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

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

Answers (1)

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 \}

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