Started definable sets handout

Advanced/Definable Sets/main.tex

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+% use [nosolutions] flag to hide solutions.
+% use [solutions] flag to show solutions.
+\documentclass[
+	solutions,
+	singlenumbering
+]{../../resources/ormc_handout}
+
+% Typewriter tabs
+\usepackage{tabto}
+\TabPositions{1cm, 2cm, 3cm, 4cm, 5cm}
+
+% for \coloneqq, a centered :=
+\usepackage{mathtools}
+
+\begin{document}
+	\maketitle
+		<Advanced 2>
+		<Spring 2023>
+		{Definable Sets}
+		{
+			Prepared by Mark on \today
+		}
+
+
+	\input{parts/0 logic.tex}
+	\input{parts/1 structures.tex}
+
+\end{document}

Advanced/Definable Sets/parts/0 logic.tex

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+\section{Logical Algebra}
+
+\definition{}
+Odds are, you are familiar with \textit{logical symbols}. \par
+In this handout, we'll use the following:
+\begin{itemize}
+	\item $\lnot$: not
+	\item $\land$: and
+	\item $\lor$: or
+	\item $\rightarrow$: implies
+	\item $()$, parenthesis.
+\end{itemize}
+
+The function of these is defined by \textit{truth tables}:
+\begin{center}
+\begin{tabular}{ c | c | c }
+	\multicolumn{3}{ c }{and} \\
+	\hline
+		$A$ & $B$ & $A \land B$ \\
+	\hline
+	F & F & F \\
+	F & T & F \\
+	T & F & F \\
+	T & T & T
+\end{tabular}
+\hfill
+\begin{tabular}{ c | c | c }
+	\multicolumn{3}{ c }{or} \\
+	\hline
+		$A$ & $B$ & $A \lor B$ \\
+	\hline
+	F & F & F \\
+	F & T & T \\
+	T & F & T \\
+	T & T & T
+\end{tabular}
+\hfill
+\begin{tabular}{ c | c | c }
+	\multicolumn{3}{ c }{implies} \\
+	\hline
+		$A$ & $B$ & $A \rightarrow B$ \\
+	\hline
+	F & F & T \\
+	F & T & T \\
+	T & F & F \\
+	T & T & T
+\end{tabular}
+\hfill
+\begin{tabular}{ c | c }
+	\multicolumn{2}{ c }{not} \\
+	\hline
+		$A$ & $\lnot A$ \\
+	\hline
+	T & F \\
+	F & T \\
+	~ & ~ \\
+	~ & ~ \\
+\end{tabular}
+\end{center}
+
+\vspace{2mm}
+
+$A \land B$ is only true if both $A$ and $B$ are true. $A \lor B$ is only true if $A$ or $B$ (or both) are true. \par
+$\lnot A$ is the opposite of $A$, which is why it looks like a \say{negative} sign. \par
+
+\vspace{2mm}
+
+$A \rightarrow B$ is a bit harder to understand. Read aloud, this is \say{$A$ implies $B$.} \par
+The only time $\rightarrow$ is false is when $T \rightarrow F$. Think about it: why does this make sense? \par
+
+\problem{}
+Evaluate the following.
+\begin{itemize}
+	\item $(T \land F) \lor T$
+	\item $(\lnot (F \lor \lnot T) ) \rightarrow T$
+	\item $A \rightarrow T$ for any $A$
+	\item $(\lnot (A \rightarrow B)) \rightarrow A$ for any $A,B$
+\end{itemize}
+
+\vfill
+\pagebreak
+
+\problem{}
+Show that $\lnot (A \rightarrow \lnot B)$ is equivalent to $A \land B$. \par
+\hint{Use a truth table}
+
+\vfill
+
+\problem{}
+Can you express $A \lor B$ using only $\lnot$, $\rightarrow$, and $()$?
+
+\begin{solution}
+	$((\lnot A) \rightarrow B)$
+\end{solution}
+
+\vfill
+
+Note that both $\land$ and $\lor$ can be defined using the other logical symbols. \par
+The only logical symbols we \textit{need} are $\lnot$, $\rightarrow$, and $()$. \par
+We include $\land$ and $\lor$ to simplify our logical expressions.
+
+\pagebreak

Advanced/Definable Sets/parts/1 structures.tex

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+\section{Structures}
+
+\definition{}<def:language>
+A \textit{language} is a set of meaningless symbols. Here are a few examples:
+\begin{itemize}
+	\item $\{a, b, ..., z\}$
+	\item $\{0, 1\}$
+	\item $\mathbb{Z}$, $\mathbb{R}$, etc.
+\end{itemize}
+
+Every language comes with the equality check $=$, which checks if two elements are the same.
+
+
+\definition{}
+A \textit{structure} over a language $\mathcal{L}$ consists of three sets:
+\begin{itemize}
+	\item A set of \textit{constant symbols} $\mathcal{C}$ \par
+		Constant symbols let us specify specific elements of our language. \par
+		$\mathcal{C}$ must thus be a subset of $\mathcal{L}$.
+		\vspace{3mm}
+
+
+	\item A set of \textit{function symbols} $\mathcal{F}$ \par
+		Function symbols let us navigate between elements of our language. \par
+		$+$, $-$ are functions, as are $\sin{x}$, $\cos{x}$, and $\sqrt{x}$ \par
+		Functions take inputs in $\mathcal{L}$ and produce outputs in $\mathcal{L}$.
+		\vspace{3mm}
+
+	\item A set of \textit{relation symbols} $\mathcal{R}$ \par
+		Relation symbols let us compare elements of our language. \par
+		You are already familiar with this concept: $>$, $<$, and $\leq$ are relation symbols. \par
+		$=$ is \textbf{not} a relational symbol. Why? \hint{See \ref{def:language}}
+
+\end{itemize}
+
+\vspace{2mm}
+
+The purpose of a structure is to give a language meaning. This is best explained by example.
+
+\vspace{3mm}
+
+
+\example{}
+\def\structgeneric{\ensuremath{}}
+
+The first structure we'll look at is the following:
+$$
+	\Bigl(
+		\mathcal{L} ~\big|~ \{\mathcal{C}, ~ \mathcal{F}, ~ \mathcal{R}\}
+	\Bigr)
+	=
+	\Bigl( \mathbb{Z} ~\big|~ \{0, 1, ~ +, -, ~ <\} \Bigr)
+$$
+
+\vspace{2mm}
+
+This is a structure over $\mathbb{Z}$ with the following symbols:
+\begin{itemize}
+	\item $\mathcal{C} = \{0, 1\}$   \tab \note{(constants)}
+	\item $\mathcal{F} = \{+, -\}$   \tab \note{(functions)}
+	\item $\mathcal{R} = \{<\}$      \tab \note{(relations)}
+\end{itemize}
+
+\vspace{2mm}
+
+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.
+
+\vspace{1mm}
+
+Say we want the number 2. We could use the function $+$ to define it: $2 \coloneqq [x \text{ where } 1 + 1 = x]$ \par
+We would write this as $2 \coloneqq [x \text{ where } +(1, 1) = x]$ in proper \say{functional} notation.
+
+
+\problem{}
+Can we define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)$? If so, how?
+
+\vfill
+
+\problem{}
+Can we define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, +, -, <\} \Bigr)$? \par
+\hint{In this problem, $1$ has been removed from the set of constant symbols.}
+
+\vfill
+\pagebreak
+
+Let us formalize what we found in the previous two problems. \par
+\say{Definable elements} are one of the two most important ideas in this handout.
+
+\definition{}
+A \textit{formula} in a structure $S$ is a well-formed string of constants, functions, and relations. \par
+
+\vspace{2mm}
+
+You already know what a \say{well-formed} string is: $1 + 1$ is fine, $\sqrt{+}$ is nonsense. \par
+For the sake of time, I will not provide a formal definition. It isn't particularly interesting.
+
+
+\definition{Definable Elements}
+Say $S$ is a structure over a language $\mathcal{L}$. \par
+We say an element $e$ of $\mathcal{L}$ is \textit{definable in $S$} if we can write a formula that only $e$ satisfies.
+
+\problem{}
+Can we define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr)$?
+
+\begin{solution}
+	No. We could try $[x \text{ where } x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \\
+	We have no way to distinguish between negative and positive numbers.
+\end{solution}
+
+\vfill
+
+\problem{}
+Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bigr)$? \par
+\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$}
+
+\begin{solution}
+	Yes! $-2$ no longer exists, so $2$ can be defined by $[x \text{ where } x \times x = 4]$.
+\end{solution}
+
+\vfill
+
+\problem{}
+What is definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
+
+\begin{solution}
+	All powers of two, positive and negative.
+\end{solution}
+
+\vfill
+\pagebreak