Consider functions f and g such that composite g of is defined and is one-one. Are f and g both necessarily one-one.
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 of : A → C is one-one.
We are to prove that f is one-one If possible, suppose that f is not one-one.
there exists x1, x2 ∈ A such that x1 ≠ x2 but f (x1) = f (x2)
But f(x1) = f (x2) ⇒ g (f(x1)) = g (f(x2))
⇒ (gof) (x1) = (gof) (x2)
∴ x1, x2 ∈ A such that x1 ≠ x2 but (gof) (x1) = (gof) (x2)
∴ gof is not one-one, which is against the given hypothesis that g of is one-one our supposition is wrong.
∴ f is one-one.
Consider f : {1, 2, 3, 4} ≠ {1, 2, 3, 4, 5, 6} defined as f (x) = x, ∀ x and g : {1, 2, 3, 4, 5, 6} → {1, 2, 3, 4, 5, 6} as g. (x) = x, for x = 1, 2, 3, 4 and g (5) = g (6) = 5. Then, g o f (x) = x ∀ x, which shows that g o f is one-one. But g is clearly not one-one.
