# 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 âŠ† A Ã— 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 = {aâˆˆA : (a, b) âˆˆ R for some bâˆˆB}

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 = {bâˆˆB : (a, b) âˆˆ R for some a âˆˆA}

Clearly domain of R âŠ† A and range of R âŠ† 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 AÃ—B contains mn elements. Hence, AÃ—B has 2

^{mn}subsets. That is the total number of relations from A to B are 2

^{mn}. The relations Ï† (called a Void Relation) and A Ã— 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

^{â€“1}.

**Example :**

(i) Let A = {1, 2, 3} and B = {a, b, c}

If R = {(1, a), (2, a), (3, b), (3, c)} âŠ† A Ã— B

Then R^{â€“1} = {(a, 1), (a, 2), (b, 3), (c, 3)} âŠ† BÃ—A