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Suppose that f: X rightarrow Y and g: Y rightarrow Z are surjections. Prove that the composite g
f: X rightarrow Z is a surjection.
Solution
Let z be in Z
Since g is surjection there is a y in Y so that
g(y)=z
Since f is a surjection so there is x in X so that f(x)=y
Hence, g(f(x))=g(y)=z
ie (gof)(x)=z
But z is arbitrary element in Z
So for any z in Z there exists x in X so that
(gof)(x)=z
Hence, gof is a surjecxtion

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

  • 1. Suppose that f: X rightarrow Y and g: Y rightarrow Z are surjections. Prove that the composite g f: X rightarrow Z is a surjection. Solution Let z be in Z Since g is surjection there is a y in Y so that g(y)=z Since f is a surjection so there is x in X so that f(x)=y Hence, g(f(x))=g(y)=z ie (gof)(x)=z But z is arbitrary element in Z So for any z in Z there exists x in X so that (gof)(x)=z Hence, gof is a surjecxtion