CBSE Class 12-science Answered

Let R be an equivalence relation.

Reflexive (a,a) R  aA

Symmetric (a,b) R

                  ⇒ (b,a) R

Transitive (a,b), (b,c) R

                 ⇒ ac R

Let R^-1 is the inverse relation of R

Now R^-1 is reflexive

(a,a) R

⇒ (a,a) R^-1

Symmetric

(b,a) R ⇒ (a,b) R^-1

(a,b)  R ⇒ (b,a)  R^-1

(a,b), (b,a) R^-1

So R^-1 is symmetric.

Transitive

(a,b), (b,c), (a,c) R

Now, (b,a) R-1 (a,c) R^-1

(c,b) R^-1

Since R^-1 is symmetric (b,c) R^-1

So, R^-1 is transitive.

Hence R^-1 is an equivalence relation.

Note:

Domain of R = Range of R^-1

Range of R^-1 = Domain of R