Edits

Advanced/Definable Sets/main.tex

@@ -18,7 +18,7 @@
 		<Spring 2023>
 		{Definable Sets}
 		{
-			Prepared by Mark on \today
+			Prepared by Mark and Nikita on \today
 		}
 
 

Advanced/Definable Sets/parts/1 structures.tex

@@ -1,39 +1,36 @@
 \section{Structures}
 
-\definition{}<def:language>
-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}
 

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