03Proof_10

03Proof_10 - ENGG1007 Foundations of Computer Science...

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1 ENGG1007 Foundations of Computer Science Methods of Proof Methods of Proof Prof. Francis Chin, Dr SM Yiu (Chapters 1.6, 1.7, 4.1)
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2 ENGG1007 FCS 2 Logical Inference 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? p: Peter did it q: Peter is nervous when he is questioned The argument: p q q -------- p Is (p q) q p a tautology? No, if p is F, q is T, the proposition is false! An argument is called valid if whenever all the hypotheses are true, the conclusion is also true.
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3 ENGG1007 FCS 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? 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?
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4 ENGG1007 FCS 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 The argument is not valid, but the conclusion is true
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5 ENGG1007 FCS Some “rule of inference” [Textbook, Table 1, p.66] p (p q) (p q) p [p (p q)] q [ ¬ q (p q)] →¬ p [(p q) (q r)] (p r) [(p q) ∧¬ p] q [(p q) ( ¬ p r) (q r) p (p q) (p q) p [p (p q)] q [ ¬ q (p q)] →¬ p [(p q) (q r)] (p r) [(p q) ∧¬ p] q [(p q) ( ¬ p r) (q r) All these are tautologies! Do you know how to prove it? How about for predicates and quantifiers?
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6 ENGG1007 FCS x (P(x) Q(x)) P(J) ----------------------- Q(J) x P(x) ------------------ P(c) if c U x P(x) ------------------ P(c) if c U P(c) for some c U ------------------------- ∴∃ x P(x) P(c) for some c U ------------------------- ∴∃ x P(x) Is this argument valid? Yes,
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03Proof_10 - ENGG1007 Foundations of Computer Science...

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