Equivalence Relations MSU CSE 260 Outline Introduction Equivalence Relations Definition, Examples Equivalence Classes Definition
Equivalence Classes and Partitions Theorems Example Introduction Consider the relation R on the set of MSU students: a R b a and b are in the same graduating class. R is reflexive, symmetric and transitive.
Relations which are reflexive, symmetric and transitive on a set S, are of special interest because they partition the set S into disjoint subsets, within each of which, all elements are all related to each other (or equivalent.) Equivalence Relations Definition. A relation on a set A is called an equivalence relation if it is reflexive,
symmetric, and transitive. Two elements related by an equivalence relation are called equivalent. Example Consider the Congruence modulo m relation R = {(a, b) Z | a b (mod m)}. Reflexive. a Z a R a since a - a = 0 = 0 m Symmetric. a, b Z If a R b then a - b = km. So b - a = (-k) m. Therefore, b R a.
Transitive. a, b, c Z If a R b b R c then a - b = km and b - c = lm. So (a-b)+(b-c) = a-c = (k+l)m. So a R c. R is then an equivalence relation. Equivalence Classes Definition. Let R be an equivalence relation on a set A. The set of all elements related to an element a of A is called the equivalence class of a, and is denoted by [a]R. [a]R = {xA | (a, x) R}
Elements of an equivalence class are called its representatives. Example What are the equivalence classes of 0, 1, 2, 3 for congruence modulo 4? [0]4 = {, -8, -4, 0, 4, 8, } [1]4 = {, -7, -3, 1, 5, 9, } [2]4 = {, -6, -2, 2, 6, 10, } [3]4 = {, -5, -1, 3, 7, 11, }
The other equivalence classes are identical to one of the above. [a]m is called the congruence class of a modulo m. Equivalence Classes & Partitions Theorem. Let R be an equivalence relation on a set S. The following statements are logically equivalent: aRb [a] = [b]
[a][b] Equiv. Classes & Partitions - cont Definition. A partition of a set S is a collection {Ai | i I} of pairwise disjoint nonempty subsets that have S as their union. i,jI Ai Aj = , and iI Ai = S. Theorem. Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S.
Conversely, for any partition {Ai | i I} of S there is an equivalence relation that has the sets Ai as its equivalence classes. Example Every integer belongs to exactly one of the four equivalence classes of congruence modulo 4: [0]4 = {, -8, -4, 0, 4, 8, } [1]4 = {, -7, -3, 1, 5, 9, } [2]4 = {, -6, -2, 2, 6, 10, }
[3]4 = {, -5, -1, 3, 7, 11, } Those equivalence classes form a partition of Z. [0]4 [1]4 [2]4 [3]4 = Z [0]4, [1]4, [2]4 and [3]4 are pairwise disjoint.