Isomorphism (Abstract Algebra)/Examples/Addition under Doubling
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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