Give an example of a function f Z rightarrow N that is injec

Give an example of a function f: Z rightarrow N that is injective but not surjective. Explain why this is a valid example. Attach File

Solution

Suppose f:ABf:AB.

We need to choose a function which is injective, so two distinct numbers will produce distinct results, and to ensure that ff is not surjective we design it in a way that some number will surely not be in the range of the function.

Ex: f(x)=2xf(x)=2x would ensure that only even numbers are produced by ff, so f(x)=1f(x)=1is impossible.

On the other hand, if f(x)=f(y)f(x)=f(y) then 2x=2y2x=2y,

so we can divide by 22 and havex=yx=y.

Therefore f(x)=2xf(x)=2x is injective but not surjective.