College Algebra Exam Review 8

College Algebra Exam Review 8 - 18 1 ALGEBRAIC THEMES 1 3 2...

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18 1. ALGEBRAIC THEMES 1 2 3 4 5 6 7 3 2 1 7 4 5 6 1 2 3 4 5 6 7 4 3 1 2 6 5 7 D 1 2 3 4 5 6 7 7 1 3 2 5 4 6 : The set of permutations of n identical objects shares the following properties with the set of symmetries of a geometric figure: 1. The multiplication of permutations is associative. 2. There is an identity permutation e , which leaves each object in its original position. The product of e with any other permutation , in either order, is . 3. For each permutation , there is an inverse permutation 1 , which “undoes” . For all i; j , if moves an object from posi- tion i to position j , then 1 moves an object from position j to position i . The product of with 1 , in either order, is e . A slightly different point of view makes these properties even more evident. Recall that a function (or map) f W X ! Y is one to one (or injective ) if f .x 1 / ¤ f .x 2 / whenever x 1 ¤ x 2 are distinct elements of X . 6 A map f W X ! Y is said to be onto (or surjective ) if the range of f is all of Y ; that is, for each y 2 Y , there is an x 2 X
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