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.

Conditions and key results

Every method must retain the theorem's quantifiers and domain. Contradiction must negate the entire claim correctly; for $\forall x\,P(x)$ the negation is $\exists x\,\neg P(x)$. Examples can suggest a theorem, but only a general argument establishes it; one valid counterexample is enough to refute a universal statement.

A reliable strategy

1. Formalize the claim: identify domain, quantifiers, hypotheses, and exact conclusion.
2. Expand definitions and inventory useful consequences before selecting a method.
3. Choose the method whose starting assumptions expose the needed structure; state those assumptions explicitly.
4. Audit every inference, all cases, and the final quantifier; if the claim is false, present a counterexample instead.

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

Verification and reasonableness

• Translate the final sentence back into the theorem's exact logical form.
• Search for a small counterexample before investing in a universal proof.
• For cases, prove their union is the domain; for contradiction, identify precisely which assumption must therefore be false.

Practice

1. Which method naturally proves 'if $n^2$ is even, then $n$ is even'?
2. How is a universal claim disproved?
3. What must a proof by cases establish about its cases?

Further deduction

Existence and uniqueness are separate obligations. To prove 'there exists exactly one $x$,' first construct or otherwise establish at least one solution, then suppose $x_1$ and $x_2$ both satisfy the property and prove $x_1=x_2$. A formula derived from necessary algebraic conditions may give candidates but not existence until each candidate is checked. Conversely, exhibiting one solution establishes existence but says nothing about whether others occur.

Related topics

• Direct Proof
• Proof by Contradiction
• Mathematical Induction