Methods of Proof

Precise definition

A proof is a finite argument showing that a conclusion follows from definitions, hypotheses, axioms, and established results by valid inference. Different logical forms invite different methods: universal implications often allow direct or contrapositive proof; existential claims require a witness; false universal claims are refuted by one counterexample.

Notation and mathematical language

For $P\to Q$, direct proof assumes $P$ and derives $Q$; contrapositive proof assumes $\neg Q$ and derives $\neg P$; contradiction assumes the claim's negation and derives an impossibility. Induction handles integer-indexed families. Proof by cases partitions the domain into exhaustive, non-overlapping or deliberately overlapping cases whose union covers it.

Conceptual picture

Proof choice is strategy, not a change in standard. A good method exposes available structure: divisibility definitions favour direct algebra, a conclusion saying 'not' may favour contrapositive, and a least or greatest counterexample may enable contradiction or induction.

Fully worked example

Interpretation and application

Proof methods certify algorithms, mathematical models, and claims about all structures in a domain. Outside formal mathematics, the habit of stating assumptions and distinguishing evidence from deduction helps prevent a persuasive example from being mistaken for universal or causal proof.

Common mistakes