The Art of Proof

Intro to Proofs · Axiom Academy

A capstone recap of the whole course — from logic and sets to the proof techniques that turn intuition into mathematical certainty. A proof is a finite chain of statements, each justified by a hypothesis, definition, axiom, or previously proven result, ending in the conclusion. The core techniques are direct proof , contrapositive ( ), contradiction , and induction — most theorems yield to one of them. Choosing the method is a skill in itself: the structure of the claim (existence, "for all n ", an inequality, a divisibility) points to the tool. The same logical machinery powers every domain — number theory, combinatorics, abstract algebra, analysis, and topology all rest on it. Proof gives certainty : a theorem proven once is true forever, and understanding why replaces memorizing that . Propositional logic — truth tables and the connectives — fixes the grammar of reasoning so every statement is unambiguously true or false. Predicate logic adds the quantifiers and , letting us speak about infinite domains. Key idea: a precise definition turns a vague concept into an object you can reason about. Watch out for: quantifier order matters — is not the same claim as . Sets are the universal language of mathematics. Element-chasing proofs establish identities like ; power sets, relations, and functions are all built on top. When to use: prove set equality by showing mutual inclusion, and .

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