The Proof Toolkit

Intro to Proofs · Axiom Academy

Before you write a single line, you choose your weapon. Learn to read a statement's structure and pick the proof technique that fits — like a detective picking the right tool for the case. Strategy comes before the first line Most proofs don't go wrong in the algebra — they go wrong in the choice . Pick the technique that fits the statement and the proof almost writes itself; pick the wrong one and you grind into a dead end. There are only a handful of techniques, and a statement's shape usually tells you which one to reach for. Watch a single claim — “if n² is even, then n is even” — travel down a decision tree. At each fork the tree asks one question about the statement's structure , lights the branch that fits, and lands on the technique to use. Nothing about the algebra yet — just the route from shape to strategy. The structure of the statement guided us to the technique — no guessing, no grinding. That's the whole skill. Match the statement to its technique Slide through six real statements. For each one, the technique that fits lights up, and a one-line reason tells you why its structure points there. Watch the patterns emerge: “for all n” keeps pointing one way, “this is false” another. The clue is always the structure — a sequence and “for all n”, a negative claim, an “if-then” with a hard conclusion. Read the shape, name the tool. Walk a statement to its technique

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