You've helped them one at a time. The real understanding is how the ideas CONNECT — teach that next.
Contradiction Cassius (proof by contradiction) ×Contrapositive Cara (proof by contrapositive)
Both Cassius and Cara begin by assuming something is NOT true. What do they share?
Contradiction Cassius then says: “Contradiction and contrapositive are just two names for the same proof — always interchangeable.” — is that right?
The link: both are "argue from the negation" techniques — Cara flips the implication, Cassius blows it up.
Two ways to work backwards from "suppose it's false."
Induction Ida (standard induction (assume case n)) ×Strong-Induction Sten (strong induction (assume all cases ≤ n))
Ida and Sten both use a base case and a step. What's the ONE difference?
Induction Ida then says: “Strong induction doesn't need a base case — it assumes everything already.” — is that right?
The link: same base-case-plus-step skeleton — Sten just gets to lean on ALL the earlier rungs, not only the last one.
One ladder, two grips.
Capstone — three claims, three provers
A worksheet has three very different claims. Each needs a DIFFERENT kind of proof — bring in the right cast member for each.
"There EXISTS an even number that is the sum of two primes." (show one)
Who do you bring in?
"1 + 2 + … + n = n(n+1)/2 for EVERY positive n." (a pattern that never ends)
Who do you bring in?
"ALL prime numbers are odd." (a sweeping claim you doubt)
Who do you bring in?
The synthesis: existence wanted a construction, a never-ending pattern wanted induction, and a sweeping false claim fell to one counterexample — the CLAIM tells you which prover to call.
A proof isn't one tool — it's choosing the right one for the shape of the claim.
