Direct Proof Summary

Intro to Proofs · Axiom Academy

A review of how direct proof gives the most straightforward path from hypothesis to conclusion through logical deduction. Most straightforward approach. Direct proof is the simplest, most natural technique when applicable — always try it first. Clarity through structure. The linear progression from hypothesis to conclusion makes a direct proof easy to follow and verify. Definitions are key. Success often hinges on unpacking definitions to expose the underlying mathematical structure. Cases provide completeness. When the argument splits into scenarios, systematic case analysis ensures nothing is missed. Core Concept What Is a Direct Proof? A direct proof starts from the hypothesis, applies definitions and known results through logical steps, and arrives at the conclusion — an unbroken chain of reasoning with no contradiction or detour. Structure: assume P (the hypothesis) is true, then use valid inferences to show Q (the conclusion) must follow. When to use: reach for it first whenever a clear logical path runs from hypothesis to conclusion. Core Concept The Direct Proof Method Four moves carry you from assumption to conclusion in order: State assumptions — open with "Let" or "Assume"; make every variable explicit. Unpack definitions — translate key terms into their formal definitions to get concrete objects. Apply known results — use axioms, theorems, and algebraic properties to transform statements.

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