FINDING INVERSE RELATION FROM THE GIVEN RELATION

Question 1 :

If A  =  {1, 2, 3, 4} B = {2, 4, 6, 8} and f : A-> B given by f(x)  =  2x, then write f and f^-1 as a set of ordered pairs.

Solution :

f(1)  =  2, f(2)  =  4, f(3)  =  6 and f(4)  =  8

f  =  {(1, 2) (2, 4) (3, 6) (4, 8)}

For each elements we have unique images and every elements in B is associated with elements of A. Hence it is bijection.

On interchanging the components of ordered pairs in f, we obtain.

f^-1  =  {(2, 1) (4, 2) (6, 3) (8, 4)}

Question 2 :

Let S  =  {1, 2, 3}. Determine whether the function f:S -> S defined as below have inverse. Find f-1 if it exists.

(i)  f  =  {(1, 1) (2, 2) (3, 3)}

(ii)  f  =  {(1, 2) (2, 1) (3, 1)}

(iii)  f  =  {(1, 3) (3, 2) (2, 1)}

Solution :

(i)  f  =  {(1, 1) (2, 2) (3, 3)}

Clearly f : S -> S is a bijection. So f is invertible and its inverse is given by f^-1  =  {(1, 1) (2, 2) (3, 3)}.

(ii)  f  =  {(1, 2) (2, 1) (3, 1)}

f(2)  =  f(3)  =  1. Therefore f is many to one and hence it is not invertible.

(iii)  f  =  {(1, 3) (3, 2) (2, 1)}

Clearly f : S -> S is a bijection. So f is invertible and its inverse is given by f^-1  =  {(3, 1) (2, 3) (1, 2)}.

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