# Relations | Domain and Range of a Relation | Inverse Relation

## What are Relations in Math?

Consider two sets A and B such that A contains a few countries of the world, namely India, Japan, Pakistan, Thailand and France and set B contains their capitals. Thus,

Writing R for the relation ‘is the capital of’ the statement that Tokyo is the capital of Japan can be represented as ‘Tokyo R Japan’. Similarly, New Delhi R India ⇒ New Delhi is the capital of India. Now, omitting the letter R between the pairs of the names one can write them as ordered pairs like (Tokyo, Japan) etc., and the relationship can be written as a set R of ordered pairs. Thus,

Thus, if A and B be two sets, then a relation R from A to B is a subset of A x B i.e., R is a relation from A to B ⇔ R ⊆ A x B.

## Domain and Range of a Relation:

Let R be a relation from a set A to a set B. Then, the set of all the first entries of the ordered pairs belonging to R is called the domain of R and the set consisting of all the second entries of the ordered pairs in R is known as the range of R. The set B in this case is known as the codomain of R.

Consider a non-empty set A. Then, a subset of A x A is also a relation from A to itself and is known as a relation on set A.

## Inverse Relation:

If A and B be two non-empty sets and R be a relation from set A to set B then the inverse of R denoted by R-1, is a relation from set B to set A and is given as-