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Prerequisite Diagnostic

Allow 40 minutes. This checks Module 1 skills, not counting formulas or tree theorems that this module will teach. Save an attempt before reading the feedback.

  1. Negate “every x in A has some y in B related to it.” State both domains.
  2. Explain what is needed to prove a function is a bijection.
  3. Prove A∩(B∪C)=(A∩B)∪(A∩C) using membership.
  4. Prove the sum of the first n odd integers is n², including n=0.
  5. Trace nested loops through [0,1] and [2,3]; list the emitted ordered pairs.
  6. Run a short Python script containing a list and a dictionary; record input, expected output and observed output separately.

Feedback and repair​

  1. Some x in A has no related y in B: ∃x∈A ∀y∈B ¬R(x,y). Repair quantifier order in Module 1's logic cluster if the witness changes with y.
  2. A total map with a declared domain/codomain must be injective and surjective. Two inverse constructions can prove both; mere equal sizes are insufficient.
  3. Membership on either side means x∈A and either x∈B or x∈C. Explain both directions or give a chain of genuine equivalences.
  4. Empty sum is 0. Assuming n², add 2n+1 to obtain (n+1)². Review Module 1 induction if the hypothesis or base is missing.
  5. (0,2),(0,3),(1,2),(1,3) when the first list is the outer loop.
  6. Any reproducible example suffices. Repair terminal/data-structure fluency before labs; a mathematical lesson need not wait for environment repair.

Use the previous module for targeted remediation. Reattempt each missed skill with different inputs. There is no total-score shortcut around a missing proof prerequisite.