Set of Even Integers is Equivalent to Set of Integers

Theorem
Let $\Z$ denote the set of integers.

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

Then:
 * $2 \Z \sim \Z$

where $\sim$ denotes set equivalence.

Proof
To demonstrate set equivalence, it is sufficient to construct a bijection between the two sets.

Let $f: \Z \to 2 \Z$ defined as:
 * $\forall x \in \Z: \map f x = 2 x$

demonstrating injectivity.

demonstrating surjectivity.

Hence by definition $f$ is a bijection.