Some review for the final
All questions below (and on the final) are in “prove or disprove” format. I guarantee
that (up to possible rephrasing):
• at least one from 8–13 will be on the final;
• at least one from 50–69 will be on the final;
• at least one from 74–86 will be on the final;
• at least one from 87–99 will be on the final.
I guarantee that a few other questions from the below (up to possible rephrasing) will be
on the final. I guarantee that at least some questions on the final will not be from the
below.
1. P ⇒ (Q ⇒ R) is logically equivalent to (P ⇒ Q) ⇒ R
2. (P ∧ Q) ∨ R is logically equivalent to P ∧ (Q ∨ R).
3. ¬(P ⇒ Q) is logically equivalent to ¬P ⇒ ¬Q.
4. If P (x) and Q(x) are predicates and D is the domain of x, then the statements
∀x ∈ D, (P (x) ∨ Q(x))
and
(∀x ∈ D, P (x)) ∨ (∀x ∈ D, Q(x))
have the same truth values.
5. If P (x) and Q(x) are predicates and D is the domain of x, then the statements
∃x ∈ D, (P (x) ∧ Q(x))
and
(∃x ∈ D, P (x)) ∧ (∃x ∈ D, Q(x))
have the same truth values.
6. If P (x) and Q(x) are predicates and D is the domain of x, then the statements
∃x ∈ D, (P (x) ⇒ Q(x))
and
(∃x ∈ D, P (x)) ⇒ (∃x ∈ D, Q(x))
have the same truth values.
7. If P (x) and Q(x) are predicates and D is the domain of x, then the statements
∀x ∈ D, (P (x) ⇒ Q(x))
and
(∀x ∈ D, P (x)) ⇒ (∀x ∈ D, Q(x))
have the same truth values.
8. The sum of two odd integers is even.
9. The sum of an odd integer and an even integer is odd.
10. The square of an odd integer is odd.
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11. The square of an even integer is even.
12. The product of two odd integers is odd.
13. The product of two even integers is odd.
14. For every integer a, if a2 is odd, then a is odd.
15. For every integer a, if a2 is even, then a is even.
16. The sum of two irrational numbers is irrational.
17. The sum of two positive irrational numbers is irrational.
18. The sum of two rational numbers is rational.
19. The difference of two rational numbers is rational.
20. The product of two irrational numbers is irrational.
21. The product of a rational number and an irrational number is irrational.
22. The product of a nonzero rational number and an irrational number is irrational.
23. The sum of any three consecutive integers is divisible by 3.
24. If n is an integer, then n2 − 3 is not divisible by 4.
25. If a and b be positive integers such that a > 1 and b is divisible by a, then a + b is not
prime.
26. If n is a natural number, then n! − n is not a prime.
27. For any integer n and prime number p, if p | n then p - n + 1.
28. For all integers a, b and c, if a | b and a | c, then a | b − c.
29. For all integers a, b and c, if a | b + c, then a | b or a | c.
30. For all integers a, b and c, if a | bc, then a | b or a | c.
31. For all integers a and b, if a | b, then a2 | b2 .
32. There is a largest real number.
33. There is a smallest real number.
34. There is a smallest positive real number.
35. There is a smallest positive rational number.
36. There is a largest negative real number.
37. There is a largest negative rational number.
38. Every integer is even or odd.
√
39. 2 is irrational.
40. For all nonnegative integers n,
n
X
k=0
k=
n(n + 1)
.
2
2
41. For all positive integers n,
n
X
i=1
1
n
=
.
i(i + 1)
n+1
42. If n is a positive integer, then 5n is divisible by 4.
43. If n is a non-negative integer, then 22n − 1 is divisible by 3.
P
44. 2n+1 = 1 + ni=0 2i for all nonnegative integers n.
45. For all positive integers n,
(n + 1)! = 1 +
n
X
j(j!) .
j=1
46. There are infinitely many prime numbers.
47. For each positive integer n, there exist integers k and m such that m > 0, k is odd and
n = 2m k.
48. For all natural numbers a and d, there exist integers q and r such that a = qd + r and
0 ≤ r ≤ d − 1.
49. Every integer greater than 1 is divisible by a prime.
50. If A, B and C are sets, then A \ (B \ C) = (A \ B) \ C.
51. If A and B are sets such that A ∪ B = (A \ B) ∪ (B \ A), then A ∩ B = ∅.
52. If A, B and C are sets such that A ⊆ B, then A ∪ C ⊆ B ∪ C.
53. If A, B and C are sets such that A ⊆ B, then A ∩ C ⊆ B ∩ C.
54. If A, B and C are sets such that A ∩ B = A ∩ C, then B = C.
55. If A, B and C are sets such that A ∪ B = A ∪ C, then B = C.
56. If A, B and C are sets such that A \ B = A \ C, then B = C.
57. If A, B and C are sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C, then B = C.
58. If A, B, C are sets with B ∩ C ⊆ A, then (A \ B) ∩ (A \ C) = ∅.
59. If A and B are disjoint sets, then the power sets P(A) and P(B) are disjoint.
60. If A and B are sets, then P(A ∪ B) = P(A) ∪ P(B), where P indicates taking the
power set.
61. If A and B are sets, then P(A ∩ B) = P(A) ∩ P(B), where P indicates taking the
power set.
62. If A and B are sets, then P(A \ B) = P(A) \ P(B), where P indicates taking the power
set.
63. If A and B are sets, then P(A × B) = P(A) × P(B), where P indicates taking the
power set.
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64. For all sets A, B, C and D,
(A \ C) × (B \ D) ⊆ (A × B) \ (C × D).
65. For all sets A, B and C, if A × C ⊆ B × C, then A ⊆ B.
66. For all sets A, B and C, if A ⊆ B, then A × C ⊆ B × C.
67. For all sets A and B, (A ∪ B) × (A ∪ B) ⊇ (A × A) ∪ (B × B).
68. For all sets A and B, (A ∪ B) × (A ∪ B) = (A × A) ∪ (B × B).
69. For all sets A and B, if (A ∪ B) × (A ∪ B) 6= (A × A) ∪ (B × B), then both A \ B and
B \ A are nonempty.
70. For all sets A, B and C, if f : A → B and g : B → C are injections, then g ◦ f is an
injection.
71. For all sets A, B and C, if f : A → B and g : B → C are functions such that g ◦ f is
an injection, then f and g are injections.
72. For all sets A, B and C, if f is a surjection from A to B and g is a surjection from B
to C, then g ◦ f is a surjection from A to C.
73. For all sets A, B and C, if f : A → B and g : B → C are functions such that g ◦ f is a
surjection from A to C, then f and g are surjections.
74. For all sets A, B, C, D such that A ⊆ B ⊆ C, if f : C → D is a function, then
f [A] ⊆ f [B].
75. For all sets A, B, C, D such that A, B ⊆ C, if f : C → D is a function, then f [A \ B] =
f [A] \ f [B].
76. For all sets A, B, C, D such that A, B ⊆ C, if f : C → D is a function, then f [A ∪ B] =
f [A] ∪ f [B].
77. For all sets A, B, C, D such that A, B ⊆ C, if f : C → D is a function, then f [A ∩ B] =
f [A] ∩ f [B].
78. For all sets A, B, C, D such that A ⊆ B ⊆ D, if f : C → D is a function, then
f −1 [A] ⊆ f −1 [B].
79. For all sets A, B, C, D such that A, B ⊆ D, if f : C → D is a function, then f −1 [A∩B] =
f −1 [A] ∩ f −1 [B].
80. For all sets A, B, C, D such that A, B ⊆ D, if f : C → D is a function, then f −1 [A∪B] =
f −1 [A] ∪ f −1 [B].
81. For all sets A, B, C, D such that A, B ⊆ D, if f : C → D is a function, then f −1 [A\B] =
f −1 [A] \ f −1 [B].
82. For all sets A, B, C such that C ⊆ A, if f : A → B is a function, then f −1 [f [C]] = C.
83. For all sets A, B, C such that C ⊆ B, if f : A → B is a function, then f [f −1 [C]] = C.
84. For all sets A, B, C, D such that C, D ⊆ A, if f : A → B is a function and f [C] = f [D],
then C = D.
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85. For all sets A, B, C, D such that C, D ⊆ B, if f : A → B is a function and f −1 [C] =
f −1 [D], then C = D.
86. If f : A → B and g : B → C are functions, then (g ◦ f )−1 [D] = f −1 [g −1 [D]] for all
D ⊆ C.
87. The relation A defined on R by
x A y ⇔ ∃k ∈ Z, x − y = 2k
is an equivalence relation.
88. The relation A defined on R by x A y ⇔ x − y ∈ Q is an equivalence relation.
89. The relation A defined on Z by x A y ⇔ 5 | y 2 − x2
is an equivalence relation.
90. Let c be a positive real number. Define the relation S on R by
x S y ⇔ there exists q ∈ Q such that x = ycq .
Prove or disprove. S is an equivalence relation.
91. Let c be a positive rational number. Define the relation R on Q by
x R y ⇔ there exists k ∈ Z such that x = yck .
Prove or disprove. R is an equivalence relation.
92. The relation A defined on R by
x A y ⇔ ∃q ∈ Q, (q 6= 0 ∧ y = qx)
is an equivalence relation.
93. Let R be a symmetric, transitive relation on a set A such that for every x ∈ A there
exists y ∈ A with x R y.
Prove or disprove. R is an equivalence relation.
94. If R and S are equivalence relations on a set A, then R ∩ S is an equivalence relation
on A.
95. If R and S are equivalence relations on a set A, then R ∪ S is an equivalence relation
on A.
96. If R and S are partial order relations on a set A, then R ∩ S is a partial order relation
on A.
97. If R and S are partial order relations on a set A, then R ∪ S is a partial order relation
on A.
98. Let R be a partial order on a set A. Let a, b ∈ A be such that b is a greatest element
of A \ {a}.
Prove or disprove. At least one of a or b is a greatest element of A.
99. The relation E on R defined by x E y iff x2 ≥ y 2 is a partial order.
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