College Algebra Exam Review 79

College Algebra Exam Review 79 - 89 2.1. FIRST RESULTS but...

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Unformatted text preview: 89 2.1. FIRST RESULTS but by the associative law, the first two and the last two are equal. Thus there are at most three different product of four elements: a.bcd /; .ab/.cd /; .abc/d: Using the associative law, we see that all three are equal: a.bcd / D a.b.cd // D .ab/.cd / D ..ab/c/d D .abc/d: Thus there is a well-defined product of four elements, which is independent of the way the elements are grouped for multiplication. There are 14 ways to group five elements for multiplication; we won’t bother to list them. Because there is a well-defined product of four or less elements, independent of the way the elements are grouped for multiplication, there are at most four distinct products of five elements: a.bcde/; .ab/.cde/ .abc/.de/; .abcd /e: Using the associative law, we can show that all four products are equal, a.bcde/ D a.b.cde// D .ab/.cde/; etc. Thus the product of five elements at a time is well-defined, and independent of the way that the elements are grouped for multiplication. Continuing in this way, we obtain the following general associative law: Proposition 2.1.19. (General associative law) Let M be a set with an associative operation, M M ! M , denoted by juxtaposition. For every n 1, there is a unique product M n ! M , .a1 ; a2 ; : : : ; an / 7! a1 a2 an ; such that (a) The product of one element is that element .a/ D a. (b) The product of two elements agrees with the given operation .ab/ D ab . (c) For all n 2, for all a1 ; : : : an 2 M , and for all 1 Ä k Ä n 1, a1 a2 an D .a1 ak /.ak C1 an /: Proof. For n Ä 2 the product is uniquely defined by (a) and (b). For n D 3 a unique product with property (c) exists by the associative law. Now let n > 3 and suppose that for 1 Ä r < n, a unique product of r elements exists satisfying properties (a)-(c). Fix elements a1 ; : : : an 2 M . By the induction hypothesis, the n 1 products pk D .a1 ak /.ak C1 an /; ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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