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CHAPTER 5: EQUIVALENCE RELATIONS AND EQUIVALENCE
CLASSES
Section 5.1: Equivalence Relations
Relations
Examples of relations on the set of real numbers include "=", "<", and "?". Examples of
relations on P(R), the power set of R, include "=" and "?".
Definition 1:Arelation on a set S is subset of S × S.
Comments: At first glance, there appears to be a disconnect between the examples of
relations given above and the definition of a relation. To make the connection, consider the
relation "<"onR. Technically, "<" is a subset of R×R. For instance, (1, 2) ?<. Our
practice, however, is to write 1 < 2, and we will continue that practice, even in the
abstract.
If S is a set, we will use the symbol "similarequal" to denote either an abstract relation or a specific
relation for which there is no standard notation. For a, b ? S we will write a similarequal b,not
(a,b) ?similarequal, to indicate that a and b are related.
Definition 2:Letsimilarequal be a relation of a set S.Wesaythatsimilarequal is reflexiv...

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