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Unformatted text preview: 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 ....
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This note was uploaded on 06/03/2010 for the course MATH 135 taught by Professor Peter during the Winter '07 term at University of the West.
 Winter '07
 peter
 Math

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