Problem 1
Define the objects precisely, apply the governing rule for Algorithm Correctness with Preconditions, Postconditions and Invariants, show each step, and verify the conclusion with a second representation, proof, or algorithmic check.
Faculty solution guide
Algorithm Correctness with Preconditions, Postconditions and Invariants
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 Algorithm Correctness with Preconditions, Postconditions and Invariants.
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.