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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 tha...

Aptitude for ICET, CAT, Campus places - Relations

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 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}
          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}.
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.