Indirect Proof

Abstract

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• When Direct Proof is hard to derive, we can try indirect proof

Disproof by Counterexample (反例法)

• An example that shows that a Mathematical Statement isn’t always true

Important

Useful for disproving Universal Statement

Example

• Disproof the product of two irrational numbers is always irrational.

Proof by Contradiction (矛盾证明法)

• An Indirect Proof, to prove a mathematical statement is false, prove its negation is true, and vice versa. This is useful when direct proof is difficult, and the negated statement has a form that is easier to prove.

Example

• Proof square root of 2 is irrational
• There isn’t a greatest integer

Proof by Contraposition (逆否命证明法)

• Use when the Contrapositive (逆否命题) is easier to proof