Equivalence Relations

1. Equivalence Relations

2. Equivalence Relations • A relation R on a set A is an equivalence relation if it is reflexive, symmetric, and transitive • a~b denotes that a and b (a,b∈ A) are equivalent elements with respect to R • Intuition: an equivalence relation ties together elements into groups such that all are symmetrically interrelated (each to all within the group)

3. Equivalence Relations • Example: Let R be the relation on real numbers such that aRbiff a–b is an integer • a–a=0 is an integer for all values of a so R is reflexive • For any aRb, a–b is an integer, b–a is also an integer and hence bRa holds. So R is symmetric • If aRb and bRc then a–b and b–c are integers. Therefore (a–b) + (b–c) = a–c is also an integer. Hence aRc holds. So R is transitive • Consequently, R is an equivalence relation

4. Equivalence Classes • Let R be an equivalence relation on a set A. The equivalence class of a is the set of all elements that are related to a (and also to each other because of symmetry and transitivity) • Notation: [a]R is the equivalence class of a with respect to R • [a]R= {s | (a,s) ∈ R} • Example: Let R be the relation on integers such that aRbiffa=b or a=–b • [a] = {a, –a} • …, [–2]={–2,2}, [–1]={–1,1}, [0]={0}, [1]={1,–1}, [2]={2,–2}, …

5. Equivalence Classes and Partitions • An equivalence relation on a set A partitions A into disjoint subsets, such that: • Example: A = {1, 2, 3, 4, 5, 6} The following sets form a partition of A: A1 = {1, 2, 3}, A2 = {4, 5}, A3 = {6}