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# L9W09 - MATH 135 Lecture IX Notes Winter 2009 If and only...

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MATH 135 Winter 2009 Lecture IX Notes If and only if In mathematics, we often see statements of the form “ A if and only if B ” ( A B ). (See Assignment 1.) This means “(If A then B ) and (If B then A )”. The parentheses are here for mathematical reasons, not English language ones! Sometimes we say “The truth of A is equivalent to the truth of B ” or “ A is equivalent to B ”, since if A B has been proven then if A is TRUE, B is TRUE, and if A is FALSE, B cannot be TRUE (otherwise A would be). Can you see why? To prove these statements, we have two directions to prove, since there are two “If...then...” state- ments that must be proven to be TRUE. Example Suppose x, y 0. Then x = y if and only if x + y 2 = xy . Proof If x = y 0, then x + y 2 = 2 x 2 = x and xy = x 2 = x (since x 0) so x + y 2 = xy . If 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 . Example In ABC , b = c cos( A ) if and only if C = 90 .

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L9W09 - MATH 135 Lecture IX Notes Winter 2009 If and only...

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