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handout9 - ECS20 Handout 9 Proof Techniques I 1 Direct...

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ECS20 Handout 9 January 27, 2011 Proof Techniques – I 1. Direct proof . The implication p q can be proved by showing that if p is true, then q must also be true. A proof of this kind is called a direct proof . Example: Prove that “If n is odd, then n 2 is odd” 2. Indirect proof . Since the implication p q is equivalent to its contrapositive ¬ q → ¬ p , the implication p q can be proved by showing that its contrapositive ¬ q → ¬ p is true. This related implication is usually proved directly. An argument of this type is called an indirect proof . Example: Prove that “if 3 n + 2 is odd, then n is odd”. 3. Proof by contradiction . By assuming that the hypothesis p is true, and that the conclusion q is fals, then using p and ¬ q as well as other axioms, definitions, and previously derived theorems, derives a contradiction. Proof by contradiction can be justified by noting that ( p q ) ( p ∧ ¬ q r ∧ ¬ r ) Examples: (a) Prove that 2 is irrational (b) Prove that: For all real numbers x and y
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