Question
Give examples of two functions f : N → Z and g : Z → Z such that g o f is injective but g is not injective.
Solution
Let f(x) = x, g(x) | x |
where f : N → Z and g : Z → Z
Now g(– 1) = | – | | = 1, g(1) = | 1 | = 1
g is not one-one i.e. injective.
Now f : N → Z and g : Z → Z
∴ g o f : N → Z
Let x1 , x2 ∈ N such that
(g o f) (x1) = (g o f)( x2) ∴ g (f(x1)) = g(f(x2))
∴ g(x1) = g(x2)
∴ |x1| = | x2|
⇒ x1 = x2 [∵ x1 > 0, x2 > 0 ]
∴ (g o f)(x1) = (g o f) (x2) ⇒ x1 = x2 ∴ g o f is onto.
