Function Composition Important Questions - with Proof | Oscar Levin | GO Classes

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20. Let f : X → Y and g : Y → Z be functions. We can define the composition of f and g to be the function g ◦ f : X → Z for which the image of each x ∈ X is g( f (x)). That is, plug x into f , then plug the result into g (just like composition in algebra and calculus).
(a) If f and g are both injective, must g ◦ f be injective? Explain.
(b) If f and g are both surjective, must g ◦ f be surjective? Explain.
(c) Suppose g ◦ f is injective. What, if anything, can you say about f and g? Explain.
(d) Suppose g ◦ f is surjective. What, if anything, can you say about f and g? Explain.

Composition of Injective Functions is Injective.
Composition of Surjective Functions is Surjective.
Composition of Bijective Functions is Bijective.
Let g : A → B and f : B → C be functions. Show that if f ◦ g is bijective, then g is one to one and f is onto.

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