Practice: Rewrite & Convert Proofs

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

A powerful exercise in learning proof techniques is rewriting the same proof using different logical strategies.

Try converting:

A direct proof → a contrapositive proof.
A contrapositive proof → a contradiction proof.
A contradiction proof → a direct proof (when possible).

Example:

If n is odd, then n² is odd.

Direct proof: n = 2k + 1 → n² = 4k² + 4k + 1 → odd.
Contrapositive: If n² is even → n is even.
Contradiction: Assume n² is even and n is odd → contradiction in algebra.

Rewriting proofs strengthens your understanding of logical equivalence and proof structure.

TL;DR — key idea

Rewriting proofs in different forms reveals logical structure and improves versatility in mathematical problem solving.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
Rewrite ONE of your previous proofs from Direct Proofs or Logic in: 1. Contrapositive form 2. Contradiction form Then compare the three versions and reflect: which method produced the clearest argument?

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.