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.

(Hint: Consider f(x) = x and g(x) =)

Answer 1

Define f: N → Z as f(x) = x and g: Z → Z as g(x) =.

We first show that g is not injective.

It can be observed that:

g(−1) =

g(1) =

∴ g(−1) = g(1), but −1 ≠ 1.

∴ g is not injective.

Now, gof: N → Z is defined as.

Let x, y ∈ N such that gof(x) = gof(y).

⇒

Since x and y ∈ N, both are positive.

Hence, gof is injective