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10a - Theorems Introduction and Proofs Discrete Mathematics...

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Introduction Theorems and Proofs Discrete Mathematics Andrei Bulatov
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Discrete Mathematics – Theorems and Proofs 10-2 Previous Lecture Axioms and theorems Rules of inference for quantified statements Direct proofs
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Discrete Mathematics – Theorems and Proofs 10-3 Methods of Proving – Proof by Contraposition Sometimes direct proofs do not work Prove that if 3 n + 2 is even, then n is also even That is 2200 x (E( 3 x + 2 ) E( x )) Let us try the direct approach: As for the generic value n the number 3 n + 2 is even, for some k we have 3 n + 2 = 2 k . Therefore 3 n = 2( k + 1). Now what? What if instead of 2200 x (E( 3 x + 2 ) E( x )) we prove the contrapositive, 2200 x ( ¬ E( x ) → ¬ E( 3 x + 2 )) ? Definition: n is even if and only if there is k such that n = 2 k
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Discrete Mathematics – Theorems and Proofs 10-4 Methods of Proving – Proof by Contraposition (cntd) So assume that n is odd , that is there is k such that n = 2 k + 1 . Then 3 n + 2 = 3 (2 k + 1) + 2 = 6 k + 5 = 2(3 k + 2) + 1 .
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