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Question
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
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Solution
Given: A → A, g : A → A are two bijections.
Then, fog : A → A
Surjectivity of fog:
Let z be an element in the co-domain of fog (A).
Now, z ∈ A (co-domain of f) and f is a surjection.
So, z = f(y), where y ∈ A (domain of f) ...(1)
Now, y∈A (co-domain of g) and g is a surjection.So, y=g(x), where x∈A (domain of g) ...(2)
From (1) and (2),
z = f (y) = f (g (x)) = (fog) (x), where x ∈ A (domain of fog)
So, fog is a surjection.
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