mathematical reasoning sets and functions Suppose that f X r

mathematical reasoning

sets and functions

Suppose that f: X rightarrow Y and g: Y rightarrow Z are surjections. Prove that the composite g middot f: X rightarrow Z is a surjection.

Solution

Solutions :

Proof :

Let z Z. Since g is a surjection, there exists y Y such that g(y) = z. Since f is a surjection, there exists x X such that y = f(x). Then (g o f)(x) = g(f(x)) = g(y) = z, and so  g o f : X Z is a surjection.