**Ordered Pair:**

An ordered pair,
usually denoted by (x, y) is a pair of elements x and y of some sets, which is
ordered in the sense that (x, y) ≠ (y, x) whenever x ≠ y.

Here x is known as the first co-ordinate
and y, as the second coordinate of the ordered pair (x, y).

Ex: The ordered
pairs (1, 2) and (2, 1) though consist of the same elements 1 and 2, are
different because they represent different points in the co-ordinate plane.

**Cartesian product:**

The Cartesian
product of two sets A and B is the set of all those pairs whose first
co-ordinate is an element of A and the second co-ordinate is an element B. The set is denoted by A × B and is read as ‘A
cross B’ or ‘Product set of A and B’. i.e.,

A×B = {(x, y): x A ^ y B}

Ex: Let A={1, 2, 3}, and B={3, 5}

Q A × B ={1, 2, 3} × {3, 5}

={(1, 3}, (1, 5), (2, 3), (2, 5), (3, 3), (3, 5)}

B × A = {(3, 1),
(3, 2), (3, 3), (5, 1), (5, 2), (5, 3)}

Q A × B ≠B × A

**Relations:**

Consider a set,

A=
{1, 2, 3}, so that the Cartesian product set is

A×A={(1, 1), (1,
2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

Consider a subset from this set in
such a way that the first coordinate is ‘less than’ the second coordinate. If
we denote this relation by R, then R is described by

R=
{(x, y) A×A: x<y }

In which (x,
y) is a member of the relation R.

**Relation in a Set:**

A
relation between two sets A and B is a subset of A×B and id denoted by R.

Thus R subset A×B.

We write xRy, if and only if (x, y) R.

xRy is read as ‘x is related to y’

if A=B then we
say that R is a relation in the set A.

Ex: Consider the realtion

R={(x, y) N×N: x>y, N is a set of natural
numbers. }

Obviously, R subset N × N and so R is a relation N.

**Domain and Range of a Relation**

**:**

The
domain D of the relation R is defined as the set of all first elements of the
ordered pairs which belong to R, i.e.,

D={x: (x, y) R, for x A}

The range E of the relation R is
defined as the set of all second elements of the ordered pairs which belong to R,
i.e.,

E={y: (x, y) R, for yB}

Obviously, D subset
A and E subset B

Ex:
Let A={1, 2, 3, 4} and B={a, b, c}, Every subset of A×B is a relation from A to
B. So, if R={(2, a), (4, a), (4, c)|, then the domain of R is the set (2, 4)
and the range of R is the set (a, c).

**Total number of distinct relations from a set A to set B:**

Let the number of elements of A and B
be m and n repsectively. Then the number of elements of A×B is mn. Therefore
the number of elements of the power set of A×B is 2

^{mn}. Thus A×B has 2^{mn}different subsets. Now every subset of A×B is a relation from A to B. Hence the number of different relations A to B is 2^{mn}.**Types of Relations in a Set:**

Consider some special types of relations in a set A.

**Inverse Relation:**

**Let R be a relation from the set A to the set B, then the inverse relation R**

^{-1}from set B to the set A is defined by

{ (b, a) : (a, b) R}

In other words, the inverse relation R

^{-1}consists of those ordered pairs which when reversed belong to R. Thus every relation R from the set A to the set B has an inverse relation R^{-1}from B to A.
Ex:
(i) Let A={1, 2, 3}, B={a, b} and R={(1, a), (1, b), (3, a), (2, b)} be a
relation from A to B. The inverse relation R is

R

^{-1}={(a, 1), (b, 1), (a, 3), (b, 2)}
(ii)
Let B={2, 3, 4}, B={2, 3, 4} and R={(x,y) : |x-y|=1} be realtion from A to B.
That is , R ={ (3,2), (2, 3), (4, 3), (3, 4)}. The inverse relation of R is.

R

^{-1}={(3, 2), (2, 3), (4, 3), (3, 4)}
It may be noted that R=R

^{-1}.
Here
every relation has an inverse relation. If R be a relation from A to B, then

(R

^{-1})^{-1}=R.**Theorem:**

**If R be a relation from A to B, then the domain of R is the range of R**

^{-1}and the range of R is the domain of R

^{-1}.

**Identity Relation:**

**A relation R in a set A is said to be identity relation, generally denoted by IA, if**

I

_{A}={(x, x): xA}.
Ex: Let A={2, 4, 6}, then I

_{A}={(2, 2), (4, 4), (6, 6)} is an identity relation in A.**Universal Relation:**

**A relation R in a set A is said to be universal realtion if R is equal to A×A, i.e., if R = A×A.**

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

R =A×A={(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3,
1), (3, 2), (3, 3)} is a univeral realtion in A.

**Void Relation:**

**A relation R in a set A is said to be a void relation if R is a null set i.e., if R=Ã˜ observe theat R=Ã˜ÃŒA×A is a void relation.**