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L09_Inference - COMP170 Discrete Mathematical Tools for...

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1-1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 3.3, pp. 117-124 Inference Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen
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2-1 3.3 Inference Direct Inference (Modus Ponens) Rules of Inference for Direct Proofs Contrapositive Rule of Inference Proof by Contradiction What is a Proof?
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3-1 What is a Mathematical Proof , really ?
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3-2 What is a Mathematical Proof , really ? In this section we will introduce various tech- niques used to develop mathematical proofs.
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3-3 What is a Mathematical Proof , really ? In this section we will introduce various tech- niques used to develop mathematical proofs. Some of these techniques will actually be variations on similar ideas (so don’t get con- fused if they look similar to each other).
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3-4 What is a Mathematical Proof , really ? In this section we will introduce various tech- niques used to develop mathematical proofs. Some of these techniques will actually be variations on similar ideas (so don’t get con- fused if they look similar to each other). We start by examining a simple mathematical proof and its components
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4-1 Prove that if m is even, then m 2 is even.
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4-2 Prove that if m is even, then m 2 is even. Let m be an integer.
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4-3 Prove that if m is even, then m 2 is even. Let m be an integer. Suppose that m is even.
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4-4 Prove that if m is even, then m 2 is even. Let m be an integer. Suppose that m is even. If m is even, then k with m = 2 k .
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4-5 Prove that if m is even, then m 2 is even. Let m be an integer. Suppose that m is even. If m is even, then k with m = 2 k . Then k such that m = 2 k .
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4-6 Prove that if m is even, then m 2 is even. Let m be an integer. Suppose that m is even. If m is even, then k with m = 2 k . Then k such that m 2 = 4 k 2 . Then k such that m = 2 k .
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4-7 Prove that if m is even, then m 2 is even. Let m be an integer. Suppose that m is even. If m is even, then k with m = 2 k . Then k such that m 2 = 4 k 2 . Then, there is an integer h = 2 k 2 s.t. m 2 = 2 h . Then k such that m = 2 k .
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4-8 Prove that if m is even, then m 2 is even. Let m be an integer. Suppose that m is even. If m is even, then k with m = 2 k . Then k such that m 2 = 4 k 2 . Then, there is an integer h = 2 k 2 s.t. m 2 = 2 h . Thus, if m is even, then m 2 is even. Then k such that m = 2 k .
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5-1 3.3 Inference Direct Inference (Modus Ponens) Rules of Inference for Direct Proofs Contrapositive Rule of Inference Proof by Contradiction What is a Proof?
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6-1 1) Suppose that m is even. 2) If m is even, then k with m = 2 k . 3) Then k such that m = 2 k . Consider the statements
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6-2 1) Suppose that m is even. 2) If m is even, then k with m = 2 k . 3) Then k such that m = 2 k .
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