125 lines
3.6 KiB
TeX
125 lines
3.6 KiB
TeX
\section{Structures}
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\definition{}<def:language>
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A \textit{language} is a set of meaningless objects. Here are a few examples:
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\begin{itemize}
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\item $\{a, b, ..., z\}$
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\item $\{0, 1\}$
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\item $\mathbb{Z}$, $\mathbb{R}$, etc.
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\end{itemize}
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Every language comes with the equality check $=$, which checks if two elements are the same.
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\definition{}
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A \textit{structure} over a language $\mathcal{L}$ consists of a set of symbols. \par
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The purpose of a structure is to give a language meaning.
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\vspace{2mm}
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Symbols generally come in three types:
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\begin{itemize}
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\item Constant symbols, which let us specify specific elements of our language. \par
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Examples: $0, 1, \frac{1}{2}, \pi$
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\vspace{2mm}
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\item Function symbols, which let us navigate between elements of our language. \par
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Examples: $+, \times, \sin{x}, \sqrt{x}$
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\vspace{2mm}
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\item Relation symbols, which let us compare elements of our language. \par
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Examples: $<, >, \leq, \geq$ \par
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The symbol $=$ is \textbf{not} a relation. Why? \hint{See \ref{def:language}}
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\vspace{2mm}
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\end{itemize}
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\vspace{3mm}
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\example{}
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\def\structgeneric{\ensuremath{}}
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The first structure we'll look at is the following:
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$$
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\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)
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$$
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\vspace{2mm}
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This is a structure over $\mathbb{Z}$ with the following symbols:
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\begin{itemize}
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\item Constants: \tab $\{0, 1\}$
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\item Functions: \tab $\{+, -\}$
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\item Relations: \tab $\{<\}$
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\end{itemize}
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\vspace{2mm}
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If you look at our set of constant symbols, you'll see that the only integers we can directly refer to in this structure are 0 and 1. If we want any others, we must define them using the tools the structure offers.
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\vspace{1mm}
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Say we want the number 2. We could use the function $+$ to define it: $2 \coloneqq [x \text{ where } 1 + 1 = x]$ \par
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We would write this as $2 \coloneqq [x \text{ where } +(1, 1) = x]$ in proper \say{functional} notation.
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\problem{}
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Can we define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)$? If so, how?
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\vfill
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\problem{}
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Can we define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, +, -, <\} \Bigr)$? \par
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\hint{In this problem, $1$ has been removed from the set of constant symbols.}
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\vfill
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\pagebreak
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Let us formalize what we found in the previous two problems. \par
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\definition{}
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A \textit{formula} in a structure $S$ is a well-formed string of constants, functions, and relations. \par
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\vspace{2mm}
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You already know what a \say{well-formed} string is: $1 + 1$ is fine, $\sqrt{+}$ is nonsense. \par
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For the sake of time, I will not provide a formal definition. It isn't particularly interesting.
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\definition{Definable Elements}
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Say $S$ is a structure over a language $\mathcal{L}$. \par
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We say an element $e$ of $\mathcal{L}$ is \textit{definable in $S$} if we can write a formula that only $e$ satisfies.
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\problem{}
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Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bigr)$? \par
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\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$}
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\begin{solution}
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$-2 \notin \mathbb{Z}^+$, so $2$ can be defined by $[x \text{ where } x \times x = 4]$.
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\end{solution}
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\vfill
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\problem{}
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Can we define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr)$?
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\begin{solution}
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No. We could try $[x \text{ where } x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \\
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We have no way to distinguish between negative and positive numbers.
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\end{solution}
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\vfill
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\problem{}
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What numbers are definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
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\begin{solution}
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With the tools we have so far, we can only define powers of two, positive and negative.
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\end{solution}
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\vfill
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\pagebreak |