1.4, #2 . Claim : For all integers p and q , if their product pq is even, then p

Question

1.4, #2. Claim: For all integers p and q, if their product pq is even, then p and q are even.

Consider the following "proofs" of the claim.

Proof A:

Suppose p and q are integers and pq is even.

By definition of even, $ integers m and n such that  pq = (2m)(2n).

Then p = 2m and q = 2n for integers m and n.

By definition of even, p and q are even integers.

Proof B:

(By contraposition; i.e., proving the contrapositive)

Suppose p and q are odd. We want to show that pq is odd.

By definition of odd, $ integer m such that  p = 2m + 1 and $ integer n such that  q = 2n + 1.

Then pq = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1.

Let k = 2mn + m + n, which is an integer.

Thus, pq = 2k + 1 for some integer k, and by definition of odd, pq is odd.

Proof C:

Suppose p and q are any even integers. By definition of even, $ integer k such that  p = 2k and q = 2k.

Then pq = (2k)(2k) = 2(2k2). Let m = 2k2, which is an integer.

Thus, pq = 2m for some integer m, and by definition of even, pq is even.

Proof D:

Suppose p and q are any even integers.

By definition of even, $ integer m such that  p = 2m and $ integer n such that  q = 2n.

Then pq = (2m)(2n) = 2(2mn). Let k = 2mn, which is an integer.

Thus, pq = 2k for some integer k, and by definition of even, pq is even.

INSTRUCTIONS:

(a) Critique each proof (A, B, C, D). For each proof, is it logically valid or not? What are the flaws, if any?

(b) Is the Claim true or false? Explain.

Explanation / Answer

I did not mean to submit yet.

Proof B. The mistake is made in stating the contrapositive. The contrapositive should have been if p or q is odd then pq is odd. The contrapositive here had p and q are odd, not p or q are odd. Using this contrapositive, it becomes clear that this is not the case.

Proof C. This proves the converse, that p and q are even, therefore pq are even. This does not prove the statement.

Proof D. This problem makes the same mistake, trying to prove by proving the converse, that p and q are even, therefore pq are even.

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