12-ProofTechniques

# 12-ProofTechniques - Discussion#12 Deduction Proofs and Proof Techniques Discussion#12 Chapter 1 Sections 1.6.6 1/21 Topics Proofs = Sound

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Discussion #12 Chapter 1, Sections 1.6.6 1/21 Discussion #12 Deduction, Proofs and Proof Techniques

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Discussion #12 Chapter 1, Sections 1.6.6 2/21 Topics Proofs = Sound Arguments, Derivations, Deduction Proof Techniques 1. Exhaustive 2. Equivalence to Truth 3. If and only if (iff) 4. Contrapositive Proof 5. Contradiction (or Indirect) 6. Conditional 7. Case Analysis 8. Induction 9. Direct
Discussion #12 Chapter 1, Sections 1.6.6 3/21 1. Exhaustive Truth Proof Show true for all cases. e.g. Prove x 2 + 5 < 20 for integers 0 x 3 0 2 + 5 = 5 < 20 T 1 2 + 5 = 6 < 20 T 2 2 + 5 = 9 < 20 T 3 2 + 5 = 14 < 20 T Thus, all cases are exhausted and true. Same as using truth tables when all combinations yield a tautology.

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Discussion #12 Chapter 1, Sections 1.6.6 4/21 2. Equivalence to Truth 2a. Transform logical expression to T. Prove: R S ¬ ( ¬ R ¬ S) R S ¬ ( ¬ R ¬ S) R S ¬¬ R ¬¬ S de Morgan’s law R S R S double negation ¬ (R S) (R S) implication (P Q ¬ P Q) T law of excl. middle (P ¬ P) T
Discussion #12 Chapter 1, Sections 1.6.6 5/21 2. Equivalence to Truth (continued…) 2b. Or, transform lhs to rhs or vice versa. Prove: P Q ¬ P Q Q P Q ¬ P Q (P ¬ P) Q distributive law (factoring) T Q law of excluded middle Q identity

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Discussion #12 Chapter 1, Sections 1.6.6 6/21 3. If and only if Proof Also called necessary and sufficient If we have P Q, then we can create two standard deductive proofs, namely, (P Q) and (Q P). i.e. P Q (P Q) (Q P) Transforms proof to a standard deductive proof actually two of them
Discussion #12 Chapter 1, Sections 1.6.6 7/21 3. If and only if Proof: Example Prove: 2x – 4 > 0 iff x > 2. Thus, we can do two proofs: (1) If 2x – 4 > 0 then x > 2. (2) If x > 2 then 2x – 4 > 0. Proof: (1) Suppose 2x – 4 > 0. Then 2x > 4 and thus x > 2. (2) Suppose x > 2.

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## This note was uploaded on 03/02/2012 for the course C S 236 taught by Professor Michaelgoodrich during the Winter '12 term at BYU.

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12-ProofTechniques - Discussion#12 Deduction Proofs and Proof Techniques Discussion#12 Chapter 1 Sections 1.6.6 1/21 Topics Proofs = Sound

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