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Domain And Range Of A Relation

 

Let A and B are two sets and R is a relation from A to B, i.e. R A × B


The set of all the first components of the ordered pairs of the relation R is called the DOMAIN of R. Thus
domain of R = {aA : (a, b)  R for some bB}


The set of all the second components of the ordered pairs of the relation R is called the RANGE of R. Thus, 
range of R = {bB : (a, b)  R for some a A}


Clearly domain of R
A and range of R  B


The set B is called the CO-DOMAIN of R

 

Example :
(i) If A = {1,2,3} and B = {a, b, c} let R = {(1,a) (1,c), (2, b)

Then domain of R = {1, 2} range of R = {a, b, c}


Number of Relations : 
Let A contains m elements and B contains n element. Then A×B contains mn elements. Hence, A×B has 2mn subsets. That is the total number of relations from A to B are 2mn. The relations φ (called a Void Relation) and A × B (called an Universal Relation) are said to be Trivial Relations from A to B.


Inverse Relation : 
The inverse relation of a relation R is the set obtained by reversing each of the ordered pairs of R and is denoted by R–1.

 

Example :
(i) Let A = {1, 2, 3} and B = {a, b, c}
If R = {(1, a), (2, a), (3, b), (3, c)} ⊆ A × B

Then R–1 = {(a, 1), (a, 2), (b, 3), (c, 3)} ⊆ B×A





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