Math 3200 Final Exam Practice Problems
1. Suppose |A| = m and |B| = n for m, n ∈ N.
(a) What is |P(A × B)|?
(b) What is |P(A × P(B))|?
2. For each of the following, determine whether the statement is true or false. Explain your reasoning.
(a) (R × Z) ∩ (Z × R) = Z × Z
(b) (R × Z) ∪ (Z × R) = R × R
3. Determine the negations of the following statements.
(a) “If 3 is prime, then 3 is even.”
(b) “If r is a nonzero rational number, then tan(r) is irrational.”
4. What is the contrapositive of the statement “If you can’t beat them, join them”?
5. Let S = {0, 1, 2, 3, 4, 5} and let Q(x, y) be the statement that x is divisible by y. Translate the following
into English and then determine whether it is true or false:
∃y ∈ S, ∀x ∈ S, Q(x, y).
6. Give the truth tables for the following:
(a) ∼ (P ⇒ Q)
(b) P ∨ (Q ∧ (∼ R))
(c) (P ∧ (∼ P )) ⇒ Q
7. (a) Is P ∧ Q logically equivalent to ∼ ((∼ P ) ∨ (∼ Q))?
(b) Is P ⇒ Q logically equivalent to (P ∧ (∼ Q)) ⇒ (Q ∧ (∼ Q))?
(c) Is ∼ (P ∨ Q) logically equivalent to (∼ P ) ∧ (∼ Q)?
8. √
Prove
√ or give a counterexample for each of the following statements (you can use the fact that
2, 3, π, e, etc. are irrational without having to give a proof).
(a) If x is a nonzero rational number and y is irrational, then xy is irrational.
(b) If x and y are irrational, then xy is irrational.
9. Prove that if x ∈ Z is odd, then x3 is odd (it may be helpful to recall that (a+b)3 = a3 +3a2 b+3ab2 +b3
for any a and b).
10. Prove that if x ∈ Z with x3 − 1 even, then x is odd.
11. Prove that if n ∈ Z, then 4 6 | (n2 − 3).
12. Let A and B be sets. Prove that (A ∩ B) − C = (A − C) ∩ (B − C).
13. Suppose A and B are sets so that A − B = ∅. Does this imply that B 6= ∅? Prove or give a
counterexample.
14. Either prove the following statement or give a counterexample: the product of any two consecutive
integers is even.
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15. Either prove the following statement or give a counterexample: every odd integer is the sum of three
odd integers.
16. Recall that for any n ∈ N, the factorial of n is n! = 1 · 2 · 3 · · · · · (n − 1) · n. For example, 3! = 1 · 2 · 3 = 6
and 4! = 1 · 2 · 3 · 4 = 24.
Prove that
2
n
1
1
+ + ... +
=1−
2! 3!
(n + 1)!
(n + 1)!
P∞
i
= 1.)
for all n ∈ N. (Notice: this implies that i=1 (i+1)!
17. Use induction to prove that for all integers n ≥ 0, the number n2 + 5n − 4 is even.
18. Suppose R is an equivalence relation on an infinite set S. Does R necessarily have infinitely many
equivalence classes? Either prove that it does or find a counterexample.
19. Give an example of a relation R on a set S so that R is reflexive but not transitive.
20. Define the function f : Z × Z → Z by f (m, n) = 3n − 4m.
(a) Is f injective?
(b) Is f surjective? (Hint: Can you find m and n so that f (m, n) = 1? How can you use your answer
to this question to shed light on the problem? )
21. Consider Z6 = {[0], [1], [2], [3], [4], [5]}, the set of equivalence classes of integers modulo 6. Define
g : Z6 → Z6 by g([a]) = [a2 ]. Let A = {[0], [2], [4]} and let B = {[0], [1], [2]}.
(a) What is g(A)?
(b) What is g −1 (B)?
22. Recall (from Exam 3) the function h : Z → Z given by
(
h(a) =
a
2
3a + 1
if a is even
if a is odd
Is h injective? Prove or give a counterexample.
23. Suppose f : A → B and g : B → C so that g ◦ f is surjective. For each of the following statements,
either prove or give a counterexample.
(a) g is surjective.
(b) f is surjective.
(c) g is injective.
24. Suppose A and B are sets so that there exists a surjective function f : A → B. If B is uncountable,
what can we say about the cardinality of A?
25. Prove or give a counterexample: If S is an uncountable set, then |S| = |R|.
26. Prove that {0, 1} × N is countable.
Challenge Problem: We know that the set of all subsets of N is uncountable (and, in fact, has the same
cardinality as R). On the other hand, we’ve seen that the set of all two-element subsets of N is
countable. So here’s something in the middle: let F be th set of all finite subsets of N. Is F countable
or uncountable? Justify your answer.
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