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 = {x  A | (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: –a R b –[a] = [b] –[a]  [b]  

Equiv. Classes & Partitions - cont Definition. A partition of a set S is a collection {A i | i  I} of pairwise disjoint nonempty subsets that have S as their union. –  i,j  I A i  A j = , and  i  I A i = 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 {A i | i  I} of S there is an equivalence relation that has the sets A i 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.