Isomorphism (Abstract Algebra)/Examples/Addition under Doubling

Example of Isomorphism

Let $\N$ denote the set of natural numbers.

Let $2 \N$ denote the set of even non-negative integers:

$2 \N := \set {0, 2, 4, 6, \ldots}$

Let $\struct {\N, +}$ and $\struct {2 \N, +}$ be the algebraic structures formed from the above with addition.

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

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

Then $f$ is an isomorphism.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): isomorphism
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): isomorphism