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

1.	Show that if f : A → B and g : B → C are onto, then gof is also onto.
Answer» 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) (gof) x = g(f(x)) = g(y) ……[From (ii)] = z …..[From(i)] i.e., for every z ∈ C, there is x in A such that (gof) x = z ∴ gof is surjective (onto).

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

Answer»

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)

(gof) x = g(f(x))

= g(y) ……[From (ii)]

= z …..[From(i)]

i.e., for every z ∈ C, there is x in A such that

(gof) x = z

∴ gof is surjective (onto).