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09+Slides--More+proofs%2C+SAT+Solving

# 09+Slides--More+proofs%2C+SAT+Solving - CS103 HO#9 More...

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CS103 HO#9 More proofs, SAT Solving 4/6/11 1 CS103 Mathematical Foundations of Computing 4/6/11 Monday Kevin 10 - 12 Gates 200 Bob 3:45 - 5:00 Gates 178 Tuesday Ryan 1:15 - 3:15 ??? Hrysoula 7 - 9 Gates B12 Wednesday Conal 12:15 - 2:15 Gates 200 Bob 3:45 - 5:30 Gates 178 Thursday Evan 2:15 - 4:15 Gates 400 Mingyu 7 - 9 200-201 Sunday Neel 1 - 3 Gates B12 Karl 5 - 7 Gates B12 Office Hours Proof by Contradiction Indirect Proof Suppose ¬Q ... ... ¬S Contradiction: S and ¬S Q S ... ... Trying to prove Q S is a premise or a derived statement Suppose n is prime and n > 2. Prove that n is odd. PROOF by contradiction : Suppose that n is even. Then there is an integer k such that n = 2 k , and since n > 2, k > 1. But that means that n is divisible by 2 and k , and thus n is not prime, contradicting the hypothesis. Thus n must be odd. n is prime n > 2 Suppose n is even Then n = 2 k for some integer k 3, def. of even k > 1 2, 4 algebra n is divisible by 2 and k 4, def. of divides n is not prime 5, 6 def. of prime Contradiction 1, 7 n is odd 3 - 8 1 2 3 4 5 6 7 8 9 So ¬ Q follows from P 1 , …, P n Proof by Contradiction Indirect Proof Suppose that from premises P 1 , …, P n we wish to show ¬ Q . One possibility is to temporarily assume Q , and see if we can show that P 1 , …, P n , Q is impossible , i.e., that these claims can’t be true simultaneously. Then in any circumstances where P 1 , …, P n are true, Q is false ¬ Q is true

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