Rayat Shikshan Sanstha’s S.M.Joshi College, Hadapsar -28 Department of Mathematics Power Point Presentation Topic – Equivalence Class Prof. Darekar S.R

Introduction Equivalence Relations Equivalence Classes Equivalence Classes and Partitions

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]  

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,jI Ai  Aj = , and iI 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.

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