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Mathematical Reasoning — One Question at a Time

Mathematical Reasoning — One Question at a Time

Practise 176 Mathematical Reasoning multiple-choice questions covering sets, propositional logic, inference rules, quantifiers, relations, functions and sequences. Work through one question at a time, check the correct answer with an explanation, and select Next Question to continue. This is self-paced practice without automatic scoring.

Before you practise: Read Mathematical Reasoning study material and examples. You can also browse all 1,576 interactive MCQs.

Practice Set 1 · Question 1 of 176

Question 1. If A has 3 distinct elements, how many subsets does A have?

  1. 8
  2. 4
  3. 9
  4. 7
Show Answer and Explanation

Correct answer: A. 8

Each element is either included or excluded, giving 2^3 = 8 subsets.

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Practice Set 1 · Question 2 of 176

Question 2. How many proper subsets does a 3-element set have?

  1. 8
  2. 3
  3. 4
  4. 7
Show Answer and Explanation

Correct answer: D. 7

There are 2^3 subsets; exclude the set itself.

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Practice Set 1 · Question 3 of 176

Question 3. What is the cardinality of the power set of a set with 2 elements?

  1. 3
  2. 5
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: C. 4

The power set contains all 2^2 subsets.

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Practice Set 1 · Question 4 of 176

Question 4. If A has 4 distinct elements, how many subsets does A have?

  1. 15
  2. 17
  3. 16
  4. 8
Show Answer and Explanation

Correct answer: C. 16

Each element is either included or excluded, giving 2^4 = 16 subsets.

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Practice Set 1 · Question 5 of 176

Question 5. How many proper subsets does a 4-element set have?

  1. 8
  2. 15
  3. 16
  4. 4
Show Answer and Explanation

Correct answer: B. 15

There are 2^4 subsets; exclude the set itself.

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Practice Set 1 · Question 6 of 176

Question 6. What is the cardinality of the power set of a set with 3 elements?

  1. 8
  2. 3
  3. 6
  4. 7
Show Answer and Explanation

Correct answer: A. 8

The power set contains all 2^3 subsets.

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Practice Set 1 · Question 7 of 176

Question 7. If A has 5 distinct elements, how many subsets does A have?

  1. 32
  2. 16
  3. 25
  4. 31
Show Answer and Explanation

Correct answer: A. 32

Each element is either included or excluded, giving 2^5 = 32 subsets.

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Practice Set 1 · Question 8 of 176

Question 8. How many proper subsets does a 5-element set have?

  1. 32
  2. 5
  3. 16
  4. 31
Show Answer and Explanation

Correct answer: D. 31

There are 2^5 subsets; exclude the set itself.

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Practice Set 1 · Question 9 of 176

Question 9. What is the cardinality of the power set of a set with 4 elements?

  1. 8
  2. 15
  3. 16
  4. 4
Show Answer and Explanation

Correct answer: C. 16

The power set contains all 2^4 subsets.

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Practice Set 1 · Question 10 of 176

Question 10. If A has 6 distinct elements, how many subsets does A have?

  1. 36
  2. 63
  3. 64
  4. 32
Show Answer and Explanation

Correct answer: C. 64

Each element is either included or excluded, giving 2^6 = 64 subsets.

Continue to Practice Set 2

Practice Set 2 · Question 11 of 176

Question 11. How many proper subsets does a 6-element set have?

  1. 32
  2. 63
  3. 64
  4. 6
Show Answer and Explanation

Correct answer: B. 63

There are 2^6 subsets; exclude the set itself.

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Practice Set 2 · Question 12 of 176

Question 12. What is the cardinality of the power set of a set with 5 elements?

  1. 32
  2. 5
  3. 10
  4. 31
Show Answer and Explanation

Correct answer: A. 32

The power set contains all 2^5 subsets.

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Practice Set 2 · Question 13 of 176

Question 13. If A has 7 distinct elements, how many subsets does A have?

  1. 128
  2. 64
  3. 49
  4. 127
Show Answer and Explanation

Correct answer: A. 128

Each element is either included or excluded, giving 2^7 = 128 subsets.

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Practice Set 2 · Question 14 of 176

Question 14. How many proper subsets does a 7-element set have?

  1. 128
  2. 7
  3. 64
  4. 127
Show Answer and Explanation

Correct answer: D. 127

There are 2^7 subsets; exclude the set itself.

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Practice Set 2 · Question 15 of 176

Question 15. What is the cardinality of the power set of a set with 6 elements?

  1. 12
  2. 63
  3. 64
  4. 6
Show Answer and Explanation

Correct answer: C. 64

The power set contains all 2^6 subsets.

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Practice Set 2 · Question 16 of 176

Question 16. If A has 8 distinct elements, how many subsets does A have?

  1. 64
  2. 255
  3. 256
  4. 128
Show Answer and Explanation

Correct answer: C. 256

Each element is either included or excluded, giving 2^8 = 256 subsets.

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Practice Set 2 · Question 17 of 176

Question 17. How many proper subsets does a 8-element set have?

  1. 128
  2. 255
  3. 256
  4. 8
Show Answer and Explanation

Correct answer: B. 255

There are 2^8 subsets; exclude the set itself.

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Practice Set 2 · Question 18 of 176

Question 18. What is the cardinality of the power set of a set with 7 elements?

  1. 128
  2. 7
  3. 14
  4. 127
Show Answer and Explanation

Correct answer: A. 128

The power set contains all 2^7 subsets.

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Practice Set 2 · Question 19 of 176

Question 19. If A has 9 distinct elements, how many subsets does A have?

  1. 512
  2. 256
  3. 81
  4. 511
Show Answer and Explanation

Correct answer: A. 512

Each element is either included or excluded, giving 2^9 = 512 subsets.

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Practice Set 2 · Question 20 of 176

Question 20. How many proper subsets does a 9-element set have?

  1. 512
  2. 9
  3. 256
  4. 511
Show Answer and Explanation

Correct answer: D. 511

There are 2^9 subsets; exclude the set itself.

Continue to Practice Set 3

Practice Set 3 · Question 21 of 176

Question 21. What is the cardinality of the power set of a set with 8 elements?

  1. 16
  2. 255
  3. 256
  4. 8
Show Answer and Explanation

Correct answer: C. 256

The power set contains all 2^8 subsets.

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Practice Set 3 · Question 22 of 176

Question 22. If A has 10 distinct elements, how many subsets does A have?

  1. 100
  2. 1023
  3. 1024
  4. 512
Show Answer and Explanation

Correct answer: C. 1024

Each element is either included or excluded, giving 2^10 = 1024 subsets.

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Practice Set 3 · Question 23 of 176

Question 23. How many proper subsets does a 10-element set have?

  1. 512
  2. 1023
  3. 1024
  4. 10
Show Answer and Explanation

Correct answer: B. 1023

There are 2^10 subsets; exclude the set itself.

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Practice Set 3 · Question 24 of 176

Question 24. What is the cardinality of the power set of a set with 9 elements?

  1. 512
  2. 9
  3. 18
  4. 511
Show Answer and Explanation

Correct answer: A. 512

The power set contains all 2^9 subsets.

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Practice Set 3 · Question 25 of 176

Question 25. If |A|=10, |B|=8, and |A∩B|=3, find |A∪B|.

  1. 3
  2. 15
  3. 18
  4. 21
Show Answer and Explanation

Correct answer: B. 15

Inclusion–exclusion: |A∪B| = 10+8−3 = 15.

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Practice Set 3 · Question 26 of 176

Question 26. In a universal set of 24 elements, A has 11 elements. Find |Aᶜ|.

  1. 13
  2. 11
  3. 24
  4. 35
Show Answer and Explanation

Correct answer: A. 13

The complement contains 24−11 = 13 elements.

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Practice Set 3 · Question 27 of 176

Question 27. If |A|=9, |B|=7, and |A∩B|=2, how many elements belong to A only?

  1. 9
  2. 5
  3. 11
  4. 7
Show Answer and Explanation

Correct answer: D. 7

A only means |A\B| = |A|−|A∩B| = 7.

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Practice Set 3 · Question 28 of 176

Question 28. Which expression is equal to (A∪B)ᶜ (De Morgan’s law), for arbitrary sets A and B?

  1. A∩B
  2. A∪B
  3. Aᶜ∩Bᶜ
  4. Aᶜ∪Bᶜ
Show Answer and Explanation

Correct answer: C. Aᶜ∩Bᶜ

Complement of a union is the intersection of the complements.

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Practice Set 3 · Question 29 of 176

Question 29. Which identity holds for any set A and any universe U?

  1. A∩U = ∅
  2. A∪Aᶜ = U
  3. A∩Aᶜ = U
  4. A∪∅ = U
Show Answer and Explanation

Correct answer: B. A∪Aᶜ = U

Every element of U lies either in A or in its complement.

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Practice Set 3 · Question 30 of 176

Question 30. If |A|=11, |B|=9, and |A∩B|=4, find |A∪B|.

  1. 20
  2. 24
  3. 4
  4. 16
Show Answer and Explanation

Correct answer: D. 16

Inclusion–exclusion: |A∪B| = 11+9−4 = 16.

Continue to Practice Set 4

Practice Set 4 · Question 31 of 176

Question 31. In a universal set of 25 elements, A has 12 elements. Find |Aᶜ|.

  1. 25
  2. 37
  3. 13
  4. 12
Show Answer and Explanation

Correct answer: C. 13

The complement contains 25−12 = 13 elements.

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Practice Set 4 · Question 32 of 176

Question 32. If |A|=10, |B|=8, and |A∩B|=2, how many elements belong to A only?

  1. 12
  2. 8
  3. 10
  4. 6
Show Answer and Explanation

Correct answer: B. 8

A only means |A\B| = |A|−|A∩B| = 8.

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Practice Set 4 · Question 33 of 176

Question 33. If A={1,2,3,4} and B={3,4,5}, what is A\B?

  1. {1,2}
  2. {3,4}
  3. {5}
  4. {1,2,5}
Show Answer and Explanation

Correct answer: A. {1,2}

A\B contains elements of A not in B.

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Practice Set 4 · Question 34 of 176

Question 34. If A={1,2,3} and B={3,4}, what is the symmetric difference A△B?

  1. {3}
  2. {1,2,4}
  3. {1,2,3,4}
  4. {1,2}
Show Answer and Explanation

Correct answer: B. {1,2,4}

Symmetric difference contains elements belonging to exactly one of the sets.

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Practice Set 4 · Question 35 of 176

Question 35. If |A|=12, |B|=10, and |A∩B|=5, find |A∪B|.

  1. 5
  2. 17
  3. 22
  4. 27
Show Answer and Explanation

Correct answer: B. 17

Inclusion–exclusion: |A∪B| = 12+10−5 = 17.

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Practice Set 4 · Question 36 of 176

Question 36. In a universal set of 26 elements, A has 13 elements. Find |Aᶜ|.

  1. 13
  2. 26
  3. 39
  4. 14
Show Answer and Explanation

Correct answer: A. 13

The complement contains 26−13 = 13 elements.

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Practice Set 4 · Question 37 of 176

Question 37. If |A|=11, |B|=9, and |A∩B|=2, how many elements belong to A only?

  1. 11
  2. 7
  3. 13
  4. 9
Show Answer and Explanation

Correct answer: D. 9

A only means |A\B| = |A|−|A∩B| = 9.

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Practice Set 4 · Question 38 of 176

Question 38. If |A|=3 and |B|=4, how many ordered pairs are in A×B?

  1. 7
  2. 12
  3. 1
  4. 24
Show Answer and Explanation

Correct answer: B. 12

The Cartesian product contains |A||B|=3×4=12 ordered pairs.

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Practice Set 4 · Question 39 of 176

Question 39. If A⊆B and B⊆A, which conclusion follows?

  1. A∩B=∅
  2. A=B
  3. A∪B=∅
  4. A≠B
Show Answer and Explanation

Correct answer: B. A=B

Mutual inclusion is the defining criterion for equality of sets.

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Practice Set 4 · Question 40 of 176

Question 40. If |A|=13, |B|=11, and |A∩B|=6, find |A∪B|.

  1. 24
  2. 30
  3. 6
  4. 18
Show Answer and Explanation

Correct answer: D. 18

Inclusion–exclusion: |A∪B| = 13+11−6 = 18.

Continue to Practice Set 5

Practice Set 5 · Question 41 of 176

Question 41. In a universal set of 27 elements, A has 14 elements. Find |Aᶜ|.

  1. 27
  2. 41
  3. 13
  4. 14
Show Answer and Explanation

Correct answer: C. 13

The complement contains 27−14 = 13 elements.

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Practice Set 5 · Question 42 of 176

Question 42. If |A|=12, |B|=10, and |A∩B|=2, how many elements belong to A only?

  1. 14
  2. 10
  3. 12
  4. 8
Show Answer and Explanation

Correct answer: B. 10

A only means |A\B| = |A|−|A∩B| = 10.

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Practice Set 5 · Question 43 of 176

Question 43. For arbitrary sets A and B, what is A∩(A∪B)?

  1. B
  2. A∪B
  3. A
  4. ∅
Show Answer and Explanation

Correct answer: C. A

The absorption law gives A∩(A∪B)=A.

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Practice Set 5 · Question 44 of 176

Question 44. For arbitrary sets A and B, what is A∪(A∩B)?

  1. A
  2. B
  3. A∩B
  4. U
Show Answer and Explanation

Correct answer: A. A

The absorption law gives A∪(A∩B)=A.

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Practice Set 5 · Question 45 of 176

Question 45. If |A|=14, |B|=12, and |A∩B|=7, find |A∪B|.

  1. 7
  2. 19
  3. 26
  4. 33
Show Answer and Explanation

Correct answer: B. 19

Inclusion–exclusion: |A∪B| = 14+12−7 = 19.

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Practice Set 5 · Question 46 of 176

Question 46. In a universal set of 28 elements, A has 15 elements. Find |Aᶜ|.

  1. 13
  2. 15
  3. 28
  4. 43
Show Answer and Explanation

Correct answer: A. 13

The complement contains 28−15 = 13 elements.

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Practice Set 5 · Question 47 of 176

Question 47. If |A|=13, |B|=11, and |A∩B|=2, how many elements belong to A only?

  1. 13
  2. 9
  3. 15
  4. 11
Show Answer and Explanation

Correct answer: D. 11

A only means |A\B| = |A|−|A∩B| = 11.

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Practice Set 5 · Question 48 of 176

Question 48. Which identity is the distributive law of intersection over union?

  1. A∩(B∪C)=(A∩B)∪(A∩C)
  2. A∩(B∪C)=A∪B∪C
  3. A∩(B∪C)=A∩B∩C
  4. A∩(B∪C)=A∪(B∩C)
Show Answer and Explanation

Correct answer: A. A∩(B∪C)=(A∩B)∪(A∩C)

Intersection distributes over union.

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Practice Set 5 · Question 49 of 176

Question 49. Which expression equals (A∩B)ᶜ for sets in a fixed universe?

  1. Aᶜ∩Bᶜ
  2. A∪B
  3. Aᶜ∪Bᶜ
  4. A∩B
Show Answer and Explanation

Correct answer: C. Aᶜ∪Bᶜ

By De Morgan’s second law, the complement of an intersection is the union of complements.

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Practice Set 5 · Question 50 of 176

Question 50. If |A|=15, |B|=13, and |A∩B|=8, find |A∪B|.

  1. 28
  2. 36
  3. 8
  4. 20
Show Answer and Explanation

Correct answer: D. 20

Inclusion–exclusion: |A∪B| = 15+13−8 = 20.

Continue to Practice Set 6

Practice Set 6 · Question 51 of 176

Question 51. In a universal set of 29 elements, A has 16 elements. Find |Aᶜ|.

  1. 29
  2. 45
  3. 13
  4. 16
Show Answer and Explanation

Correct answer: C. 13

The complement contains 29−16 = 13 elements.

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Practice Set 6 · Question 52 of 176

Question 52. If |A|=14, |B|=12, and |A∩B|=2, how many elements belong to A only?

  1. 16
  2. 12
  3. 14
  4. 10
Show Answer and Explanation

Correct answer: B. 12

A only means |A\B| = |A|−|A∩B| = 12.

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Practice Set 6 · Question 53 of 176

Question 53. If A and B are disjoint, |A|=6 and |B|=5, find |A∪B|.

  1. 30
  2. 6
  3. 11
  4. 1
Show Answer and Explanation

Correct answer: C. 11

Disjoint sets have empty intersection, so |A∪B|=6+5=11.

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Practice Set 6 · Question 54 of 176

Question 54. If A={a,b,c}, how many subsets contain a?

  1. 2
  2. 4
  3. 6
  4. 8
Show Answer and Explanation

Correct answer: B. 4

Fix a as included, then independently choose whether to include b and c: 2²=4.

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Practice Set 6 · Question 55 of 176

Question 55. If |A|=16, |B|=14, and |A∩B|=9, find |A∪B|.

  1. 9
  2. 21
  3. 30
  4. 39
Show Answer and Explanation

Correct answer: B. 21

Inclusion–exclusion: |A∪B| = 16+14−9 = 21.

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Practice Set 6 · Question 56 of 176

Question 56. In a universal set of 30 elements, A has 17 elements. Find |Aᶜ|.

  1. 13
  2. 17
  3. 30
  4. 47
Show Answer and Explanation

Correct answer: A. 13

The complement contains 30−17 = 13 elements.

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Practice Set 6 · Question 57 of 176

Question 57. If |A|=15, |B|=13, and |A∩B|=2, how many elements belong to A only?

  1. 15
  2. 11
  3. 17
  4. 13
Show Answer and Explanation

Correct answer: D. 13

A only means |A\B| = |A|−|A∩B| = 13.

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Practice Set 6 · Question 58 of 176

Question 58. For any set A, which statement about A∩∅ is true?

  1. A∩∅=A
  2. A∩∅=U
  3. A∩∅=∅
  4. A∩∅=Aᶜ
Show Answer and Explanation

Correct answer: C. A∩∅=∅

No element belongs to the empty set, so its intersection with any set is empty.

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Practice Set 6 · Question 59 of 176

Question 59. If |A∪B|=18, |A|=12 and |B|=10, find |A∩B|.

  1. 4
  2. 8
  3. 22
  4. 6
Show Answer and Explanation

Correct answer: A. 4

Rearrange inclusion-exclusion: |A∩B|=12+10−18=4.

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Practice Set 6 · Question 60 of 176

Question 60. If |A|=17, |B|=15, and |A∩B|=10, find |A∪B|.

  1. 32
  2. 42
  3. 10
  4. 22
Show Answer and Explanation

Correct answer: D. 22

Inclusion–exclusion: |A∪B| = 17+15−10 = 22.

Continue to Practice Set 7

Practice Set 7 · Question 61 of 176

Question 61. In a universal set of 31 elements, A has 18 elements. Find |Aᶜ|.

  1. 31
  2. 49
  3. 13
  4. 18
Show Answer and Explanation

Correct answer: C. 13

The complement contains 31−18 = 13 elements.

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Practice Set 7 · Question 62 of 176

Question 62. If |A|=16, |B|=14, and |A∩B|=2, how many elements belong to A only?

  1. 18
  2. 14
  3. 16
  4. 12
Show Answer and Explanation

Correct answer: B. 14

A only means |A\B| = |A|−|A∩B| = 14.

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Practice Set 7 · Question 63 of 176

Question 63. If A={2,4,6} and B={1,2,3}, what is A∩B?

  1. {1,3}
  2. {2}
  3. {2,4,6}
  4. ∅
Show Answer and Explanation

Correct answer: B. {2}

Only the element 2 is common to both sets.

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Practice Set 7 · Question 64 of 176

Question 64. If A has five elements, how many 2-element subsets does it have?

  1. 5
  2. 20
  3. 10
  4. 25
Show Answer and Explanation

Correct answer: C. 10

Choose any two of five elements: C(5,2)=10.

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Practice Set 7 · Question 65 of 176

Question 65. Let p be true and q be true. What is p ∧ q?

  1. True
  2. False
  3. Both true and false
  4. Undetermined
Show Answer and Explanation

Correct answer: A. True

A conjunction is true only when both component propositions are true.

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Practice Set 7 · Question 66 of 176

Question 66. Let p be true and q be true. What is p ∨ q (inclusive OR)?

  1. False
  2. Neither
  3. Cannot be determined
  4. True
Show Answer and Explanation

Correct answer: D. True

Inclusive OR is false only if both propositions are false.

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Practice Set 7 · Question 67 of 176

Question 67. Which is the contrapositive of "If an integer n is divisible by 2, then n has property P"?

  1. If n does not have property P, then n is not divisible by 2.
  2. If n has property P, then n is divisible by 2.
  3. If n is not divisible by 2, then n does not have property P.
  4. n is divisible by 2 and has property P.
Show 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.

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Practice Set 7 · Question 68 of 176

Question 68. Let p be false and q be false. What is p ∧ q?

  1. Both true and false
  2. Undetermined
  3. False
  4. True
Show Answer and Explanation

Correct answer: C. False

A conjunction is true only when both component propositions are true.

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Practice Set 7 · Question 69 of 176

Question 69. Let p be false and q be false. What is p ∨ q (inclusive OR)?

  1. Cannot be determined
  2. False
  3. True
  4. Neither
Show Answer and Explanation

Correct answer: B. False

Inclusive OR is false only if both propositions are false.

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Practice Set 7 · Question 70 of 176

Question 70. Which is the contrapositive of "If an integer n is divisible by 3, then n has property P"?

  1. If n is not divisible by 3, then n does not have property P.
  2. n is divisible by 3 and has property P.
  3. If n does not have property P, then n is not divisible by 3.
  4. If n has property P, then n is divisible by 3.
Show 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.

Continue to Practice Set 8

Practice Set 8 · Question 71 of 176

Question 71. Let p be true and q be false. What is p ∧ q?

  1. False
  2. True
  3. Both true and false
  4. Undetermined
Show Answer and Explanation

Correct answer: A. False

A conjunction is true only when both component propositions are true.

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Practice Set 8 · Question 72 of 176

Question 72. Let p be true and q be false. What is p ∨ q (inclusive OR)?

  1. False
  2. Neither
  3. Cannot be determined
  4. True
Show Answer and Explanation

Correct answer: D. True

Inclusive OR is false only if both propositions are false.

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Practice Set 8 · Question 73 of 176

Question 73. Which is the contrapositive of "If an integer n is divisible by 4, then n has property P"?

  1. If n does not have property P, then n is not divisible by 4.
  2. If n has property P, then n is divisible by 4.
  3. If n is not divisible by 4, then n does not have property P.
  4. n is divisible by 4 and has property P.
Show 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.

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Practice Set 8 · Question 74 of 176

Question 74. Let p be false and q be true. What is p ∧ q?

  1. Both true and false
  2. Undetermined
  3. False
  4. True
Show Answer and Explanation

Correct answer: C. False

A conjunction is true only when both component propositions are true.

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Practice Set 8 · Question 75 of 176

Question 75. Let p be false and q be true. What is p ∨ q (inclusive OR)?

  1. Cannot be determined
  2. True
  3. False
  4. Neither
Show Answer and Explanation

Correct answer: B. True

Inclusive OR is false only if both propositions are false.

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Practice Set 8 · Question 76 of 176

Question 76. Which is the contrapositive of "If an integer n is divisible by 5, then n has property P"?

  1. If n is not divisible by 5, then n does not have property P.
  2. n is divisible by 5 and has property P.
  3. If n does not have property P, then n is not divisible by 5.
  4. If n has property P, then n is divisible by 5.
Show 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.

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Practice Set 8 · Question 77 of 176

Question 77. If p is true and q is false, what is the truth value of p ⊕ q (exclusive OR)?

  1. True
  2. False
  3. Undefined
  4. Both
Show Answer and Explanation

Correct answer: A. True

Exclusive OR is true when exactly one proposition is true.

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Practice Set 8 · Question 78 of 176

Question 78. If p is true and q is true, what is ¬(p∧q)?

  1. True
  2. False
  3. Undetermined
  4. p
Show Answer and Explanation

Correct answer: B. False

Both propositions are true, so their conjunction is true and its negation is false.

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Practice Set 8 · Question 79 of 176

Question 79. Which is the contrapositive of "If an integer n is divisible by 6, then n has property P"?

  1. If n does not have property P, then n is not divisible by 6.
  2. If n has property P, then n is divisible by 6.
  3. If n is not divisible by 6, then n does not have property P.
  4. n is divisible by 6 and has property P.
Show 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.

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Practice Set 8 · Question 80 of 176

Question 80. If p is false and q is true, what is ¬p∧q?

  1. False
  2. p
  3. True
  4. Undetermined
Show Answer and Explanation

Correct answer: C. True

Since p is false, ¬p is true, and q is true.

Continue to Practice Set 9

Practice Set 9 · Question 81 of 176

Question 81. If p is false and q is false, what is ¬(p∨q)?

  1. False
  2. True
  3. Undetermined
  4. p∨q
Show Answer and Explanation

Correct answer: B. True

The disjunction is false, so its negation is true.

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Practice Set 9 · Question 82 of 176

Question 82. Which is the contrapositive of "If an integer n is divisible by 7, then n has property P"?

  1. If n is not divisible by 7, then n does not have property P.
  2. n is divisible by 7 and has property P.
  3. If n does not have property P, then n is not divisible by 7.
  4. If n has property P, then n is divisible by 7.
Show 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.

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Practice Set 9 · Question 83 of 176

Question 83. For p=true and q=false, evaluate ¬p∨q.

  1. True
  2. False
  3. Neither
  4. Undefined
Show Answer and Explanation

Correct answer: B. False

Both ¬p and q are false, making their disjunction false.

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Practice Set 9 · Question 84 of 176

Question 84. For p=false and q=true, evaluate p∧¬q.

  1. True
  2. False
  3. Both
  4. Undefined
Show Answer and Explanation

Correct answer: B. False

Both p and ¬q are false, so their conjunction is false.

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Practice Set 9 · Question 85 of 176

Question 85. Which is the contrapositive of "If an integer n is divisible by 8, then n has property P"?

  1. If n does not have property P, then n is not divisible by 8.
  2. If n has property P, then n is divisible by 8.
  3. If n is not divisible by 8, then n does not have property P.
  4. n is divisible by 8 and has property P.
Show 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.

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Practice Set 9 · Question 86 of 176

Question 86. For p=true and q=true, evaluate (p∨q)∧¬q.

  1. True
  2. False
  3. Undefined
  4. Neither
Show Answer and Explanation

Correct answer: B. False

The disjunction is true but ¬q is false, so the conjunction is false.

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Practice Set 9 · Question 87 of 176

Question 87. For p=false and q=false, evaluate ¬p∨¬q.

  1. False
  2. Cannot tell
  3. True
  4. Neither
Show Answer and Explanation

Correct answer: C. True

Both negations are true, hence their disjunction is true.

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Practice Set 9 · Question 88 of 176

Question 88. Which is the contrapositive of "If an integer n is divisible by 9, then n has property P"?

  1. If n is not divisible by 9, then n does not have property P.
  2. n is divisible by 9 and has property P.
  3. If n does not have property P, then n is not divisible by 9.
  4. If n has property P, then n is divisible by 9.
Show 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.

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Practice Set 9 · Question 89 of 176

Question 89. If p is true and q is true, what is the truth value of p → q?

  1. Undefined
  2. Both
  3. True
  4. False
Show Answer and Explanation

Correct answer: C. True

A conditional is false only when its antecedent is true and its consequent false.

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Practice Set 9 · Question 90 of 176

Question 90. If p is false and q is false, what is the truth value of p → q?

  1. True
  2. False
  3. Undefined
  4. Both
Show Answer and Explanation

Correct answer: A. True

A conditional is false only when its antecedent is true and its consequent false.

Continue to Practice Set 10

Practice Set 10 · Question 91 of 176

Question 91. If p is true and q is false, what is the truth value of p → q?

  1. Undefined
  2. Both
  3. False
  4. True
Show Answer and Explanation

Correct answer: C. False

A conditional is false only when its antecedent is true and its consequent false.

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Practice Set 10 · Question 92 of 176

Question 92. If p is false and q is true, what is the truth value of p → q?

  1. True
  2. False
  3. Undefined
  4. Both
Show Answer and Explanation

Correct answer: A. True

A conditional is false only when its antecedent is true and its consequent false.

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Practice Set 10 · Question 93 of 176

Question 93. For p=true and q=false, what is the truth value of q→p?

  1. False
  2. True
  3. Undefined
  4. Both
Show Answer and Explanation

Correct answer: B. True

A conditional with a false antecedent is true.

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Practice Set 10 · Question 94 of 176

Question 94. For p=false and q=true, what is p→¬q?

  1. False
  2. True
  3. Undetermined
  4. Neither
Show Answer and Explanation

Correct answer: B. True

The antecedent p is false, so the implication is true.

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Practice Set 10 · Question 95 of 176

Question 95. For p=true and q=true, what is ¬p→q?

  1. False
  2. True
  3. Both
  4. Undetermined
Show Answer and Explanation

Correct answer: B. True

The antecedent ¬p is false, hence the conditional is true.

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Practice Set 10 · Question 96 of 176

Question 96. For p=false and q=false, evaluate ¬p→q.

  1. True
  2. False
  3. Cannot tell
  4. Neither
Show Answer and Explanation

Correct answer: B. False

Here ¬p is true and q is false, so the implication is false.

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Practice Set 10 · Question 97 of 176

Question 97. If p is true and q is true, evaluate p ↔ q.

  1. Always true
  2. True
  3. False
  4. Not a proposition
Show Answer and Explanation

Correct answer: B. True

A biconditional is true exactly when the two truth values match.

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Practice Set 10 · Question 98 of 176

Question 98. If p is false and q is false, evaluate p ↔ q.

  1. False
  2. Not a proposition
  3. Always true
  4. True
Show Answer and Explanation

Correct answer: D. True

A biconditional is true exactly when the two truth values match.

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Practice Set 10 · Question 99 of 176

Question 99. If p is true and q is false, evaluate p ↔ q.

  1. Always true
  2. False
  3. True
  4. Not a proposition
Show Answer and Explanation

Correct answer: B. False

A biconditional is true exactly when the two truth values match.

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Practice Set 10 · Question 100 of 176

Question 100. If p is false and q is true, evaluate p ↔ q.

  1. True
  2. Not a proposition
  3. Always true
  4. False
Show Answer and Explanation

Correct answer: D. False

A biconditional is true exactly when the two truth values match.

Continue to Practice Set 11

Practice Set 11 · Question 101 of 176

Question 101. For p=true and q=false, what is p↔¬q?

  1. True
  2. False
  3. Neither
  4. Undefined
Show Answer and Explanation

Correct answer: A. True

Both p and ¬q are true, making the biconditional true.

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Practice Set 11 · Question 102 of 176

Question 102. For p=false and q=true, what is ¬p↔q?

  1. False
  2. True
  3. Cannot tell
  4. Neither
Show Answer and Explanation

Correct answer: B. True

Both ¬p and q are true, so the biconditional is true.

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Practice Set 11 · Question 103 of 176

Question 103. For p=true and q=true, evaluate p↔¬q.

  1. True
  2. Undefined
  3. False
  4. Neither
Show Answer and Explanation

Correct answer: C. False

p is true but ¬q is false, so the biconditional is false.

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Practice Set 11 · Question 104 of 176

Question 104. For p=false and q=false, evaluate ¬p↔q.

  1. True
  2. False
  3. Cannot tell
  4. Neither
Show Answer and Explanation

Correct answer: B. False

¬p is true while q is false, so the biconditional is false.

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Practice Set 11 · Question 105 of 176

Question 105. For a proposition p, which expression is always true?

  1. p ∧ ¬p
  2. p ↔ ¬p
  3. ¬(p ∨ ¬p)
  4. p ∨ ¬p
Show Answer and Explanation

Correct answer: D. p ∨ ¬p

By the law of excluded middle, p∨¬p is a tautology.

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Practice Set 11 · Question 106 of 176

Question 106. For a proposition p about a number, which expression is always true?

  1. ¬(p ∨ ¬p)
  2. p ∨ ¬p
  3. p ∧ ¬p
  4. p ↔ ¬p
Show Answer and Explanation

Correct answer: B. p ∨ ¬p

By the law of excluded middle, p∨¬p is a tautology.

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Practice Set 11 · Question 107 of 176

Question 107. Which proposition is logically equivalent to ¬(p∧q)?

  1. ¬p∧¬q
  2. ¬p∨¬q
  3. p∨q
  4. p∧q
Show Answer and Explanation

Correct answer: B. ¬p∨¬q

De Morgan’s law states ¬(p∧q) ≡ ¬p∨¬q.

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Practice Set 11 · Question 108 of 176

Question 108. Which expression is logically equivalent to p→q?

  1. p∧q
  2. p∨¬q
  3. ¬p∨q
  4. ¬p∧q
Show Answer and Explanation

Correct answer: C. ¬p∨q

A material implication is equivalent to ¬p∨q.

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Practice Set 11 · Question 109 of 176

Question 109. Which proposition is a contradiction for every p?

  1. p∨¬p
  2. p→p
  3. p∧¬p
  4. ¬(p∧¬p)
Show Answer and Explanation

Correct answer: C. p∧¬p

A proposition cannot be true and false simultaneously.

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Practice Set 11 · Question 110 of 176

Question 110. Which expression is equivalent to ¬¬p in classical logic?

  1. p
  2. ¬p
  3. p∧¬p
  4. False
Show Answer and Explanation

Correct answer: A. p

The double-negation law states ¬¬p ≡ p.

Continue to Practice Set 12

Practice Set 12 · Question 111 of 176

Question 111. Which expression is equivalent to p∧(p∨q)?

  1. q
  2. p
  3. p∧q
  4. p∨q
Show Answer and Explanation

Correct answer: B. p

The absorption law reduces p∧(p∨q) to p.

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Practice Set 12 · Question 112 of 176

Question 112. Which expression is equivalent to ¬(p∨q)?

  1. ¬p∨¬q
  2. p∧q
  3. ¬p∧¬q
  4. p∨q
Show Answer and Explanation

Correct answer: C. ¬p∧¬q

De Morgan’s law gives ¬(p∨q) ≡ ¬p∧¬q.

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Practice Set 12 · Question 113 of 176

Question 113. Which inference rule justifies: "p→q; p; therefore q" (example 1)?

  1. Affirming the consequent
  2. Denying the antecedent
  3. Modus ponens
  4. Modus tollens
Show Answer and Explanation

Correct answer: C. Modus ponens

Modus ponens applies the conditional after establishing its antecedent.

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Practice Set 12 · Question 114 of 176

Question 114. Which inference rule justifies: "p→q; p; therefore q" (example 2)?

  1. Modus ponens
  2. Modus tollens
  3. Affirming the consequent
  4. Denying the antecedent
Show Answer and Explanation

Correct answer: A. Modus ponens

Modus ponens applies the conditional after establishing its antecedent.

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Practice Set 12 · Question 115 of 176

Question 115. Which inference rule justifies: "p→q; p; therefore q" (example 3)?

  1. Affirming the consequent
  2. Denying the antecedent
  3. Modus ponens
  4. Modus tollens
Show Answer and Explanation

Correct answer: C. Modus ponens

Modus ponens applies the conditional after establishing its antecedent.

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Practice Set 12 · Question 116 of 176

Question 116. Which inference rule justifies: "p→q; p; therefore q" (example 4)?

  1. Modus ponens
  2. Modus tollens
  3. Affirming the consequent
  4. Denying the antecedent
Show Answer and Explanation

Correct answer: A. Modus ponens

Modus ponens applies the conditional after establishing its antecedent.

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Practice Set 12 · Question 117 of 176

Question 117. Which inference rule justifies: "p→q; p; therefore q" (example 5)?

  1. Affirming the consequent
  2. Denying the antecedent
  3. Modus ponens
  4. Modus tollens
Show Answer and Explanation

Correct answer: C. Modus ponens

Modus ponens applies the conditional after establishing its antecedent.

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Practice Set 12 · Question 118 of 176

Question 118. Which inference rule justifies: "p→q; p; therefore q" (example 6)?

  1. Modus ponens
  2. Modus tollens
  3. Affirming the consequent
  4. Denying the antecedent
Show Answer and Explanation

Correct answer: A. Modus ponens

Modus ponens applies the conditional after establishing its antecedent.

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Practice Set 12 · Question 119 of 176

Question 119. Which inference rule justifies: "p→q; p; therefore q" (example 7)?

  1. Affirming the consequent
  2. Denying the antecedent
  3. Modus ponens
  4. Modus tollens
Show Answer and Explanation

Correct answer: C. Modus ponens

Modus ponens applies the conditional after establishing its antecedent.

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Practice Set 12 · Question 120 of 176

Question 120. Which inference rule justifies: "p→q; p; therefore q" (example 8)?

  1. Modus ponens
  2. Modus tollens
  3. Affirming the consequent
  4. Denying the antecedent
Show Answer and Explanation

Correct answer: A. Modus ponens

Modus ponens applies the conditional after establishing its antecedent.

Continue to Practice Set 13

Practice Set 13 · Question 121 of 176

Question 121. What is the negation of "Every element of the set {1,2,…,3} has property P"?

  1. No element can be tested.
  2. There exists an element in the set without property P.
  3. Every element lacks P.
  4. Some element has P.
Show Answer and Explanation

Correct answer: B. There exists an element in the set without property P.

Negating ∀x P(x) gives ∃x ¬P(x).

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Practice Set 13 · Question 122 of 176

Question 122. Negate the statement "There exists an integer x such that x>5 and P(x)."

  1. For every integer x, x≤5 or not P(x).
  2. For every integer x, x>5 and P(x).
  3. There exists x≤5 and P(x).
  4. There exists x>5 and not P(x).
Show 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.

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Practice Set 13 · Question 123 of 176

Question 123. What is the negation of "Every element of the set {1,2,…,4} has property P"?

  1. Every element lacks P.
  2. Some element has P.
  3. No element can be tested.
  4. There exists an element in the set without property P.
Show Answer and Explanation

Correct answer: D. There exists an element in the set without property P.

Negating ∀x P(x) gives ∃x ¬P(x).

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Practice Set 13 · Question 124 of 176

Question 124. Negate the statement "There exists an integer x such that x>6 and P(x)."

  1. There exists x≤6 and P(x).
  2. There exists x>6 and not P(x).
  3. For every integer x, x≤6 or not P(x).
  4. For every integer x, x>6 and P(x).
Show 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.

Next Question →

Practice Set 13 · Question 125 of 176

Question 125. What is the negation of "Every element of the set {1,2,…,5} has property P"?

  1. No element can be tested.
  2. There exists an element in the set without property P.
  3. Every element lacks P.
  4. Some element has P.
Show Answer and Explanation

Correct answer: B. There exists an element in the set without property P.

Negating ∀x P(x) gives ∃x ¬P(x).

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Practice Set 13 · Question 126 of 176

Question 126. Negate the statement "There exists an integer x such that x>7 and P(x)."

  1. For every integer x, x≤7 or not P(x).
  2. For every integer x, x>7 and P(x).
  3. There exists x≤7 and P(x).
  4. There exists x>7 and not P(x).
Show 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.

Next Question →

Practice Set 13 · Question 127 of 176

Question 127. What is the negation of "Every element of the set {1,2,…,6} has property P"?

  1. Every element lacks P.
  2. Some element has P.
  3. No element can be tested.
  4. There exists an element in the set without property P.
Show Answer and Explanation

Correct answer: D. There exists an element in the set without property P.

Negating ∀x P(x) gives ∃x ¬P(x).

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Practice Set 13 · Question 128 of 176

Question 128. Negate the statement "There exists an integer x such that x>8 and P(x)."

  1. There exists x≤8 and P(x).
  2. There exists x>8 and not P(x).
  3. For every integer x, x≤8 or not P(x).
  4. For every integer x, x>8 and P(x).
Show 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.

Next Question →

Practice Set 13 · Question 129 of 176

Question 129. What is the negation of "Every element of the set {1,2,…,7} has property P"?

  1. No element can be tested.
  2. There exists an element in the set without property P.
  3. Every element lacks P.
  4. Some element has P.
Show Answer and Explanation

Correct answer: B. There exists an element in the set without property P.

Negating ∀x P(x) gives ∃x ¬P(x).

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Practice Set 13 · Question 130 of 176

Question 130. Negate the statement "There exists an integer x such that x>9 and P(x)."

  1. For every integer x, x≤9 or not P(x).
  2. For every integer x, x>9 and P(x).
  3. There exists x≤9 and P(x).
  4. There exists x>9 and not P(x).
Show 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.

Continue to Practice Set 14

Practice Set 14 · Question 131 of 176

Question 131. What is the negation of "Every element of the set {1,2,…,8} has property P"?

  1. Every element lacks P.
  2. Some element has P.
  3. No element can be tested.
  4. There exists an element in the set without property P.
Show Answer and Explanation

Correct answer: D. There exists an element in the set without property P.

Negating ∀x P(x) gives ∃x ¬P(x).

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Practice Set 14 · Question 132 of 176

Question 132. Negate the statement "There exists an integer x such that x>10 and P(x)."

  1. There exists x≤10 and P(x).
  2. There exists x>10 and not P(x).
  3. For every integer x, x≤10 or not P(x).
  4. For every integer x, x>10 and P(x).
Show 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.

Next Question →

Practice Set 14 · Question 133 of 176

Question 133. What is the negation of "Every element of the set {1,2,…,9} has property P"?

  1. No element can be tested.
  2. There exists an element in the set without property P.
  3. Every element lacks P.
  4. Some element has P.
Show Answer and Explanation

Correct answer: B. There exists an element in the set without property P.

Negating ∀x P(x) gives ∃x ¬P(x).

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Practice Set 14 · Question 134 of 176

Question 134. Negate the statement "There exists an integer x such that x>11 and P(x)."

  1. For every integer x, x≤11 or not P(x).
  2. For every integer x, x>11 and P(x).
  3. There exists x≤11 and P(x).
  4. There exists x>11 and not P(x).
Show 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.

Next Question →

Practice Set 14 · Question 135 of 176

Question 135. What is the negation of "Every element of the set {1,2,…,10} has property P"?

  1. Every element lacks P.
  2. Some element has P.
  3. No element can be tested.
  4. There exists an element in the set without property P.
Show Answer and Explanation

Correct answer: D. There exists an element in the set without property P.

Negating ∀x P(x) gives ∃x ¬P(x).

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Practice Set 14 · Question 136 of 176

Question 136. Negate the statement "There exists an integer x such that x>12 and P(x)."

  1. There exists x≤12 and P(x).
  2. There exists x>12 and not P(x).
  3. For every integer x, x≤12 or not P(x).
  4. For every integer x, x>12 and P(x).
Show 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.

Next Question →

Practice Set 14 · Question 137 of 176

Question 137. On the integers, consider aRb iff a−b is divisible by 2. Which description is correct?

  1. R is not reflexive
  2. R is not symmetric
  3. R is not transitive
  4. R is an equivalence relation
Show Answer and Explanation

Correct answer: D. R is an equivalence relation

Congruence modulo 2 is reflexive, symmetric and transitive.

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Practice Set 14 · Question 138 of 176

Question 138. On the integers, consider aRb iff a−b is divisible by 3. Which description is correct?

  1. R is not transitive
  2. R is an equivalence relation
  3. R is not reflexive
  4. R is not symmetric
Show Answer and Explanation

Correct answer: B. R is an equivalence relation

Congruence modulo 3 is reflexive, symmetric and transitive.

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Practice Set 14 · Question 139 of 176

Question 139. On the integers, consider aRb iff a−b is divisible by 4. Which description is correct?

  1. R is not reflexive
  2. R is not symmetric
  3. R is not transitive
  4. R is an equivalence relation
Show Answer and Explanation

Correct answer: D. R is an equivalence relation

Congruence modulo 4 is reflexive, symmetric and transitive.

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Practice Set 14 · Question 140 of 176

Question 140. On the integers, consider aRb iff a−b is divisible by 5. Which description is correct?

  1. R is not transitive
  2. R is an equivalence relation
  3. R is not reflexive
  4. R is not symmetric
Show Answer and Explanation

Correct answer: B. R is an equivalence relation

Congruence modulo 5 is reflexive, symmetric and transitive.

Continue to Practice Set 15

Practice Set 15 · Question 141 of 176

Question 141. On the integers, consider aRb iff a−b is divisible by 6. Which description is correct?

  1. R is not reflexive
  2. R is not symmetric
  3. R is not transitive
  4. R is an equivalence relation
Show Answer and Explanation

Correct answer: D. R is an equivalence relation

Congruence modulo 6 is reflexive, symmetric and transitive.

Next Question →

Practice Set 15 · Question 142 of 176

Question 142. On the integers, consider aRb iff a−b is divisible by 7. Which description is correct?

  1. R is not transitive
  2. R is an equivalence relation
  3. R is not reflexive
  4. R is not symmetric
Show Answer and Explanation

Correct answer: B. R is an equivalence relation

Congruence modulo 7 is reflexive, symmetric and transitive.

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Practice Set 15 · Question 143 of 176

Question 143. On the integers, consider aRb iff a−b is divisible by 8. Which description is correct?

  1. R is not reflexive
  2. R is not symmetric
  3. R is not transitive
  4. R is an equivalence relation
Show Answer and Explanation

Correct answer: D. R is an equivalence relation

Congruence modulo 8 is reflexive, symmetric and transitive.

Next Question →

Practice Set 15 · Question 144 of 176

Question 144. On the integers, consider aRb iff a−b is divisible by 9. Which description is correct?

  1. R is not transitive
  2. R is an equivalence relation
  3. R is not reflexive
  4. R is not symmetric
Show Answer and Explanation

Correct answer: B. R is an equivalence relation

Congruence modulo 9 is reflexive, symmetric and transitive.

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Practice Set 15 · Question 145 of 176

Question 145. On the set of integers, which relation is a partial order?

  1. a−b is odd
  2. a+b is even
  3. a≤b
  4. a≠b
Show Answer and Explanation

Correct answer: C. a≤b

The usual ≤ relation is reflexive, antisymmetric and transitive.

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Practice Set 15 · Question 146 of 176

Question 146. On positive integers, which relation defines a partial order?

  1. a divides b
  2. a+b is even
  3. a≠b
  4. a−b is odd
Show Answer and Explanation

Correct answer: A. a divides b

Divisibility is reflexive, antisymmetric and transitive on positive integers.

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Practice Set 15 · Question 147 of 176

Question 147. On the power set P({1,2}), which relation is a partial order?

  1. A intersects B
  2. A⊆B
  3. A≠B
  4. A∩B≠∅
Show Answer and Explanation

Correct answer: B. A⊆B

Set inclusion is reflexive, antisymmetric and transitive.

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Practice Set 15 · Question 148 of 176

Question 148. Which property fails for the usual strict order < on integers when viewed as a partial order?

  1. Transitivity
  2. Irreflexivity
  3. Reflexivity
  4. Asymmetry
Show Answer and Explanation

Correct answer: C. Reflexivity

A partial order must be reflexive, whereas a

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Practice Set 15 · Question 149 of 176

Question 149. For the usual ≤ on integers, which is true of 3 and 5?

  1. 5≤3
  2. 3≤5
  3. 3=5
  4. They are incomparable
Show Answer and Explanation

Correct answer: B. 3≤5

Three is less than five, so 3≤5.

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Practice Set 15 · Question 150 of 176

Question 150. Under divisibility on positive integers, is 2 comparable with 3?

  1. Yes, 2 divides 3
  2. Yes, 3 divides 2
  3. No, neither divides the other
  4. Yes, both divide each other
Show Answer and Explanation

Correct answer: C. No, neither divides the other

Neither 2 divides 3 nor 3 divides 2.

Continue to Practice Set 16

Practice Set 16 · Question 151 of 176

Question 151. Which relation on integers is an equivalence relation but not a partial order?

  1. a≤b
  2. a=b
  3. a−b is even
  4. a divides b
Show Answer and Explanation

Correct answer: C. a−b is even

Congruence modulo 2 is an equivalence relation, but distinct same-parity integers violate antisymmetry.

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Practice Set 16 · Question 152 of 176

Question 152. In a partially ordered set, what does a≤b and b≤a imply?

  1. a≠b
  2. a=b
  3. a
  4. No conclusion
Show Answer and Explanation

Correct answer: B. a=b

Antisymmetry of a partial order implies a=b.

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Practice Set 16 · Question 153 of 176

Question 153. For f(x)=2x+1, what is f(3)?

  1. 5
  2. 7
  3. 4
  4. 6
Show Answer and Explanation

Correct answer: B. 7

Substitute x=3: 2(3)+1 = 7.

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Practice Set 16 · Question 154 of 176

Question 154. Which map on the real numbers is one-to-one? (variant 1)

  1. f(x)=1x+1
  2. f(x)=x²
  3. f(x)=|x|
  4. f(x)=0
Show Answer and Explanation

Correct answer: A. f(x)=1x+1

A linear function with nonzero slope is injective on the real numbers.

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Practice Set 16 · Question 155 of 176

Question 155. For f(x)=2x+2, what is f(4)?

  1. 6
  2. 9
  3. 11
  4. 10
Show Answer and Explanation

Correct answer: D. 10

Substitute x=4: 2(4)+2 = 10.

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Practice Set 16 · Question 156 of 176

Question 156. Which map on the real numbers is one-to-one? (variant 2)

  1. f(x)=|x|
  2. f(x)=0
  3. f(x)=2x+1
  4. f(x)=x²
Show Answer and Explanation

Correct answer: C. f(x)=2x+1

A linear function with nonzero slope is injective on the real numbers.

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Practice Set 16 · Question 157 of 176

Question 157. For f(x)=2x+3, what is f(5)?

  1. 7
  2. 13
  3. 8
  4. 12
Show Answer and Explanation

Correct answer: B. 13

Substitute x=5: 2(5)+3 = 13.

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Practice Set 16 · Question 158 of 176

Question 158. Which map on the real numbers is one-to-one? (variant 3)

  1. f(x)=3x+1
  2. f(x)=x²
  3. f(x)=|x|
  4. f(x)=0
Show Answer and Explanation

Correct answer: A. f(x)=3x+1

A linear function with nonzero slope is injective on the real numbers.

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Practice Set 16 · Question 159 of 176

Question 159. For f(x)=2x+4, what is f(6)?

  1. 10
  2. 15
  3. 8
  4. 16
Show Answer and Explanation

Correct answer: D. 16

Substitute x=6: 2(6)+4 = 16.

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Practice Set 16 · Question 160 of 176

Question 160. Which map on the real numbers is one-to-one? (variant 4)

  1. f(x)=|x|
  2. f(x)=0
  3. f(x)=4x+1
  4. f(x)=x²
Show Answer and Explanation

Correct answer: C. f(x)=4x+1

A linear function with nonzero slope is injective on the real numbers.

Continue to Practice Set 17

Practice Set 17 · Question 161 of 176

Question 161. For f(x)=2x+5, what is f(7)?

  1. 9
  2. 19
  3. 12
  4. 18
Show Answer and Explanation

Correct answer: B. 19

Substitute x=7: 2(7)+5 = 19.

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Practice Set 17 · Question 162 of 176

Question 162. Which map on the real numbers is one-to-one? (variant 5)

  1. f(x)=5x+1
  2. f(x)=x²
  3. f(x)=|x|
  4. f(x)=0
Show Answer and Explanation

Correct answer: A. f(x)=5x+1

A linear function with nonzero slope is injective on the real numbers.

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Practice Set 17 · Question 163 of 176

Question 163. For f(x)=2x+6, what is f(8)?

  1. 14
  2. 21
  3. 10
  4. 22
Show Answer and Explanation

Correct answer: D. 22

Substitute x=8: 2(8)+6 = 22.

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Practice Set 17 · Question 164 of 176

Question 164. Which map on the real numbers is one-to-one? (variant 6)

  1. f(x)=|x|
  2. f(x)=0
  3. f(x)=6x+1
  4. f(x)=x²
Show Answer and Explanation

Correct answer: C. f(x)=6x+1

A linear function with nonzero slope is injective on the real numbers.

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Practice Set 17 · Question 165 of 176

Question 165. For f(x)=2x+7, what is f(9)?

  1. 11
  2. 25
  3. 16
  4. 24
Show Answer and Explanation

Correct answer: B. 25

Substitute x=9: 2(9)+7 = 25.

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Practice Set 17 · Question 166 of 176

Question 166. Which map on the real numbers is one-to-one? (variant 7)

  1. f(x)=7x+1
  2. f(x)=x²
  3. f(x)=|x|
  4. f(x)=0
Show Answer and Explanation

Correct answer: A. f(x)=7x+1

A linear function with nonzero slope is injective on the real numbers.

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Practice Set 17 · Question 167 of 176

Question 167. For f(x)=2x+8, what is f(10)?

  1. 18
  2. 27
  3. 12
  4. 28
Show Answer and Explanation

Correct answer: D. 28

Substitute x=10: 2(10)+8 = 28.

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Practice Set 17 · Question 168 of 176

Question 168. Which map on the real numbers is one-to-one? (variant 8)

  1. f(x)=|x|
  2. f(x)=0
  3. f(x)=8x+1
  4. f(x)=x²
Show Answer and Explanation

Correct answer: C. f(x)=8x+1

A linear function with nonzero slope is injective on the real numbers.

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Practice Set 17 · Question 169 of 176

Question 169. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a3.

  1. 3
  2. 9
  3. 7
  4. 6
Show Answer and Explanation

Correct answer: D. 6

The recurrence sums 1+2+…+3 = 3(3+1)/2 = 6.

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Practice Set 17 · Question 170 of 176

Question 170. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a4.

  1. 16
  2. 10
  3. 6
  4. 14
Show Answer and Explanation

Correct answer: B. 10

The recurrence sums 1+2+…+4 = 4(4+1)/2 = 10.

Continue to Practice Set 18

Practice Set 18 · Question 171 of 176

Question 171. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a5.

  1. 10
  2. 20
  3. 25
  4. 15
Show Answer and Explanation

Correct answer: D. 15

The recurrence sums 1+2+…+5 = 5(5+1)/2 = 15.

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Practice Set 18 · Question 172 of 176

Question 172. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a6.

  1. 36
  2. 21
  3. 15
  4. 27
Show Answer and Explanation

Correct answer: B. 21

The recurrence sums 1+2+…+6 = 6(6+1)/2 = 21.

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Practice Set 18 · Question 173 of 176

Question 173. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a7.

  1. 21
  2. 35
  3. 49
  4. 28
Show Answer and Explanation

Correct answer: D. 28

The recurrence sums 1+2+…+7 = 7(7+1)/2 = 28.

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Practice Set 18 · Question 174 of 176

Question 174. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a8.

  1. 64
  2. 36
  3. 28
  4. 44
Show Answer and Explanation

Correct answer: B. 36

The recurrence sums 1+2+…+8 = 8(8+1)/2 = 36.

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Practice Set 18 · Question 175 of 176

Question 175. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a9.

  1. 36
  2. 54
  3. 81
  4. 45
Show Answer and Explanation

Correct answer: D. 45

The recurrence sums 1+2+…+9 = 9(9+1)/2 = 45.

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Practice Set 18 · Question 176 of 176

Question 176. A sequence is defined by a₁=1 and aₙ=aₙ₋₁+n for n≥2. Find a10.

  1. 100
  2. 55
  3. 45
  4. 65
Show Answer and Explanation

Correct answer: B. 55

The recurrence sums 1+2+…+10 = 10(10+1)/2 = 55.

Restart from Question 1

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