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}
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 }
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.
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. }
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}
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}
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 2mn. Thus A×B has
2mn 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 2mn.
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}
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
IA={(x,
x): xA}.
A}.
Ex: Let A={2, 4, 6}, then IA={(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=Ø observe theat R=ØÌA×A is a void relation.