If f : A → A and g : A → A are two bijections, then prove that
(i) fog is an injection
(ii) fog is a surjection
Given f : A → A and g : A → A are two bijections. So, both f and g are one-one and onto functions.
We know (fog)(x) = f(g(x))
Thus, fog is also defined from A to A.
(i) First, we will prove that fog is an interjection.
Let x1, x2ϵ A (domain) such that (fog)(x1) = (fog)(x2)
⇒ f(g(x1)) = f(g(x2))
⇒ g(x1) = g(x2) [since f is one-one]
∴ x1 = x2 [since g is one-one]
So, we have (fog)(x1) = (fog)(x2) ⇒ x1 = x2.
Thus, function fog is an interjection.
(ii) Now, we will prove that fog is a surjection.
Let z ϵ A, the co-domain of fog.
As f is onto, we have y ϵ A (domain of f) such that f(y) = z.
However, as g is also onto and y belongs to the co-domain of g, we have x ϵ A (domain of g) such that g(x) = y.
Hence, (fog)(x) = f(g(x)) = f(y) = z.
Here, x belongs to the domain of fog (A) and z belongs to the co-domain of fog (A).
Thus, function fog is a surjection.