6 - Examples . If the units digit of n is 5, then n is...

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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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6 - Examples . If the units digit of n is 5, then n is...

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