Bijection between Integers and Even Integers

Theorem

Let $\Z$ be the set of integers.

Let $2 \Z$ be the set of even integers.

Then there exists a bijection $f: \Z \to 2 \Z$ between the two.

Proof

Let $f: \Z \to 2 \Z$ be the mapping defined as:

$\forall n \in \Z: \map f n = 2 n$

Let $m, n \in \Z$ such that $\map f m = \map f n$.

\(\ds \map f m\)

\(=\)

\(\ds \map f n\)

\(\ds \leadsto \ \ \)

\(\ds 2 m\)

\(=\)

\(\ds 2 n\)

Definition of $f$

\(\ds \leadsto \ \ \)

\(\ds m\)

\(=\)

\(\ds n\)

dividing both sides by $2$

Thus $f$ is an injection.

Let $n \in 2 Z$.

Then by definition:

$n = 2 r$

for some $r \in \Z$

That is:

$n = \map f r$

and so $f$ is a surjection.

So $f$ is both an injection and a surjection.

Hence by definition, $f$ is a bijection.

$\blacksquare$