INVERSE RELATION

Let R be a relation defined on the set A such that

R = {(a, b) / a, b ∈ A}

Then, the inverse relation R^-1 on A is given by

R^-1 = {(b, a) / (a, b) ∈ R}

That is, in the given relation, if "a" is related to "b", then "b" will be related to "a" in the inverse relation .

Example :

Let R be a relation defined as given below.

R = {(1, 2), (2, 2), (3, 1), (3, 2)}

Find R^-1.

Solution :

R^-1 = {(2, 1), (2, 2), (1, 3), (2, 3)}

Domain and Range of Inverse Relation

Let us consider the relation R such that

R = {(1, 1), (2, 3), (3, 4), (2, 7)}

Then, the domain and range of R :

Domain (R) = {1, 2, 3}

Range (R) = {1, 3, 4, 7}

Find inverse relation R^-1:

R^-1 = {(1, 1), (3, 2), (4, 3), (7, 2)}

Then, the domain and range of R^-1 :

Domain (R^-1) = {1, 3, 4, 7}

Range (R^-1) = {1, 2, 3}

Clearly,

Domain (R^-1) = Range (R)

Range (R^-1) = Domain (R)