Problem 1
If r>0 were smallest, then r/2 would be a smaller positive rational, a contradiction.
Faculty solution guide
Proof by Contradiction and Counterexample
Accept equivalent mathematically correct reasoning. Require definitions, proof or algorithm, edge cases, and verification.
Problem 2
A common error is to rely on a plausible pattern, diagram, or program output without checking definitions, domain conditions, quantifiers, edge cases, proof obligations, or algorithm invariants.
Problem 3
A complete response identifies the objects, domain, definitions, notation, hypotheses, and desired conclusion for Proof by Contradiction and Counterexample.
Problem 4
Repair the first invalid inference, hidden overlap, counterexample, violated graph condition, recurrence assumption, or algorithm invariant; then restate the corrected conclusion.
Problem 5
A strong application defines the computing context, mathematical model, proof or algorithm, correctness evidence, complexity or scale, and limitations.
Problem 6
Use at least two of: independent enumeration, truth table, symbolic equivalence, counterexample search, induction, invariant, alternate graph traversal, brute-force test on small cases, recurrence substitution, or second algorithm.