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Math 321 Real Analysis
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Chapter 1 Metric Spaces 1. Equivalence Relations De nition 1.1. A relation on a set S is a subset R S S. If (a; b) 2 R, we write a R b (or just a b). Example 1.2. (1) Let S be the set of members of our class, and let (a; b) 2 R if person a is shorter than person b. (2) Let S = Z and R = f(a; b) 2 Z j b = 2ag. Thus, 4 8 and 8 16. De nition 1.3. A relation on a set S is an equivalence relation if for all a; b; c 2 S, (1) a a (re exivity) (2) a b =) b a (symmetry) (3) a b; b c =) a c (transitivity). Example 1.4. Examples of equivalence relations: (1) a b in Z if a and b have the same parity. (2) a b in Z if a
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