# L9 - MATH 135 Fall 2008 Lecture IX Notes If and only if In...

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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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L9 - MATH 135 Fall 2008 Lecture IX Notes If and only if In...

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