College Algebra Exam Review 60

# College Algebra Exam Review 60 - 70 1 ALGEBRAIC THEMES...

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70 1. ALGEBRAIC THEMES Examples we have considered so far are as follows: The set of symmetries of a geometric figure with composition of symmetries as the product. The set of permutations of a (finite) set, with composition of per- mutations as the product. The set of integers with addition as the operation. Z n with addition as the operation. Indeed, Proposition 1.7.7 , parts (a), (b), and (c), says that addition in Z n is associative and has an identity OE0Ł , and that all elements of Z n have an additive inverse. KOExŁ with addition as the operation. In fact, Proposition 1.8.2 , parts (a), (b), and (c), says that addition in KOExŁ is associative, that 0 is an identity element for addition, and that all elements of KOExŁ have an additive inverse. It is convenient and fruitful to make a concept out of the common char- acteristics of these several examples. So we make the following definition: Definition 1.10.1. A group is a (nonempty) set G with a product, denoted here simply by juxtaposition, satisfying the following properties: (a)
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