## Overview

**Introduction:** In order to understand the concept of relations, let’s consider the following example:

1. 100 is greater than 99

2. 9 is divisible by 3

3. Jaipur is the capital of Rajasthan

In each of the sentence, there is a relationship between two objects.

Hence, we can define the relation as below-

We define a relation in terms of ordered pairs-

An order pair of elements of x and y, where x is the first element and y is the second element, we denote it by (x, y),

(x, y) = (p, q)

If and only if x = p and y = q

Thus unless x = y

**Product sets-**

Suppose we have two sets X and Y. the set of all ordered pairs (x, y) where is called the Cartesian product of X and Y.

We represent it as follows-

And we read it as X cross Y.

**Relation-**

Let X and Y are two sets, A relation from X to Y is a subset of .

Now suppose R is a relation form X to Y, then R is a set of ordered pairs where each first element comes from X and each second element comes from Y.

For each pair , one the following rule is true-

The domain of a relation R is the set of all first elements of the ordered pairs which belongs to R, and range is the second element.

**Pictorial representation of relations-**

Let A = {1, 2, 3}, B = {a, b, c} and R ={(1, a), (1, b), (2, c)}

we represent this relation in picture format as-

**Inverse relation-**

Let R be any relation from set X to set Y. the inverse of R is the relation form Y o X which consists of those ordered pairs which, when reversed, belong to R, such that-

Example- If R = {(1, x),(1, y),(2, z)} then

**Composition of relations-**

Let X, Y and Z are three sets and R be a relation from X to Y and S is a relation from Y to Z.

So that R is a subset of and S is a subset of .

Then R and S give a relation form X to Z which is denoted by RoS,

And can be defined as follows-

The relation RoS is called the composition of R and S.

**Note- If R is a relation on a set A, that is, R is a relation from a set A to itself. Then the composition of R is RoR. Denoted by R^2 .**

**Types of relations-**

**1. Reflexive relations **

A relation R on a set X is reflexive if xRx for every x **∈** X. that is is (x, x) for every x **∈ **X. thus R is not reflexive if there exists x belongs to X such that (x, x) does not belongs to R.

**2. Symmetric and anti-symmetric relations-**

A relation R on a set X is said to be symmetric if whenever xRy then yRx, that is if whenever (x, y)∈R then (y, x)∈R

R is said to be anti-symmetric if (x, y)∈X such that (x, y)∈R but (y, x) does not belong to R.

**3. Transitive relation-**

A relation R on a set X is said to be transitive if whenever xRy and yRzthen xRz, that is, if whenever (x, y), (y, z)∈R then (x, z)∈R

**Equivalence relation-**

A relation in a set R is said to be an equivalence relation in A, If R is reflexive, symmetric and transitive.

**Irreflexive relation**

A relation R on a set A is irreflexive if aRafor every a ∈A

Example-

Let A = {1, 2, 3} and

R = {(1, 2), (2, 3), (3, 1), (2, 1)}

Then the relation R is irreflexive on A.

**Asymmetric relation-**

A relation R defined on a set A is asymmetric if whenever aRb, then .

Example:

Let A = {a, b, c} and R = {(a, b), (b, c)} be a relation on A. then R is a symmetric.

**Compatible relation-**

A relation R in A is said to be a compatible relation if it is reflexive and symmetric.

If R is an equivalence relation on A, then R is compatible relation on A.

**Universal relation-**

A relation R in a set A is said to be universal relation if

R = A × A

Example:

Let A = {1, 2, 3}, then

R = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

is a universal relation on A.

Complementary relation-

Let R be a relation from A to B, then the complement of R denoted by R′and is

Expressed in terms of R as follows;

aRb if

The inverse of a relation-

Let R be a relation from A to B. Then the relation from B to A is called the inverse of R.

**Equivalence relations**

**Definition-**

A relation R in a set A is said to be an equivalence relation in A, if R is reflexive, symmetric and transitive.

Example- Let A = {a, b, c}, and R = {(a, a), (a, b), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)} then R is an equivalence relation in A.

**Example: Let Z denote the set of integers and the relation R in Z be defined by “aRb” iffa – b is an even integer”. Then show that R is an equivalence relation.**

**Sol.**

1. R is reflexive; since

∅ = a −a is even, hence aRafor every a∈Z.

2. R is symmetric:

If a – b is even then b – a = – (a – b) is also even hence aRb⇒bRa

3. R is transitive: for if aRband bRcthen both a – b and b – c are even.

Consequently, a – c = (a – b) + (b – c) is also even.

∴aRband bRc⇒aR c

Thus, R is an equivalence relation.

**Transitive closure of relations**

Let R be a relation on the set A. R1 denote the transitive extension of R, R2 denote the transitive extension of and in general denote the transitive extension of then the transitive closure of R is defined as the set union of

It is denoted by

Thus