# Set of Integers under Addition is Isomorphic to Set of Even Integers under Addition

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## Theorem

Let $\struct {\Z, +}$ be the algebraic structure formed by the set of integers under the operation of addition.

Let $\struct {2 \Z, +}$ be the algebraic structure formed by the set of even integers under the operation of addition.

Then $\struct {\Z, +}$ and $\struct {2 \Z, +}$ are isomorphic.

## Proof

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

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

From Bijection between Integers and Even Integers, $f$ is a bijection.

Let $m, n \in \Z$.

Then:

\(\displaystyle \map f {m + n}\) | \(=\) | \(\displaystyle 2 \paren {m + n}\) | Definition of $f$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 m + 2 n\) | Integer Multiplication Distributes over Addition | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map f m + \map f n\) | Integer Multiplication Distributes over Addition |

Thus $f$ is an isomorphism by definition.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 6$: Example $6.4$