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.

$\blacksquare$


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