Unit-2
Relations, Functions and their graphs
Relation:
Ordered pair: A pair having one element as the first element and another
element is taken as second element is known as ordered pair. Taking “a” as first
element and “b” as second element , the ordered pair is written as ( a, b).
Examples:
( 2, 3) , (-5 , 7), (9 ,6 ) , etc.
Note that (a , b)= ( c, d) if and only if a= c and b= d.
So we have , (a, b) ≠ (b, a).
Cartesian product:
The Cartesian product of two sets A and B , denoted by A×B is defined as ,
A×B = { (a, b) : a ϵ A and b ϵ B}
Similarly , B×A = { (b, a) : a ϵ A and b ϵ B}
Note that, A×B ≠ B×A Also,
Example: Let A ={ a ,b} and B ={ 1,2}. B×B = {(1,1), (1 ,2), (2,1), (2,2)}
A×B = { (a, 1), (a ,2), (b, 1), (b, 2)}
Also, B×A ={ (1, a), (1 ,b), (2, a), (2, b)}
Relation:
Any subset of A×B is known as relation from A to B. Thus, R⊆ A×B.
Examples: Let A×B = { (1,2), (1 ,3 ), (1 ,4) , (2,2) , (2,3 ) , (2 ,4) }
∴ R= { (1,2), (2 ,2) } ,
R= { (2,2),(2,3), (2,4)} are some relations.
Domain of R : The set of the first elements of the ordered pairs of the relation R.
Range of R : The set of the second elements of the ordered pairs of the relation R.
Inverse relation:
Let R = { (a , b) : a ϵ A and b ϵ B} be a relation . Then the inverse relation of R ,
denoted by 𝑅−1 is defined as
𝑅−1 = { (b , a) : a ϵ A and b ϵ B}
Example:
Let R = {(1,4 ) ,(1, 5), (2, 4) ,(2 ,5)}
∴ Domain of R = { 1, 2 }
Range of R = { 4 ,5} and 𝑅 −1 = {(4,1), (5,1) , (4,2), (5,2)}.
