Math 415, Test 1, October 20, 2003
Instructions: Do problem 1and any three of probems 2,3,4,5. Please do your best, and
show all appropriate details in your solutions.
1. (20 pts) (a) State the contrapositive of “If nkis odd, then nis odd.”
(b) Let nbe an integer, and let k∈IN, prove that nis odd if nkis odd.
(c) Let aand bbe odd integers, prove that their product ab is odd.
(d) Let nbe an odd integer. Use the Principle of Mathematical Induction to prove that nkis
odd for all k∈IN.
(e) Let nbe an integer, and let k∈IN. Is it true that nkis odd if and only if nis odd?
2. (10 pts) (a) Complete the truth table for ¬(P∧Q)←→ ¬P∨ ¬Q.
(b) Is the statement in (a) a tautology? Explain.
(c) State the negation of x > 0 and y= 5.
3. (10 pts) Let A={1,2,3,4},B={2,3,5}and C={1,5}.
(a) Find C−B.
(b) Find (A∩B)∪C.
(c) Find A−(C−B).
(d) Find P(A) the power set of A.
4. (10 pts) (a) Prove or disprove that A−(B−C) = (A−B)∪(A∩C).
(b) Prove or disprove that P(A∪B) = P(A)∪ P(B).
5. (10 pts) Prove the following versions of De Morgan’s Laws.
(a) \
i∈I
Ai!c
=[
i∈I
Ac
i
(b) [
i∈I
Ai!c
=\
i∈I
Ac
i
1