Provide solution for RD Sharma Maths Class 12 Chapter Functions Exercise 2.2 Question 4.

Answer : $g\; o\; f=\left \{ (a,a),(b,b),(c,c) \right \}\; \text {and}$

$f\; o\; g=\left \{ (u,u),(v,v),(w,w) \right \}$

Hint : For any function to be a bijective, the given function should be one-one and onto.\

Given : $A=\left \{ a,b,c \right \}$

$B=\left \{ u,v,w \right \}$

Solution :

f and g be two functions from A to B and from B to A ; $A\rightarrow B\; \text {and}\; g:B\rightarrow A$

$f=\left \{ (a,v),(b,u),(c,w) \right \}$$f=\left \{ (a,v),(b,u),(c,w)\right \}$

$g=\left \{ (u,b),(v,a),(w,c)\right \}$

For both f and g, Different elements of the domain have different images.

$\therefore$ f and f are onto

Again, for each element in co-domain of f and g, there in a preimage in domain

$\therefore$ f and f are onto

Thus,              f and g are bijectives.

Now,

$g\; o\; f=\left \{ (a,a),(b,b),(c,c) \right \} \text {and}$

$f\; o\; g=\left \{ (u,u),(v,v),(w,w) \right \}$

Hence prove, f and g both are bijections

$g\; o\; f=\left \{ (a,a),(b,b),(c,c) \right \}\; \text {and}$ $f\; o\; g=\left \{ (u,u),(v,v),(w,w) \right \}$