L09_Inference

L09_Inference - 1-1COMP170Discrete Mathematical Toolsfor...

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Unformatted text preview: 1-1COMP170Discrete Mathematical Toolsfor Computer ScienceDiscrete Math for Computer ScienceK. Bogart, C. Stein and R.L. DrysdaleSection 3.3, pp. 117-124InferenceVersion 2.0: Last updated, May 13, 2007Slidesc2005 by M. J. Golin and G. Trippen2-13.3 Inference•Direct Inference (Modus Ponens)•Rules of Inference for Direct Proofs•Contrapositive Rule of Inference•Proof by Contradiction•What is a Proof?3-1What is aMathematical Proof, really?3-2What is aMathematical Proof, really?•In this section we will introduce various tech-niques used to develop mathematical proofs.3-3What is aMathematical Proof, really?•In this section we will introduce various tech-niques used to develop mathematical proofs.Some of these techniques will actually bevariations on similar ideas (so don’t get con-fused if they look similar to each other).3-4What is aMathematical Proof, really?•In this section we will introduce various tech-niques used to develop mathematical proofs.Some of these techniques will actually bevariations on similar ideas (so don’t get con-fused if they look similar to each other).•We start by examining a simple mathematicalproof and its components4-1Prove that ifmis even, thenm2is even.4-2Prove that ifmis even, thenm2is even.Letmbe an integer.4-3Prove that ifmis even, thenm2is even.Letmbe an integer.Suppose thatmis even.4-4Prove that ifmis even, thenm2is even.Letmbe an integer.Suppose thatmis even.Ifmis even, then∃kwithm= 2k.4-5Prove that ifmis even, thenm2is even.Letmbe an integer.Suppose thatmis even.Ifmis even, then∃kwithm= 2k.Then∃ksuch thatm= 2k.4-6Prove that ifmis even, thenm2is even.Letmbe an integer.Suppose thatmis even.Ifmis even, then∃kwithm= 2k.Then∃ksuch thatm2= 4k2.Then∃ksuch thatm= 2k.4-7Prove that ifmis even, thenm2is even.Letmbe an integer.Suppose thatmis even.Ifmis even, then∃kwithm= 2k.Then∃ksuch thatm2= 4k2.Then, there is an integerh= 2k2s.t.m2= 2h.Then∃ksuch thatm= 2k.4-8Prove that ifmis even, thenm2is even.Letmbe an integer.Suppose thatmis even.Ifmis even, then∃kwithm= 2k.Then∃ksuch thatm2= 4k2.Then, there is an integerh= 2k2s.t.m2= 2h.Thus, ifmis even, thenm2is even.Then∃ksuch thatm= 2k.5-13.3 Inference•Direct Inference (Modus Ponens)•Rules of Inference for Direct Proofs•Contrapositive Rule of Inference•Proof by Contradiction•What is a Proof?6-11) Suppose thatmis even.2) Ifmis even, then∃kwithm= 2k.3) Then∃ksuch thatm= 2k.Consider the statements6-21) Suppose thatmis even.2) Ifmis even, then∃kwithm= 2k.3) Then∃ksuch thatm= 2k.Consider the statementsLetp∼(mis even)andq∼(∃kwithm= 2k)6-31) Suppose thatmis even.2) Ifmis even, then∃kwithm= 2k.3) Then∃ksuch thatm= 2k.Consider the statementsLetp∼(mis even)andq∼(∃kwithm= 2k)Then we can rewrite the three statements as6-41) Suppose thatmis even....
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L09_Inference - 1-1COMP170Discrete Mathematical Toolsfor...

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