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Unformatted text preview: MATH 135 Fall 2008 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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