# 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:

$\forall x \in \Z: \map f x = 2 x$
 $\ds \map f x$ $=$ $\ds \map f y$ $\ds \leadsto \ \$ $\ds 2 x$ $=$ $\ds 2 y$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds y$

demonstrating injectivity.

 $\ds y$ $\in$ $\ds 2 \Z$ $\ds \leadsto \ \$ $\, \ds \exists x \in \Z: \,$ $\ds y$ $=$ $\ds 2 x$ Definition of Even Integer $\ds \leadsto \ \$ $\ds y$ $\in$ $\ds f \sqbrk \Z$

demonstrating surjectivity.

Hence by definition $f$ is a bijection.

$\blacksquare$