• Overview
• Details
• Exercises

Core mathematical proof methods build correctness from assumptions to conclusions: direct proofs apply definitions and lemmas, contradiction assumes the opposite to derive impossibility, contrapositive proves an equivalent implication, and induction lifts base facts to infinite families.

• Direct: Show P ⇒ Q by chaining definitions, algebraic steps, and known results.
• Contradiction: Assume P ∧ ¬Q and derive ⊥ (inconsistency), concluding P ⇒ Q.
• Contrapositive: Prove ¬Q ⇒ ¬P; logically equivalent to P ⇒ Q and sometimes simpler.
• Induction: Base case + inductive step; strong induction when the step uses all previous cases.
• Common pitfalls: hidden assumptions, circular reasoning, and incomplete inductive hypotheses.

1. Prove √2 is irrational by contradiction.
2. Show: if a·b is even, then a is even or b is even (contrapositive recommended).
3. Prove by induction: 1 + 2 + ··· + n = n(n + 1)/2.