Universal Proof Patterns

Intro to Proofs · Axiom Academy

Almost every proof you'll ever write runs on one of a few patterns. Learn to spot which one a claim is asking for, and the proof half-writes itself. A proof has a shape before it has any content A blank page is the scariest part of a proof — until you notice that the claim itself tells you the shape to fill in. "If P , then Q " claims almost always pour into the same skeleton: assume P , do some honest work, arrive at Q . That skeleton is the direct pattern, and most first proofs are just it. Watch it happen in two beats. First the bare shape draws itself — four empty slots strung along the proof's spine, no content yet. Then a fill-point descends and each slot fills in, top to bottom: the assumption, the algebra, the conclusion. Nothing is invented; the claim is just poured into a shape that was waiting for it. The direct pattern: assume the hypothesis, work forward, land on the conclusion. Recognize it and the page stops being blank. When the front door is locked, prove the contrapositive Some claims fight you head-on. "If n^2 is even, then n is even" — try it directly and you're stuck factoring. But every "If P , then Q " has a logically identical twin: "If not Q , then not P ." Drag the dial to flip the claim and watch the same statement turn into one you can just compute. The two statements are equivalent — proving either one proves the claim. The contrapositive pattern just picks whichever door opens. Induction: tip one domino, rig the rest to fall

This is the written version of the interactive lesson above. See the full Intro to Proofs course.