Skip to main content

Sets, Functions, and Relations Lab

Attempt P6–P12 after clusters 3–4. Allow 7–10 hours. State domains before using a diagram or formula; feedback follows your attempt.

  1. P6 — Set equality. Prove A∖(B∪C)=(A∖B)∩(A∖C) by double inclusion. Use one fixed universe. Contrast ∅ with {∅} and list their power sets.
  2. P7 — Typed functions. Classify x↦x² as R→R and as nonnegative R→nonnegative R. Provide a collision or missing output when relevant and explicit witnesses when claiming surjectivity. Analyze the empty-domain cases too.
  3. P8 — Image versus preimage. Prove preimage preserves intersections. For f(x)=x², S={−1}, T={1}, compute f(S∩T) and f(S)∩f(T). Prove that injectivity restores the image-intersection identity for every pair of subsets.
  4. P9 — Function composition. Prove that injectivity of g∘f forces f injective. Give typed finite maps showing g need not be injective. State a sufficient condition on f's image for g∘f to be defined.
  5. P10 — Relations. Classify the relation on {0,1,2} with pairs (0,0),(1,1), (2,2),(0,1),(1,0),(1,2). Supply a violating witness for each failed property. Compute S∘R for R={(1,2)}, S={(2,3)} with carriers {1},{2},{3}.
  6. P11 — Equivalence. On integers define a~b when 3 divides a−b. Prove all three axioms and explain why the distinct classes partition the integers. Decide whether [a]↦a and [a]↦a mod 3 define functions to the integers, using remainders 0,1,2.
  7. P12 — Partial order. Use divisibility on {2,3,6}. Give all pairs and covers, minimal/maximal elements and least/greatest elements. Explain why the order is not total. Recheck after adding 1 to the carrier.

Tie at least one hand-derived classification to lab output. A report of booleans without the definitions or witnesses is incomplete evidence.