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

2. Introduction
Equivalence Relations
Equivalence Classes
Equivalence Classes and Partitions

3. 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.)

4. 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.

5. 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.

6. 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.

7. 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.

8. 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]  

9. 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.

10. 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.

11. Thank You