연습장

Ex. 1

Existence)
Let $w$ be any string over an $\mathrm{\Sigma }$.
By the definition, the domain of $w$ is a finite initial segment of $\mathbb{\mathbb{N}}$.
This means that $\mathrm{\exists }n\in \mathbb{\mathbb{N}}$ s.t. w is defined at $i\leftrightarrow i<n$

Uniquness)
Suppose that $\mathrm{\exists }n,m$ s.t. both satisfy the condition for the string $w$.
That means, w is defined at position $i$ if and only if $(i<n)\wedge (i<m)$.
By definition, $\{i\in \mathbb{\mathbb{N}}|i<n\}$ and $\{i\in \mathbb{\mathbb{N}}|i<m\}$ are both domain of $w$, which means they are same set.
So the only way that $\{i\in \mathbb{\mathbb{N}}|i<n\}$ and $\{i\in \mathbb{\mathbb{N}}|i<m\}$ are same is their endpoints are same. Therefore, $n=m$.

Hence, there exists a unique natural number $n$ with the desired property.

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