연습장

Ex. 1

Existence)
Let [math]\displaystyle{ w }[/math] be any string over an [math]\displaystyle{ \Sigma }[/math].
By the definition, the domain of [math]\displaystyle{ w }[/math] is a finite initial segment of [math]\displaystyle{ \mathbb{N} }[/math].
This means that [math]\displaystyle{ \exist n \in \mathbb{N} }[/math] s.t. w is defined at [math]\displaystyle{ i \leftrightarrow i \lt n }[/math]

Uniquness)
Suppose that [math]\displaystyle{ \exist n,\,\, m }[/math] s.t. both satisfy the condition for the string [math]\displaystyle{ w }[/math].
That means, w is defined at position [math]\displaystyle{ i }[/math] if and only if [math]\displaystyle{ (i \lt n) \land (i \lt m) }[/math].
By definition, [math]\displaystyle{ \{i \in \mathbb{N}|i \lt n\} }[/math] and [math]\displaystyle{ \{i \in \mathbb{N}|i \lt m\} }[/math] are both domain of [math]\displaystyle{ w }[/math], which means they are same set.
So the only way that [math]\displaystyle{ \{i \in \mathbb{N}|i \lt n\} }[/math] and [math]\displaystyle{ \{i \in \mathbb{N}|i \lt m\} }[/math] are same is their endpoints are same. Therefore, [math]\displaystyle{ n = m }[/math].

Hence, there exists a unique natural number [math]\displaystyle{ n }[/math] with the desired property.

Ex. 2

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