ABSTRACT ALGEBRA “EQUIVALENCE RELATION”

May 28th, 2015
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A relation on a set S is called an equivalence relation if, for all a, b, c S, it satisfies:(a) a-reflexibity (b) a-b implies that b-a (symmetry) (c) a-b, b-c implies that a-c (transitivity)

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Abstract Algebra Notes Version 04Spring, 2012 1. PRELIMINARIESConsider the binary operation defined on the set as follows.Figure 1. Cayley table for a binary operationThis Cayley table defines the product and the product . Definition 1. A set is closed under a binary operation iff for all .Definition 2. A binary operation defined set is associative iff for all .Note that the set is closed under a binary operation defined by the table since the interior of the table contains only symbols from . In addition, if we wanted to verify that the given operation is associative, then we would need to verify the identity given in Definition 2, where we could substitute any of the six elements in for , any of the six for , and any of the six for . So, by the fundamental counting principle, the number of identities we would need to verify would be .Its likely intuitive that not all six by six tables would define associative operations. As a matter of fact, the number of ways a table could be completed is determined as follows. If we were to take the table given above, we can observe that there are 6 choices for the definition of the product. Similarly, there are six different choices for each of the other 35 products, or a total of tables. Likewise the number of tables is which, as it turns out is more than the estimated number of atoms in the universe. (Dont believe me? Google it!) Definition 3. A set that is closed under a

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