연습장 문서 원본 보기

==Ex. 1==
Existence)<br>
Let <math>w</math> be any string over an <math>\Sigma</math>.<br>
By the definition, the domain of <math>w</math> is a finite initial segment of <math>\mathbb{N}</math>.<br>
This means that <math>\exist n \in \mathbb{N} </math> s.t. w is defined at <math>i \leftrightarrow i < n</math>

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

Hence, there exists a unique natural number <math>n</math> with the desired property.
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>
<math></math>