# Isomorphism to Closed Interval

## Theorem

Let $m, n \in \N$ such that $m < n$.

Then:

- $\left|{\left[{m + 1 \,.\,.\, n}\right]}\right| = n - m$

Let $h: \N_{n - m} \to \left[{m + 1 \,.\,.\, n}\right]$ be the mapping defined as:

- $\forall x \in \N_{n - m}: h \left({x}\right) = x + m + 1$

Let the orderings on $\left[{m + 1 \,.\,.\, n}\right]$ and $\N_{n - m}$ be those induced by the ordering of $\N$.

Then $h$ a unique order isomorphism.

## Proof

From Unique Isomorphism between Finite Totally Ordered Sets, it suffices to show that $h$ is an order isomorphism.

To this end, remark that, for all $x, y \in \N_{n - m}$:

\(\displaystyle h \left({x}\right)\) | \(=\) | \(\displaystyle h \left({y}\right)\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \iff \ \ \) | \(\displaystyle x + m + 1\) | \(=\) | \(\displaystyle y + m + 1\) | $\quad$ | $\quad$ | ||||||||

\(\displaystyle \iff \ \ \) | \(\displaystyle x\) | \(=\) | \(\displaystyle y\) | $\quad$ Natural Number Addition is Cancellable | $\quad$ |

proving $h$ is an injection, and so a bijection, from Equivalence of Mappings between Sets of Same Cardinality.

By Ordering on Natural Numbers is Compatible with Addition and Natural Number Addition is Cancellable for Ordering, it follows that:

- $x \le y \iff h \left({x}\right) \le h \left({y}\right)$

so $h$ is an order isomorphism.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 17$: Theorem $17.11$