Question
Are/and g both necessarily onto, if g o f is onto ?
Solution
Let f : A → B and g : B → C be two functions such that g o f : A → C is defined.
We are given that g o f : A → C is onto. We now prove that g is onto.
Let z ∈C.
Since gof : A → C is onto, so there exists x ∈ A such that (g o f) (x) = z
⇒ g (f (x)) = z
⇒ g (y) = z where y = f(x)
Since a ∈ A and f is a map from A to B ∴ f (x) ∈ B ⇒ y ∈
∴ for given z ∈ C, we have determined y ∈ B such that g (y) = z ∴ g : B → C is onto.
Consider f : {1, 2, 3, 4} → {1, 2, 3, 4} and g : {1, 2, 3, 4} → {1, 2, 3} defined as f (1) = 1, f(2) = 2 , f(3) = f(4) = 3, g (1) = 1, g (2) = 2 and g (3) = g (4) = 3. So, g of is onto but f is not onto.
