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Question
Show that if f : A → B and g : B → C are onto, then g ° f is also onto
Solution
Since g is surjective (onto),
there exists y ∈ B for every z ∈ C such that
g(y) = z ....(i)
Since f is surjective,
there exists x ∈ A for every y ∈ B such that
f(x) = y ....(ii)
(g ° f)x = g(f(x))
= g(y) ...[From (ii)]
= z ...[From (i)]
i.e., for every z ∈ C, there is x in A such that
(g ° f) x = z
∴ g ° f is surjective
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