The Direct Proof Method

Intro to Proofs · Axiom Academy

LESSON The Direct Proof Method Assume the hypothesis, apply definitions and known results, and walk a justified chain of steps straight to the conclusion. Every direct proof follows the same shape. You assume the hypothesis is true, then apply definitions, axioms, and theorems to transform it — one justified step at a time — until you arrive at the conclusion. Each stage rests on the one above it. Assume: state that the hypothesis P is true Apply: use definitions, axioms, and known results Deduce: make logical steps that follow from what came before Conclude: arrive at the conclusion Q Think of a direct proof as a chain where each box holds a statement and each arrow is a logical implication. The proof is valid only if every statement is forced by the ones before it — no jumping ahead, no skipping links. Definitions: unpacking what mathematical terms mean Axioms: fundamental truths we accept without proof Theorems: previously proven results we can reuse Algebraic manipulation: valid operations on equations Logical inference: rules like modus ponens Direct proofs lean on a few moves that become second nature with practice. Two of the most important are choosing an arbitrary element and unpacking a definition . To prove something "for all," pick one element with no special properties and prove it for that element — what holds for it holds for every one.

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