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Independent Module Checkpoint

This checkpoint contains questions only. Allow 2–3 hours for an unaided first attempt after practice, then a separate review session. This time is part of the assessment allowance. Calculators or the finite lab may check examples afterward; they cannot replace the requested general proofs. Disclose any assistance. Use assessment for scoring and remediation and open the separate key only after saving your attempt.

  1. A1 — Logic. Over integers, negate ∀x ∃y: y>x and y is even, and prove the original. Explain why the witness can depend on x. Decide whether ∃y ∀x: y>x and y is even holds. Include the both-domains-empty case.
  2. A2 — Proof construction. Prove, for integers a,b, that a+b is odd iff exactly one of a,b is odd. Make the case coverage and both directions clear. Refute the similar claim “ab even implies both a and b even.”
  3. A3 — Sets. Prove (A∪B)∖C=(A∖C)∪(B∖C) for arbitrary sets in one universe. Explain why proving one inclusion alone leaves a gap. Supply an example of strict inclusion between two unequal sets.
  4. A4 — Functions. Let A={0,1,2}, B={10,11}, C={20,21}, with f values 10,11,11 in A order, and g(10)=20,g(11)=21. Classify f,g,g∘f, compute f⁻¹({11}), and justify every classification. Prove in general that if g∘f is surjective onto C then g is surjective onto C. Does f have to be surjective onto B? Give a typed counterexample if not.
  5. A5 — Relations. On {0,1,2}, let R contain all self-pairs and (0,1),(1,2). Classify all four properties with evidence. Compute R∘R. Distinguish R, its positive-length closure and its reflexive-transitive closure for this fixture; explain why the two closures need not agree for another fixture.
  6. A6 — Quotients and orders. On {−2,−1,0,1,2}, use equal absolute value as equivalence. Prove the axioms and list the classes. Does [x]↦x define a function? Does [x]↦x²? Order classes by their absolute values; identify covers, minimal/maximal and least/greatest classes and justify well-definedness.
  7. A7 — Induction. Prove sum(k=0..n) 2^k = 2^(n+1)−1 for n≥0. Then prove every n≥8 is 3a+5b for nonnegative integers a,b. State bases and the exact hypothesis scope. Explain why a decreasing positive real measure does not suffice to prove a loop terminates.
  8. A8 — Structural proof and execution. Prove that mirroring a finite full binary tree twice returns the original tree, defining mirror on each constructor. Explain why this theorem alone does not prove a recursive Python implementation runs successfully on a 1,500-level tree. Give a way to handle that fixture, and distinguish tree occurrences from shared-object identities.