Assign a grade of A (correct), C (partially correct), or F (failure) to each. Ju

Question

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Suppose m is an integer. Claim. If m^2 is odd, then m is odd. "Proof." Assume that m^2 is not odd. Then m^2 is even and m^2 = 2k for some integer k. Thus 2k is a perfect square; that is, squareroot 2k is an integer. If squareroot k is odd, then squareroot 2k = 2n + 1 for some integer n, which means m^2 = 2k = (2n + 1)^2 = 4n^2 + 4n + 1 = 2(2n^2 + 2n) + 1. Thus m^2 is odd, contrary to our assumption. Therefore squareroot 2k = m must be even. Thus if m^2 is not odd, then m is not odd. Hence if m^2 is odd, then m is odd.

Explanation / Answer

The grade assigned to this proof is F (failure).

This is because the proof is not correctly started. The proof uses contraposition to suffice itself. But the contraposition of given claim is given s follows:

"If m is not odd, then m^2 is not odd".

Hence the proof starts with the wrong note. Instead of assuming that m is not odd, the given proof assumes that m^2 is not odd.

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