132 lines
4.0 KiB
TeX
132 lines
4.0 KiB
TeX
\section{Structures}
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\definition{}<def:language>
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A \textit{language} is a set of meaningless symbols. 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 three sets:
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\begin{itemize}
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\item A set of \textit{constant symbols} $\mathcal{C}$ \par
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Constant symbols let us specify specific elements of our language. \par
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$\mathcal{C}$ must thus be a subset of $\mathcal{L}$.
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\vspace{3mm}
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\item A set of \textit{function symbols} $\mathcal{F}$ \par
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Function symbols let us navigate between elements of our language. \par
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$+$, $-$ are functions, as are $\sin{x}$, $\cos{x}$, and $\sqrt{x}$ \par
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Functions take inputs in $\mathcal{L}$ and produce outputs in $\mathcal{L}$.
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\vspace{3mm}
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\item A set of \textit{relation symbols} $\mathcal{R}$ \par
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Relation symbols let us compare elements of our language. \par
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You are already familiar with this concept: $>$, $<$, and $\leq$ are relation symbols. \par
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$=$ is \textbf{not} a relational symbol. Why? \hint{See \ref{def:language}}
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\end{itemize}
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\vspace{2mm}
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The purpose of a structure is to give a language meaning. This is best explained by example.
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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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$$
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\Bigl(
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\mathcal{L} ~\big|~ \{\mathcal{C}, ~ \mathcal{F}, ~ \mathcal{R}\}
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\Bigr)
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=
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\Bigl( \mathbb{Z} ~\big|~ \{0, 1, ~ +, -, ~ <\} \Bigr)
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$$
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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 $\mathcal{C} = \{0, 1\}$ \tab \note{(constants)}
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\item $\mathcal{F} = \{+, -\}$ \tab \note{(functions)}
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\item $\mathcal{R} = \{<\}$ \tab \note{(relations)}
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\end{itemize}
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\vspace{2mm}
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Let's look at $\mathcal{C}$, our set of constant symbols. 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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Yes! $-2$ no longer exists, 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 is definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
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\begin{solution}
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All powers of two, positive and negative.
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\end{solution}
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\vfill
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\pagebreak |