CBSE Class 11 Maths Notes: Relations and Functions – Equivalence Class and Relation

| Filed under: 11th Class,Class Notes,Maths |

Equivalence Relation:

A relation R in a set A is called an equivalence relation if

  1. R is reflexive i.e., ≤ a, a) ∈ R, ” a ∈ A
  2. R is symmetric i.e., ≤ a, b) ∈ R ⇒ ≤ b, a) ∈ R
  3. R is transitive i.e., ≤ a, b), ≤ b, c) ∈ R ⇒ ≤ a, c) ∈R

The equivalence relation is usually denoted by the symbol ~.

Equivalence Classes of an Equivalence Relation: 

Let R be equivalence relation in A ≤ ≠ ϕ). Let a ∈ A.

Then the equivalence class of a denoted by [a] or {} is defined as the set of all those points of A which are related to a under the relation R. Thus [a] = {x : x ∈ A, x R a}

It is easy to see that

  1. b ∈ [a] ⇒ a ∈ [b]
  2. b ∈ [a] ⇒ [a] = [b]
  3. Two equivalence classes are either disjoint of identical.

as an example we consider a very important relation

x º y ≤ mod n) iff n divides ≤ x –y),  is fixed positive integer. Consider n = 5 then
[0] = {x : x º 0≤ mod 5)} = {5p : p ∈ z} = {0, ±5, ±10, ±15,….}
[1] = {x : x º 1≤ mod 5)} = {x : x –1 = 5k, k ∈ z} = {5k + 1: k ∈ z} = {1, 6, 11, …., –4, –9,….}
one can easily see that there are only 5 distinct equivalence classes viz. [0], [1], [2], [3] and [4] when n = 5.

Illustration -:

N is the set of natural numbers. The relation R is defined on N x N as follows:
≤ a, b) R ≤ c, d) ↔ a + d = b + c
Prove that R is equivalence relation. 

Solution:

i) ≤ a, b) R ≤ a, b) ↔ a + b = b + a
\ R is reflexive.

ii) ≤ a, b) R ≤ c, d) ⇒ a + d = b + c
⇒ c + b = d + a
⇒ ≤ c, d) R ≤ a, b)
\ R is symmetric.

Now iii) ≤ a, b) R ≤ c, d) and ≤ c, d) R ≤ e, f) ⇒ a + d = b + c & c + f = d + e
⇒ a + d + c + f = b + c + d + e
⇒ a + f = b + e ⇒ ≤ a, b) R ≤ e, f)
\ R is transitive. Thus R is an equivalence relation on N x N.

CBSE Class 11 Maths Notes: Relations and Functions   Equivalence Class and Relation

CBSE Class 11 Maths Notes: Relations and Functions   Equivalence Class and Relation

CBSE Class 11 Maths Notes: Relations and Functions   Equivalence Class and Relation

« Click Here for Previous Topic Click Here for Next Topic »

Click Here to Go Back to CBSE Class 11 Maths Sets Relations and Functions All Topic Notes

Click Here for CBSE Class 11 Maths All Chapters Notes

This page is about:

relations and functions

Tags: , , , , , ,

Stay Updated. Get All Information in Your Inbox. Enter your e-Mail below: