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03-proof

# 03-proof - CSIS1118 Foundations of Computer Science Methods...

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CSIS1118 oundations of Computer Science Foundations of Computer Science Methods of Proof Hubert Chan ([O1,O2]; Chapters 1.6, 1.7, 4.1) 1

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Logical Inference xample: If Peter did it he will be nervous when he is questioned Peter Example: If Peter did it, he will be nervous when he is questioned. Peter was very nervous when he was questioned. Therefore, Peter did it. Is the “argument” valid? An argument is called valid if whenever all the hypotheses are true, the conclusion is also true. p: Peter did it q: Peter is nervous when he is questioned he argument: The argument: p q q -------- Is (p q) q p a tautology? p No, if p is F, q is T, the proposition is false! 2
Exercise: If horses fly or cows eat plastic, then the mosquito is the national bird. If the mosquito is the national bird, then peanut butter tastes good on hot dogs. But peanut butter tastes terrible on hot dogs. Therefore, cows don’t eat plastic. Does the conclusion “cow don’t eat plastic” follow the arguments? et p denote “horses fly” ) Let p denote horses fly q denote “cows eat plastic” r denote “mosquito is the national bird” s denote “peanut butter tastes good on hot dogs” (p q) r r s s ---------------  q Q: Is it possible that the argument is valid, but the conclusion is not true? 3

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Example 1. If 2000 is divisible by 9, then (2000) 2 is divisible by 9. 2. 2000 is divisible by 9 So, (2000) 2 is divisible by 9 Is the argument valid? Is the statement “(2000) 2 is divisible by 9” true? No, because a false proposition is used in the argument! Q: If the argument is not valid, can the conclusion be true? Yes, then the proof (argument) is wrong!! Example: 1. If 12 is divisible by 3, then 12 is divisible by 4 2. 12 is divisible by 4 So, 12 is divisible by 3 he argument is not valid but 4 ,y The argument is not valid, but the conclusion is true
Some “rule of inference” [Textbook, Table 1, p.66] p (p q) (p q) p [p (p q)] q [ q (p q)]  p All these are tautologies! Do you know how to prove it? [(p q) (q r)] (p r) [(p q)  p] q [(p q) ( p r) (q r) ow about for predicates and quantifiers? How about for predicates and quantifiers? 5

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Universal Instantiation x P(x) Existential Generalization P(c) for some c U Universal Generalization P(c) for any c U ------------------ P(c) if c U -------------------------  x P(x) -------------------------  x P(x) xistential Instantiation U: universe of discourse Existential Instantiation x P(x) ------------------------- P(c) for some c U Example: Every man has two legs. John is a man.
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03-proof - CSIS1118 Foundations of Computer Science Methods...

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