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Unformatted text preview: Examples . “If the units digit of n is 5, then n is divisible by 5.” (TRUE or FALSE?) Converse: “If n is divisible by 5, then the units digit of n is 5.” (TRUE or FALSE?) “If n and n + 1 are both prime numbers, then n = 2.” (TRUE or FALSE?) Converse: “If n = 2, then n and n +1 are both primes.” (TRUE or FALSE?) 1 “If and only if” statements “ A ⇔ B ” (“ A if and only if B ”, “ A is equivalent to B ” ) means “( A ⇒ B ) and ( B ⇒ A )”. Example . Let x,y ≥ 0. Then x = y if and only if x + y 2 = √ xy . Proof . “ ⇒ ” Let x = y ≥ 0. Then x + y 2 = 2 x 2 = x and √ xy = √ x 2 = x, so that x + y 2 = √ xy. 2 “ ⇐ ” Let x + y 2 = √ xy . Then x + y = 2 √ xy ( x + y ) 2 = 4 xy x 2 + 2 xy + y 2 = 4 xy x 2 2 xy + y 2 = 0 ( x y ) 2 = 0 x = y Therefore, x = y if and only if x + y 2 = √ xy . 3 Proof by Contradiction Here we list all possibilities including the one that is to be proved and show that all of the “other” possibilities lead to contradictions.“other” possibilities lead to contradictions....
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This note was uploaded on 02/05/2010 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.
 Spring '08
 ANDREWCHILDS

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