# dm-lec5-proofs2-s - Discrete Mathematics CS204 Lecture #4...

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Discrete Mathematics CS204 Lecture #4 September 11, 2009 at KAIST 1

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Review Recall that - A proof is a sequence of purely logical arguments from hypotheses to a conclusion. - A long but non-exhaustive list of veriﬁed examples or cases does not count as a proof. - In the reasoning and those arguments, you can assume theorems that are already proved before. - An example which shows the purported statement is false, is called a counterexample. 2
Proof by Contradiction For all propositions p , q , and r , we have the following equivalence. p q ≡ ¬ p q ≡ ¬ ( p ∧ ¬ q ) ( p ∧ ¬ q ) ( r ∧ ¬ r ) Therefore to prove p q , it sufﬁces to show that assuming p and ¬ q at the same causes a contradiction. 3

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Example Theorem 2 is irrational. Proof. Suppose that 2 = p/q for some integers p and q for contradiction. Further assume that at least one of p and q is odd by reducing the fraction if necessary. By taking the squares of both sides, we have
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## This note was uploaded on 02/04/2010 for the course COMPUTER S Cs206 taught by Professor Unkown during the Spring '08 term at Korea Advanced Institute of Science and Technology.

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dm-lec5-proofs2-s - Discrete Mathematics CS204 Lecture #4...

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