Problem 1
Assume the negation of the target and derive an impossibility from definitions and valid prior results.
Faculty solution guide
Proof by Contradiction
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 Contradiction.
Problem 4
The argument must address assuming the negation of a claim and deriving an impossibility 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.