From 62fc066e69afea79d3c55602d32f61d9f286cdad Mon Sep 17 00:00:00 2001 From: Mark Date: Sat, 6 May 2023 17:05:30 -0700 Subject: [PATCH] Started definable sets handout --- Advanced/Definable Sets/main.tex | 28 ++++ Advanced/Definable Sets/parts/0 logic.tex | 102 ++++++++++++++ .../Definable Sets/parts/1 structures.tex | 130 ++++++++++++++++++ 3 files changed, 260 insertions(+) create mode 100755 Advanced/Definable Sets/main.tex create mode 100644 Advanced/Definable Sets/parts/0 logic.tex create mode 100644 Advanced/Definable Sets/parts/1 structures.tex diff --git a/Advanced/Definable Sets/main.tex b/Advanced/Definable Sets/main.tex new file mode 100755 index 0000000..992b3c6 --- /dev/null +++ b/Advanced/Definable Sets/main.tex @@ -0,0 +1,28 @@ +% 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 + + + {Definable Sets} + { + Prepared by Mark on \today + } + + + \input{parts/0 logic.tex} + \input{parts/1 structures.tex} + +\end{document} \ No newline at end of file diff --git a/Advanced/Definable Sets/parts/0 logic.tex b/Advanced/Definable Sets/parts/0 logic.tex new file mode 100644 index 0000000..2d76a30 --- /dev/null +++ b/Advanced/Definable Sets/parts/0 logic.tex @@ -0,0 +1,102 @@ +\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 \ No newline at end of file diff --git a/Advanced/Definable Sets/parts/1 structures.tex b/Advanced/Definable Sets/parts/1 structures.tex new file mode 100644 index 0000000..e0a74c9 --- /dev/null +++ b/Advanced/Definable Sets/parts/1 structures.tex @@ -0,0 +1,130 @@ +\section{Structures} + +\definition{} +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 \ No newline at end of file