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Intro to Analysis in-class_Part_3

# Intro to Analysis in-class_Part_3 - 0.2 Logic and inference...

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11 Indirect proof: Proof by contrapositive. ( A = B ) ( ¬ B = ⇒ ¬ A ) , so show ¬ B = ⇒ ¬ A directly. Example 0.2.3 (contrapositive) . 3 n + 2 odd = n odd. The contrapositive is: n even = 3 n + 2 even. 1. Assume n is an even integer. 2. Then n = 2 k , for some integer k , so 3 n + 2 = 3(2 k ) + 2 = 6 k + 2 = 2(3 k + 1) = 2 m, for some m Z . 3. Thus, 3 n + 2 is even. Example 0.2.4 (contrapositive) . n 2 even = n even. This is just the contrapositive of the prev. example. Indirect proof: Proof by contradiction. In order to show that A is true by contradiction, 1. assume that A is false (assume ¬ A is true) 2. derive a contradiction (show that ¬ A implies something which is clearly false/impossible) Example 0.2.5 (contradiction) . 2 is irrational. 1. Assume the negative of the statement: 2 = m n , for some m,n Z . 2. If m,n have a common factor, we can cancel it out to obtain 2 = a b , in lowest terms ( * ) 2 = a 2 b

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Intro to Analysis in-class_Part_3 - 0.2 Logic and inference...

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