and explanation
Correct answer: B. p
The absorption law reduces p∧(p∨q) to p.
Q112. Which expression is equivalent to ¬(p∨q)?
- A. ¬p∨¬q
- B. p∧q
- C. ¬p∧¬q
- D. p∨q
Show correct answer and explanation
Correct answer: C. ¬p∧¬q
De Morgan’s law gives ¬(p∨q) ≡ ¬p∧¬q.
Arguments
Q113. Which inference rule justifies: "p→q; p; therefore q" (example 1)?
- A. Affirming the consequent
- B. Denying the antecedent
- C. Modus ponens
- D. Modus tollens
Show correct answer and explanation
Correct answer: C. Modus ponens
Modus ponens applies the conditional after establishing its antecedent.
Q114. Which inference rule justifies: "p→q; p; therefore q" (example 2)?
- A. Modus ponens
- B. Modus tollens
- C. Affirming the consequent
- D. Denying the antecedent
Show correct answer and explanation
Correct answer: A. Modus ponens
Modus ponens applies the conditional after establishing its antecedent.
Q115. Which inference rule justifies: "p→q; p; therefore q" (example 3)?
- A. Affirming the consequent
- B. Denying the antecedent
- C. Modus ponens
- D. Modus tollens
Show correct answer and explanation
Correct answer: C. Modus ponens
Modus ponens applies the conditional after establishing its antecedent.
Q116. Which inference rule justifies: "p→q; p; therefore q" (example 4)?
- A. Modus ponens
- B. Modus tollens
- C. Affirming the consequent
- D. Denying the antecedent
Show correct answer and explanation
Correct answer: A. Modus ponens
Modus ponens applies the conditional after establishing its antecedent.
Q117. Which inference rule justifies: "p→q; p; therefore q" (example 5)?
- A. Affirming the consequent
- B. Denying the antecedent
- C. Modus ponens
- D. Modus tollens
Show correct answer and explanation
Correct answer: C. Modus ponens
Modus ponens applies the conditional after establishing its antecedent.
Q118. Which inference rule justifies: "p→q; p; therefore q" (example 6)?
- A. Modus ponens
- B. Modus tollens
- C. Affirming the consequent
- D. Denying the antecedent
Show correct answer and explanation
Correct answer: A. Modus ponens
Modus ponens applies the conditional after establishing its antecedent.
Q119. Which inference rule justifies: "p→q; p; therefore q" (example 7)?
- A. Affirming the consequent
- B. Denying the antecedent
- C. Modus ponens
- D. Modus tollens
Show correct answer and explanation
Correct answer: C. Modus ponens
Modus ponens applies the conditional after establishing its antecedent.
Q120. Which inference rule justifies: "p→q; p; therefore q" (example 8)?
- A. Modus ponens
- B. Modus tollens
- C. Affirming the consequent
- D. Denying the antecedent
Show correct answer and explanation
Correct answer: A. Modus ponens
Modus ponens applies the conditional after establishing its antecedent.
Quantifiers
Q121. What is the negation of "Every element of the set {1,2,…,3} has property P"?
- A. No element can be tested.
- B. There exists an element in the set without property P.
- C. Every element lacks P.
- D. Some element has P.
Show correct answer and explanation
Correct answer: B. There exists an element in the set without property P.
Negating ∀x P(x) gives ∃x ¬P(x).
Q122. Negate the statement "There exists an integer x such that x>5 and P(x)."
- A. For every integer x, x≤5 or not P(x).
- B. For every integer x, x>5 and P(x).
- C. There exists x≤5 and P(x).
- D. There exists x>5 and not P(x).
Show correct answer and explanation
Correct answer: A. For every integer x, x≤5 or not P(x).
Negating an existential quantifier makes it universal, and ¬(A∧B) = ¬A∨¬B.
Q123. What is the negation of "Every element of the set {1,2,…,4} has property P"?
- A. Every element lacks P.
- B. Some element has P.
- C. No element can be tested.
- D. There exists an element in the set without property P.
Show correct answer and explanation
Correct answer: D. There exists an element in the set without property P.
Negating ∀x P(x) gives ∃x ¬P(x).
Q124. Negate the statement "There exists an integer x such that x>6 and P(x)."
- A. There exists x≤6 and P(x).
- B. There exists x>6 and not P(x).
- C. For every integer x, x≤6 or not P(x).
- D. For every integer x, x>6 and P(x).
Show correct answer and explanation
Correct answer: C. For every integer x, x≤6 or not P(x).
Negating an existential quantifier makes it universal, and ¬(A∧B) = ¬A∨¬B.
Q125. What is the negation of "Every element of the set {1,2,…,5} has property P"?
- A. No element can be tested.
- B. There exists an element in the set without property P.
- C. Every element lacks P.
- D. Some element has P.
Show correct answer and explanation
Correct answer: B. There exists an element in the set without property P.
Negating ∀x P(x) gives ∃x ¬P(x).
Q126. Negate the statement "There exists an integer x such that x>7 and P(x)."
- A. For every integer x, x≤7 or not P(x).
- B. For every integer x, x>7 and P(x).
- C. There exists x≤7 and P(x).
- D. There exists x>7 and not P(x).
Show correct answer and explanation
Correct answer: A. For every integer x, x≤7 or not P(x).
Negating an existential quantifier makes it universal, and ¬(A∧B) = ¬A∨¬B.
Q127. What is the negation of "Every element of the set {1,2,…,6} has property P"?
- A. Every element lacks P.
- B. Some element has P.
- C. No element can be tested.
- D. There exists an element in the set without property P.
Show correct answer and explanation
Correct answer: D. There exists an element in the set without property P.
Negating ∀x P(x) gives ∃x ¬P(x).
Q128. Negate the statement "There exists an integer x such that x>8 and P(x)."
- A. There exists x≤8 and P(x).
- B. There exists x>8 and not P(x).
- C. For every integer x, x≤8 or not P(x).
- D. For every integer x, x>8 and P(x).
Show correct answer and explanation
Correct answer: C. For every integer x, x≤8 or not P(x).
Negating an existential quantifier makes it universal, and ¬(A∧B) = ¬A∨¬B.
Q129. What is the negation of "Every element of the set {1,2,…,7} has property P"?
- A. No element can be tested.
- B. There exists an element in the set without property P.
- C. Every element lacks P.
- D. Some element has P.
Show correct answer and explanation
Correct answer: B. There exists an element in the set without property P.
Negating ∀x P(x) gives ∃x ¬P(x).
Q130. Negate the statement "There exists an integer x such that x>9 and P(x)."
- A. For every integer x, x≤9 or not P(x).
- B. For every integer x, x>9 and P(x).
- C. There exists x≤9 and P(x).
- D. There exists x>9 and not P(x).
Show correct answer and explanation
Correct answer: A. For every integer x, x≤9 or not P(x).
Negating an existential quantifier makes it universal, and ¬(A∧B) = ¬A∨¬B.
Q131. What is the negation of "Every element of the set {1,2,…,8} has property P"?
- A. Every element lacks P.
- B. Some element has P.
- C. No element can be tested.
- D. There exists an element in the set without property P.
Show correct answer and explanation
Correct answer: D. There exists an element in the set without property P.
Negating ∀x P(x) gives ∃x ¬P(x).
Q132. Negate the statement "There exists an integer x such that x>10 and P(x)."
- A. There exists x≤10 and P(x).
- B. There exists x>10 and not P(x).
- C. For every integer x, x≤10 or not P(x).
- D. For every integer x, x>10 and P(x).
Show correct answer and explanation
Correct answer: C. For every integer x, x≤10 or not P(x).
Negating an existential quantifier makes it universal, and ¬(A∧B) = ¬A∨¬B.
Q133. What is the negation of "Every element of the set {1,2,…,9} has property P"?
- A. No element can be tested.
- B. There exists an element in the set without property P.
- C. Every element lacks P.
- D. Some element has P.
Show correct answer and explanation
Correct answer: B. There exists an element in the set without property P.
Negating ∀x P(x) gives ∃x ¬P(x).
Q134. Negate the statement "There exists an integer x such that x>11 and P(x)."
- A. For every integer x, x≤11 or not P(x).
- B. For every integer x, x>11 and P(x).
- C. There exists x≤11 and P(x).
- D. There exists x>11 and not P(x).
Show correct answer and explanation
Correct answer: A. For every integer x, x≤11 or not P(x).
Negating an existential quantifier makes it universal, and ¬(A∧B) = ¬A∨¬B.
Q135. What is the negation of "Every element of the set {1,2,…,10} has property P"?
- A. Every element lacks P.
- B. Some element has P.
- C. No element can be tested.
- D. There exists an element in the set without property P.
Show correct answer and explanation
Correct answer: D. There exists an element in the set without property P.
Negating ∀x P(x) gives ∃x ¬P(x).
Q136. Negate the statement "There exists an integer x such that x>12 and P(x)."
- A. There exists x≤12 and P(x).
- B. There exists x>12 and not P(x).
- C. For every integer x, x≤12 or not P(x).
- D. For every integer x, x>12 and P(x).
Show correct answer and explanation
Correct answer: C. For every integer x, x≤12 or not P(x).
Negating an existential quantifier makes it universal, and ¬(A∧B) = ¬A∨¬B.
Relations
Q137. On the integers, consider aRb iff a−b is divisible by 2. Which description is correct?
- A. R is not reflexive
- B. R is not symmetric
- C. R is not transitive
- D. R is an equivalence relation
Show correct answer and explanation
Correct answer: D. R is an equivalence relation
Congruence modulo 2 is reflexive, symmetric and transitive.
Q138. On the integers, consider aRb iff a−b is divisible by 3. Which description is correct?
- A. R is not transitive
- B. R is an equivalence relation
- C. R is not reflexive
- D. R is not symmetric
Show correct answer and explanation
Correct answer: B. R is an equivalence relation
Congruence modulo 3 is reflexive, symmetric and transitive.
Q139. On the integers, consider aRb iff a−b is divisible by 4. Which description is correct?
- A. R is not reflexive
- B. R is not symmetric
- C. R is not transitive
- D. R is an equivalence relation
Show correct answer and explanation
Correct answer: D. R is an equivalence relation
Congruence modulo 4 is reflexive, symmetric and transitive.
Q140. On the integers, consider aRb iff a−b is divisible by 5. Which description is correct?
- A. R is not transitive
- B. R is an equivalence relation
- C. R is not reflexive
- D. R is not symmetric
Show correct answer and explanation
Correct answer: B. R is an equivalence relation
Congruence modulo 5 is reflexive, symmetric and transitive.
Q141. On the integers, consider aRb iff a−b is divisible by 6. Which description is correct?
- A. R is not reflexive
- B. R is not symmetric
- C. R is not transitive
- D. R is an equivalence relation
Show correct answer and explanation
Correct answer: D. R is an equivalence relation
Congruence modulo 6 is reflexive, symmetric and transitive.
Q142. On the integers, consider aRb iff a−b is divisible by 7. Which description is correct?
- A. R is not transitive
- B. R is an equivalence relation
- C. R is not reflexive
- D. R is not symmetric
Show correct answer and explanation
Correct answer: B. R is an equivalence relation
Congruence modulo 7 is reflexive, symmetric and transitive.
Q143. On the integers, consider aRb iff a−b is divisible by 8. Which description is correct?
- A. R is not reflexive
- B. R is not symmetric
- C. R is not transitive
- D. R is an equivalence relation
Show correct answer and explanation
Correct answer: D. R is an equivalence relation
Congruence modulo 8 is reflexive, symmetric and transitive.
Q144. On the integers, consider aRb iff a−b is divisible by 9. Which description is correct?
- A. R is not transitive
- B. R is an equivalence relation
- C. R is not reflexive
- D. R is not symmetric
Show correct answer and explanation
Correct answer: B. R is an equivalence relation
Congruence modulo 9 is reflexive, symmetric and transitive.
Partial Ordering
Q145. On the set of integers, which relation is a partial order?
- A. a−b is odd
- B. a+b is even
- C. a≤b
- D. a≠b
Show correct answer and explanation
Correct answer: C. a≤b
The usual ≤ relation is reflexive, antisymmetric and transitive.
Q146. On positive integers, which relation defines a partial order?
- A. a divides b
- B. a+b is even
- C. a≠b
- D. a−b is odd
Show correct answer and explanation
Correct answer: A. a divides b
Divisibility is reflexive, antisymmetric and transitive on positive integers.
Q147. On the power set P({1,2}), which relation is a partial order?
- A. A intersects B
- B. A⊆B
- C. A≠B
- D. A∩B≠∅
Show correct answer and explanation
Correct answer: B. A⊆B
Set inclusion is reflexive, antisymmetric and transitive.
Q148. Which property fails for the usual strict order < on integers when viewed as a partial order?
- A. Transitivity
- B. Irreflexivity
- C. Reflexivity
- D. Asymmetry
Q149. For the usual ≤ on integers, which is true of 3 and 5?
- A. 5≤3
- B. 3≤5
- C. 3=5
- D. They are incomparable
Show correct answer and explanation
Correct answer: B. 3≤5
Three is less than five, so 3≤5.
Q150. Under divisibility on positive integers, is 2 comparable with 3?
- A. Yes, 2 divides 3
- B. Yes, 3 divides 2
- C. No, neither divides the other
- D. Yes, both divide each other
Show correct answer and explanation
Correct answer: C. No, neither divides the other
Neither 2 divides 3 nor 3 divides 2.
Q151. Which relation on integers is an equivalence relation but not a partial order?
- A. a≤b
- B. a=b
- C. a−b is even
- D. a divides b
Show correct answer and explanation
Correct answer: C. a−b is even
Congruence modulo 2 is an equivalence relation, but distinct same-parity integers violate antisymmetry.
Q152. In a partially ordered set, what does a≤b and b≤a imply?
- A. a≠b
- B. a=b
- C. a
- D. No conclusion
Show correct answer and explanation
Correct answer: B. a=b
Antisymmetry of a partial order implies a=b.
Functions
Q153. For f(x)=2x+1, what is f(3)?
- A. 5
- B. 7
- C. 4
- D. 6
Show correct answer and explanation
Correct answer: B. 7
Substitute x=3: 2(3)+1 = 7.
Q154. Which map on the real numbers is one-to-one? (variant 1)
- A. f(x)=1x+1
- B. f(x)=x²
- C. f(x)=|x|
- D. f(x)=0
Show correct answer and explanation
Correct answer: A. f(x)=1x+1
A linear function with nonzero slope is injective on the real numbers.
Q155. For f(x)=2x+2, what is f(4)?
- A. 6
- B. 9
- C. 11
- D. 10
Show correct answer and explanation
Correct answer: D. 10
Substitute x=4: 2(4)+2 = 10.
Q156. Which map on the real numbers is one-to-one? (variant 2)
- A. f(x)=|x|
- B. f(x)=0
- C. f(x)=2x+1
- D. f(x)=x²
Show correct answer and explanation
Correct answer: C. f(x)=2x+1
A linear function with nonzero slope is injective on the real numbers.
Q157. For f(x)=2x+3, what is f(5)?
- A. 7
- B. 13
- C. 8
- D. 12
Show correct answer and explanation
Correct answer: B. 13
Substitute x=5: 2(5)+3 = 13.
Q158. Which map on the real numbers is one-to-one? (variant 3)
- A. f(x)=3x+1
- B. f(x)=x²
- C. f(x)=|x|
- D. f(x)=0
Show correct answer and explanation
Correct answer: A. f(x)=3x+1
A linear function with nonzero slope is injective on the real numbers.
Q159. For f(x)=2x+4, what is f(6)?
- A. 10
- B. 15
- C. 8
- D. 16
Show correct answer and explanation
Correct answer: D. 16
Substitute x=6: 2(6)+4 = 16.
Q160. Which map on the real numbers is one-to-one? (variant 4)
- A. f(x)=|x|
- B. f(x)=0
- C. f(x)=4x+1
- D. f(x)=x²
Show correct answer and explanation
Correct answer: C. f(x)=4x+1
A linear function with nonzero slope is injective on the real numbers.
Q161. For f(x)=2x+5, what is f(7)?
- A. 9
- B. 19
- C. 12
- D. 18
Show correct answer and explanation
Correct answer: B. 19
Substitute x=7: 2(7)+5 = 19.
Q162. Which map on the real numbers is one-to-one? (variant 5)
- A. f(x)=5x+1
- B. f(x)=x²
- C. f(x)=|x|
- D. f(x)=0
Show correct answer and explanation
Correct answer: A. f(x)=5x+1
A linear function with nonzero slope is injective on the real numbers.
Q163. For f(x)=2x+6, what is f(8)?
- A. 14
- B. 21
- C. 10
- D. 22
Show correct answer and explanation
Correct answer: D. 22
Substitute x=8: 2(8)+6 = 22.
Q164. Which map on the real numbers is one-to-one? (variant 6)
- A. f(x)=|x|
- B. f(x)=0
- C. f(x)=6x+1
- D. f(x)=x²
Show correct answer and explanation
Correct answer: C. f(x)=6x+1
A linear function with nonzero slope is injective on the real numbers.
Q165. For f(x)=2x+7, what is f(9)?
- A. 11
- B. 25
- C. 16
- D. 24
Show correct answer and explanation
Correct answer: B. 25
Substitute x=9: 2(9)+7 = 25.
Q166. Which map on the real numbers is one-to-one? (variant 7)
- A. f(x)=7x+1
- B. f(x)=x²
- C. f(x)=|x|
- D. f(x)=0
Show correct answer and explanation
Correct answer: A. f(x)=7x+1
A linear function with nonzero slope is injective on the real numbers.
Q167. For f(x)=2x+8, what is f(10)?
- A. 18
- B. 27
- C. 12
- D. 28
Show correct answer and explanation
Correct answer: D. 28
Substitute x=10: 2(10)+8 = 28.
Q168. Which map on the real numbers is one-to-one? (variant 8)
- A. f(x)=|x|
- B. f(x)=0
- C. f(x)=8x+1
- D. f(x)=x²
Show correct answer and explanation
Correct answer: C. f(x)=8x+1
A linear function with nonzero slope is injective on the real numbers.
Recursively Defined Functions
Q169. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a3.
- A. 3
- B. 9
- C. 7
- D. 6
Show correct answer and explanation
Correct answer: D. 6
The recurrence sums 1+2+…+3 = 3(3+1)/2 = 6.
Q170. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a4.
- A. 16
- B. 10
- C. 6
- D. 14
Show correct answer and explanation
Correct answer: B. 10
The recurrence sums 1+2+…+4 = 4(4+1)/2 = 10.
Q171. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a5.
- A. 10
- B. 20
- C. 25
- D. 15
Show correct answer and explanation
Correct answer: D. 15
The recurrence sums 1+2+…+5 = 5(5+1)/2 = 15.
Q172. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a6.
- A. 36
- B. 21
- C. 15
- D. 27
Show correct answer and explanation
Correct answer: B. 21
The recurrence sums 1+2+…+6 = 6(6+1)/2 = 21.
Q173. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a7.
- A. 21
- B. 35
- C. 49
- D. 28
Show correct answer and explanation
Correct answer: D. 28
The recurrence sums 1+2+…+7 = 7(7+1)/2 = 28.
Q174. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a8.
- A. 64
- B. 36
- C. 28
- D. 44
Show correct answer and explanation
Correct answer: B. 36
The recurrence sums 1+2+…+8 = 8(8+1)/2 = 36.
Q175. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a9.
- A. 36
- B. 54
- C. 81
- D. 45
Show correct answer and explanation
Correct answer: D. 45
The recurrence sums 1+2+…+9 = 9(9+1)/2 = 45.
Q176. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a10.
- A. 100
- B. 55
- C. 45
- D. 65
Show correct answer and explanation
Correct answer: B. 55
The recurrence sums 1+2+…+10 = 10(10+1)/2 = 55.
Interactive MCQ Practice
Practise one MCQ at a time. Reveal the correct answer and explanation before proceeding to the next question.