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Suppose that f X rightarrow Y and g Y rightarrow Z are bijections.pdf

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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)=(g1C)g1=1C(gf)(f1g1)=(g1C)g1=1C (f1g1)(gf)=(f11A)f=1A(f1g1)(gf)=(f11A)f=1A so gf has an inverse and thus is bijective. Hence proved. In the answer written above, A and C are subscripts.(please note).

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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)=(g1C)g1=1C(gf)(f1g1)=(g1C)g1=1C (f1g1)(gf)=(f11A)f=1A(f1g1)(gf)=(f11A)f=1A so gf has an inverse and thus is bijective. Hence proved. In the answer written above, A and C are subscripts.(please note)

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  • 1. 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)=(g1C)g1=1C(gf)(f1g1)=(g1C)g1=1C (f1g1)(gf)=(f11A)f=1A(f1g1)(gf)=(f11A)f=1A so gf has an inverse and thus is bijective. Hence proved. In the answer written above, A and C are subscripts.(please note)