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College Algebra Exam Review 9

College Algebra Exam Review 9 - 1.5 PERMUTATIONS 19 Now a...

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1.5. PERMUTATIONS 19 Now a permutation of n objects can be identified with a bijective func- tion on the set f 1; 2; : : : ; n g ; the permutation moves an object from position i to position j if the function maps i to j . For example, the permutation D 1 2 3 4 5 6 7 4 3 1 2 6 5 7 in S 7 is identified with the bijective map of f 1; 2; : : : ; 7 g that sends 1 to 4, 2, to 3, 3 to 1, and so on. It should be clear, upon reflection, that the multiplication of permutations is the same as the composition of bijective maps. Thus the three properties listed for permutations follow immediately from the corresponding properties of bijective maps. We generally write S n for the permutations of a set of n elements rather than Sym . f 1; 2; : : : ; n g / . It is not difficult to see that the size of S n is D n.n 1/ .2/.1/ . In fact, the image of 1 under an invertible map can be any of the n numbers 1; 2; : : : ; n ; for each of these possibilities, there are n 1 possible images for 2, and so forth. When n is moderately large, the set S n
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