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L09_Inference_print

# L09_Inference_print - COMP170 Discrete Mathematical Tools...

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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 3.3 Inference Direct Inference (Modus Ponens) Rules of Inference for Direct Proofs Contrapositive Rule of Inference Proof by Contradiction What is a Proof?
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). We start by examining a simple mathematical proof and its components

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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 . 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 .
5 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) 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 Let p ( m is even) and q ( k with m = 2 k ) Then we can rewrite the three statements as 1) p 2) If p then q ( p q ) 3) q
7 Direct Inference (Modus Ponens) Principle 3.3 (Direct inference) From p and p q we may conclude q . Why is this valid? IMPLIES p q p q T T T T F F F T T F F T

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8 Principle 3.4 (Conditional Proof) If by assuming p we may prove q , then the statement p q is true In our example proof we showed that If m is even then m 2 is even . Essentially, we assumed m is even and derived that m 2 is even . In symbols, we showed that ( m is even) ( m 2 is even) .
9 Principle 3.5 (Universal Generalization) If we can prove a statement p ( x ) about x by assuming only that x is a member of our universe, then we can conclude that p ( x ) is true for every member of our universe.

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L09_Inference_print - COMP170 Discrete Mathematical Tools...

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