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Unformatted text preview: x such that p ( x ) is true. Example: Prove that the following is not valid ﬁrstorder 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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 Winter '07
 YaoyunShi
 Mathematical proof, Proof methods Proof

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