Suppose that f X rightarrow Y and g Y rightarrow Z are bijec

Suppose that f: X rightarrow Y and g: Y rightarrow Z are bijections of sets. Prove that the composite g f: X rightarrow Z is also a bijection and that (g f)^-1 = f^-1 g^-1: z rightarrow Y rightarrow X.

Solution

As they are bijections, they have inverses f1, g1 (i.e. the respective inverses are defined) from B to A and from C to B respectively.

(gf)(f¹g¹)=(g1_C)g¹=1_C(gf)(f¹g¹)=(g1_C)g¹=1_C

(f¹g¹)(gf)=(f¹1_A)f=1_A(f¹g¹)(gf)=(f¹1_A)f=1_A

so gf has an inverse and thus is bijective.

Hence proved.

In the answer written above, A and C are subscripts.(please note)