23 - LECTURE 23: INVERSES AND PERMUTATIONS ( 9.6-9.7) Let...

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Unformatted text preview: LECTURE 23: INVERSES AND PERMUTATIONS ( 9.6-9.7) Let all things be done decently and in order. -Paul the apostle 1. More on inverses From last time: Theorem 1. Let f : A B be a function. The inverse relation f- 1 is a function from B to A if and only if f is a bijection. Corollary 2. If f- 1 is a function then f- 1 is a bijection. Further, f- 1 f = id A and f f- 1 = id B . Sometimes it is easy to figure out f- 1 because f has only finitely many pairs! In other cases, it might take more work. Example 3. The function f : R-{ 2 } R-{ 3 } given by f ( x ) = 3 x/ ( x- 2) is a bijection. Prove this, and find the inverse. Proof. First we show f is injective. Let x,y R- { 2 } and assume f ( x ) = f ( y ). Then 3 x/ ( x- 2) = 3 y/ ( y- 2). Clearing denominators, we have 3 xy- 6 x = 3 xy- 6 y . So- 6 x =- 6 y . Thus x = y . Next we show that f is surjective. Let b R- { 3 } . We want to find a R- { 2 } so that f ( a ) = b ....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

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23 - LECTURE 23: INVERSES AND PERMUTATIONS ( 9.6-9.7) Let...

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