Question

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

Answer 1

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