From 6ea7e9df9797d273ea77c7edd7fb96c401a676d6 Mon Sep 17 00:00:00 2001 From: Mark Date: Thu, 11 May 2023 14:54:18 -0700 Subject: [PATCH] Edits --- Advanced/Definable Sets/main.tex | 2 +- .../Definable Sets/parts/1 structures.tex | 38 +++++++------- Advanced/Definable Sets/parts/3 sets.tex | 49 +++++++++++++++---- 3 files changed, 62 insertions(+), 27 deletions(-) diff --git a/Advanced/Definable Sets/main.tex b/Advanced/Definable Sets/main.tex index 43b96de..0e5f7bb 100755 --- a/Advanced/Definable Sets/main.tex +++ b/Advanced/Definable Sets/main.tex @@ -18,7 +18,7 @@ {Definable Sets} { - Prepared by Mark on \today + Prepared by Mark and Nikita on \today } diff --git a/Advanced/Definable Sets/parts/1 structures.tex b/Advanced/Definable Sets/parts/1 structures.tex index e161337..7e8d71e 100644 --- a/Advanced/Definable Sets/parts/1 structures.tex +++ b/Advanced/Definable Sets/parts/1 structures.tex @@ -1,39 +1,36 @@ \section{Structures} -\definition{} -A \textit{language} is a set of meaningless objects. Here are a few examples: +\definition{} +A \textit{universe} is a set of meaningless objects. 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 a set of symbols. \par -The purpose of a structure is to give a language meaning. +A \textit{structure} consists of a universe $U$ and set of symbols. \par +A structure's symbols give meaning to the objects in its universe. \vspace{2mm} Symbols generally come in three types: \begin{itemize} - \item Constant symbols, which let us specify specific elements of our language. \par + \item Constant symbols, which let us specify specific elements of our universe. \par Examples: $0, 1, \frac{1}{2}, \pi$ \vspace{2mm} - \item Function symbols, which let us navigate between elements of our language. \par + \item Function symbols, which let us navigate between elements of our universe. \par Examples: $+, \times, \sin{x}, \sqrt{x}$ \vspace{2mm} - \item Relation symbols, which let us compare elements of our language. \par + \item Relation symbols, which let us compare elements of our universe. \par Examples: $<, >, \leq, \geq$ \par - The symbol $=$ is \textbf{not} a relation. Why? \hint{See \ref{def:language}} \vspace{2mm} - \end{itemize} +The equality check $=$ is \textbf{not} a relation symbol. It is included in every structure by default. + \vspace{3mm} @@ -47,7 +44,7 @@ $$ \vspace{2mm} -This is a structure over $\mathbb{Z}$ with the following symbols: +This is a structure with the universe $\mathbb{Z}$ that contains the following symbols: \begin{itemize} \item Constants: \tab $\{0, 1\}$ \item Functions: \tab $\{+, -\}$ @@ -86,10 +83,17 @@ A \textit{formula} in a structure $S$ is a well-formed string of constants, func 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. +\vspace{2mm} + +A formula can contain one or more \textit{free variables.} These are denoted $\varphi{(a, b, ...)}$. \par +Formulas with free variables let us define \say{properties} that certain objects have. \par + +For example, $x$ is a free variable in the formula $\varphi(x) = x > 0$. \par +$\varphi(3)$ is true and $\varphi(-3)$ is false. \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. +Say $S$ is a with a universe $U$. \par +We say an element $e \in U$ is \textit{definable in $S$} if we can write a formula that only $e$ satisfies. \problem{} @@ -104,10 +108,10 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi \problem{} -Can we define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr)$? +Try to 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$. \\ + This isn't possible. 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} diff --git a/Advanced/Definable Sets/parts/3 sets.tex b/Advanced/Definable Sets/parts/3 sets.tex index c65cbe5..51c0eb4 100644 --- a/Advanced/Definable Sets/parts/3 sets.tex +++ b/Advanced/Definable Sets/parts/3 sets.tex @@ -19,8 +19,8 @@ $$ $$ \definition{Definable Sets} -Let $S$ be a structure over a language $\mathcal{L}$. \par -We say a subset $M$ of $\mathcal{L}$ is \textit{definable} if we can write a formula that is true for some $x$ iff $x \in M$. +Let $S$ be a structure with a universe $U$. \par +We say a subset $M$ of $U$ is \textit{definable} if we can write a formula that is true for some $x$ iff $x \in M$. \vspace{4mm} @@ -57,7 +57,7 @@ Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z) \hint{$z$ is nonreal if $x \in \mathbb{C}$ and $x \notin \mathbb{R}$} \begin{solution} - $\Bigl\{ x ~\bigl|~ \lnot (\text{real}(i \times x) = 0) \Bigr\}$ + $\Bigl\{ x ~\bigl|~ \text{real}(x) = x \Bigr\}$ \end{solution} @@ -65,14 +65,13 @@ Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z) \problem{} -Define the set of integers in $\Bigl( \mathbb{R} ~\big|~ \{\times\} \Bigr)$ \par +Define $\mathbb{R}^+_0$ in $\Bigl( \mathbb{R} ~\big|~ \{\times\} \Bigr)$ \par -\hint{$\text{real}(z)$ gives the real part of a complex number: $\text{real}(3 + 2i) = 3$} -\hint{$z$ is nonreal if $x \in \mathbb{C}$ and $x \notin \mathbb{R}$} +\vfill -\begin{solution} - $\Bigl\{ x ~\bigl|~ \lnot (\text{real}(i \times x) = 0) \Bigr\}$ -\end{solution} +\problem{} +Let $\bigtriangleup$ be a relational symbol. $a \bigtriangleup b$ holds iff $a$ divides $b$. \par +Define the set of prime numbers in $\Bigl( \mathbb{Z}^+ ~\big|~ \{ \bigtriangleup \} \Bigr)$ \par \vfill @@ -87,6 +86,38 @@ Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$ \problem{} Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$ +\vfill +\pagebreak + +\problem{} +Consider the structure $S = ( \mathbb{R} ~|~ \{0, \diamond \} )$ \par +The relation $a \diamond b$ holds if $| a - b | = 1$ + +\problempart{} +Define $\{\}$ in $S$. + +\problempart{} +Define ${-1, 1}$ in $S$. + +\problempart{} +Define $\{-2, 2\}$ in $S$. + +\vfill + +\problem{} +Let $P$ be the set of all subsets of $\mathbb{Z}^+_0$. This is called a \textit{power set}. \par +Let $S$ be the stucture $( P ~|~ \{\subseteq\} )$ \par + +\problempart{} +Show that the empty set is definable in $S$. + +\problempart{} +Let $x \Bumpeq y$ be a relation on $P$. $x \Bumpeq y$ holds if $x \cap y \neq \{\}$. \par +Show that $\Bumpeq$ is definable in $S$. + +\problempart{} +Let $f$ be a function on $P$ defined by $f(x) = \mathbb{Z}^+_0 - x$. This is called the \textit{compliment} of the set $x$. \par +Show that $f$ is definable in $S$. \vfill \pagebreak \ No newline at end of file