lec05-proofs - x such that p x is true Example Prove that...

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EECS 203, Winter 2007 Discrete Mathematics Lecture 5 Proofs January 18 Reading: Rosen [1.6] January 18 Proofs, Page 1
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5.1 Proof methods Direct Proof: to prove a statement ”If p then q .” Start with the hypothesis p , Conclude q . Example: Prove that for any integer n , if n is odd then n 2 is odd. Proof by contraposition: to prove ”If p then q .” Start with the hypothesis ¬ q , Conclude ¬ p . Example: Prove that for any integer n , if n 2 is odd then n is odd. January 18 Proofs, Page 2
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5.1 Proof methods Proof by contradiction: to prove a statement p , Assuming for the purpose of getting a contradiction ¬ p , Obtain a contradiction, Conclude p . Example: Prove that there are infinite number of primes. January 18 Proofs, Page 3
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5.1 Proof methods Proof by cases: to prove ” xp ( x ) . ”, Partition the range of x to several cases and prove the statement for each case. Example: Prove that for any integer n , n ( n + 1) is even. Proof by construction: to prove ” xp ( x )”. Construct a specific
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Unformatted text preview: x such that p ( x ) is true. Example: Prove that the following is not valid first-order formula: ∃ x ( φ-→ ψ )-→ (( ∃ xφ )-→ ( ∃ ψ )) . January 18 Proofs, Page 4 5.1 Proof methods Proof by induction: to prove “For any integer n ≥ 0, P ( n ).” • Prove that P (0) is true. • Prove that for any integer i ≥ 0, P ( i )-→ P ( i + 1). • Conclude that P ( n ) is true for all n ≥ 0. Correctness: by axiom of natural numbers. Example: Prove that for any n ≥ 0, 0 + 1 + 2 + ··· + n = n ( n + 1) / 2. Prove that 2 b ≥ n 2 for all n ≥ 4. January 18 Proofs, Page 5 5.2 Proofs that go wrong . Mistaken : • To prove p : we know p-→ q and q , therefore, p . • To prove ∀ xP ( x ): here are examples of x that P ( x ) is true, thus P ( x ) must be true for all x . January 18 Proofs, Page 6...
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