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Lesson subsection

Contrapositive: The Cleanest Technique

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 contrapositive proof is one of the cleanest techniques in mathematics.

To prove:

If P, then Q,

you instead prove:

If not Q, then not P.

These two statements are logically equivalent, meaning proving either one proves the original.

Examples:

  1. If n² is even, then n is even. Contrapositive:

    If n is odd, then n² is odd — easy.

  2. If a·b = 0, then a = 0 or b = 0. Contrapositive:

    If a ≠ 0 and b ≠ 0, then a·b ≠ 0.

The contrapositive is preferred when:

  • The negation is easy to work with.
  • The original statement feels backwards.
  • The hypothesis (P) gives you little to work with, but the negation of the conclusion (¬Q) does.

Contrapositive proofs are common in analysis, algebra, number theory, and logic.

TL;DR — key idea

Contrapositives replace "If P then Q" with the equivalent but often much easier statement "If not Q then not P."

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 each statement into its contrapositive: 1. If x ≥ 5, then x² ≥ 25. 2. If a·b is even, then a is even or b is even. 3. If a sequence converges, then it is bounded. Pick one and write a contrapositive proof for it.

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.