Question

Show that if f : A → B and g : B → C are onto, then gof : A → C is

also onto.​

Answer 1

Solution: Given an arbitrary element z ∈ C, there exists a pre-image y of z under gsuch that g (y) = z, since g is onto. Further, for y ∈ B, there exists an element x in Awith f(x) = y, since f is onto. Therefore, gof(x) = g (f(x)) = g (y) = z, showing that gofis onto.