07+Proof+Methods

07+Proof+Methods - CS103 HO#7 Proof Methods 4/4/11 Doing...

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CS103 HO#7 Proof Methods 4/4/11 1 Suppose we can prove that q follows from p. There are two ways to think about the proof: p s 1 ... s n q Suppose p is true s 1 ... s n q p q Doing Real Proofs: Example Prove: If n is even, then n 2 is even. That is, Prove: Even(n) Even(n 2 ) Problem: the formula above is not a sentence in FOL, since n appears as a free variable. This is better: x (Even(x) Even(x 2 )) but what is the domain of the quantifier? Doing Real Proofs: Example Prove: If n is even, then n 2 is even. That is, Prove: Even(n) Even(n 2 ) Problem: the formula above is not a sentence in FOL, since n appears as a free variable. What we really mean: Prove: x (Even(x) Even(x 2 )), where the domain is integers So now we need rules of inference for quantifiers. x (P(x) Q(x)) z (Q(z) R(z)) x(P(x) R(x)) Can we just use the rules we have inside the quantifiers? x (P(x) Q(x)) z (Q(z) R(z)) x(P(x) R(x)) No! Example: Statement-reason proof Prove: For all integers, if n is even, then n 2 is even. 1. Let c be an arbitrary integer, and suppose Even(c). 2. k (c = 2k) Def. of Even, 1 3. c 2 = 4k 2 Algebra, 2 4. c 2 = 2m, where m = 2k 2 Algebra, 3 5. Even(c 2 ) Def. of Even, 4 6. x (Even(x) Even(x 2 )) General Conditional Proof, 1 - 5 Example: Statement-reason proof Prove: For all integers, if n is even, then n 2 is even. 1. Let c be an arbitrary integer, and suppose Even(c). 2. k (c = 2k) Def. of Even, 1 3. c 2 = 4k 2 Algebra, 2 4. c 2 = 2m, where m = 2k 2 Algebra, 3 5. Even(c 2 ) Def. of Even, 4 6. x (Even(x) Even(x 2 )) General Conditional Proof, 1 - 5
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CS103 HO#7 Proof Methods 4/4/11 2 Example: Paragraph form Prove: For all integers, if n is even, then n 2 is even. PROOF: Let c be an arbitrary integer, and suppose c is even. We will show that c 2 is even. Since c is even, there exists an integer k such that c = 2k by the definition of even. Thus c 2 = 4k 2 , and c 2 = 2m where m = 2k 2 , which means that c 2 is even. Rules of Inference for Quantified Statements x P(x) P(c) where c is an object in the domain Universal Instantiation P(c) where c is an object in the domain x P(x) Existential Generalization Rules of Inference for Quantified Statements x P(x) P(c) for some object c in the domain, where c is a new name in the proof Existential Instantiation ... x P(x) Give the name c to such an object P(c) ... Rules of Inference for Quantified Statements P(c) where c is an arbitrary object in the domain x P(x) Universal Generalization Let c be an arbitrary object in the domain ... P(c) x P(x) Every child is right-handed or intelligent No intelligent child eats liver There is a child who eats liver and onions There is a right-handed child who eats onions 1. x (R(x) I(x)) All children are right-handed or intelligent 2. x (I(x) ¬L(x)) No intelligent child eats liver 3. x (L(x) O(x)) Some child eats liver and onions 4. L(a) O(a) For some a, Existential Inst., 3
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07+Proof+Methods - CS103 HO#7 Proof Methods 4/4/11 Doing...

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