Problem 1
Prove the contrapositive: if n is odd, n=2k+1 and n² is odd.
Faculty solution guide
Proof by Contraposition
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Proof by Contraposition.
Problem 4
The argument must address proving an implication through the logically equivalent contrapositive with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in number theory, impossibility results, optimization, uniqueness, and correctness proofs, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.