The Art of Mathematical Proof

Real Analysis · Axiom Academy

LESSON The Art of Mathematical Proof A proof is a chain of certainty from what we assume to what we conclude. See the three classic ways that chain gets built. 1. Direct Proof — Follow the Chain Forward The most natural route. Assume the hypothesis P is true, then advance by valid steps — each justified by a definition, an axiom, or an already-proven theorem — until you reach the conclusion Q . Every link must follow from the one before it. Assume the hypothesis, derive the conclusion Watch the classic direct proof that the angles of any triangle sum to . Draw a line through the apex parallel to the base. The two base angles reappear at the apex as alternate interior angles — and there, beside the top angle, the three together form a straight line. 2. Proof by Contrapositive — Prove the Mirror Image Sometimes is awkward, but its contrapositive is easy. The two are logically identical — so a proof of one is a proof of the other. (Careful: this is the contrapositive, not the converse , which is a different statement.) A statement and its contrapositive are equivalent Why are they the same? Because they agree on every row of the truth table. The animation builds both columns side by side: each row resolves to an identical value, and the bracket turns green when the columns match — all four rows. To prove "if n^2 is even then n is even," the direct route stalls. Its contrapositive — "if n is odd then n^2 is odd " — is one line: , odd. Same theorem, easier road.

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