Question 3.42: Given mappings f :A → B, g : B → C, h : C → D, suppose that ......

Algebra Through Practice: Volume 1, Sets, Relations and Mappings: A Collection of Problems in Algebra with Solutions [1546655]

Given mappings f : A → B, g : B → C, h : C → D, suppose that g o f and h o g are bijections. Prove that f,g, h are all bijections.

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Since g o f is a bijection we have that g is surjective; and since h o g is a bijection we have that g is injective. Thus g is a bijection. Since g o f=k where k is a bijection, we thus have that $f=g^{-1}$ o k is also a bijection. Likewise, since h o g = m where m is a bijection, we have that $h=m\circ g^{-1}$ is also a bijection.

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