Science:Math Exam Resources/Courses/MATH220/December 2011/Question 07 (a)/Solution 1

Multiplying by any non-zero integer (other than 1 or -1) will do the trick. For example, multiplying by 2 gives the map

$\begin{array}{rl}f:\mathbb{\mathbb{Z}}& \to \mathbb{\mathbb{Z}}\\ n& \mapsto 2n\end{array}$

Proof: recall that this question didn't ask for a proof, just the answer. Since the proofs are of a difficulty that you are expected to be able to handle in this course, we present them here for completeness.

The function is injective since

$2n=2m⟺n=m$

and is not surjective since clearly no odd integer will be in the image of the function ƒ.