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Unformatted text preview: 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 cant be true simultaneously. Then in any circumstances where P 1 , , P n are true, Q is false Q is true CS103 HO#9 More proofs, SAT Solving 4/6/11 2 Demonstration of Nonconsequence To show that a conclusion does not follow from a set of premises, we try to find a counterexample : a possible circumstance where the premises are true and the conclusion is false. After stating that "The following is a counterexample", you must then show why that situation satisfies the premises but not the conclusion. And that point, you know that the conclusion does not follow from the premises. P 1 P 2 ... P n Q Demonstration of Nonconsequence To show that a conclusion does not follow from a set of premises: find a possible circumstance where the premises are true and the conclusion is false. SameRow(b, c) SameRow(a, d) SameCol(d, f) FrontOf(a,b) FrontOf(f, c) Demonstration of Nonconsequence We often use a counterexample to show that a universally quantified predicate does not hold: x P(x) is not true if we can find an object b in the domain such that P(b) is false. P Q Suppose P is true ... S Suppose Q is true ... S Then since at least one of P or Q must be true, S follows We can think if these as "Subproofs" Rule of Inference: Proof by Cases If we know P Q is true, and S follows from P , and S follows from Q , then we can infer S Rule of Inference: Proof by Cases (P S) (Q S) ( P S)...
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This note was uploaded on 06/01/2011 for the course EE 103 taught by Professor Plummer during the Spring '11 term at Stanford.
- Spring '11