Added

Advanced/Definable Sets/main.tex

@@ -7,7 +7,7 @@
 
 % Typewriter tabs
 \usepackage{tabto}
-\TabPositions{1cm, 2cm, 3cm, 4cm, 5cm}
+\TabPositions{1cm, 2cm, 3cm, 4cm, 5cm, 6cm, 7cm, 8cm}
 
 % for \coloneqq, a centered :=
 \usepackage{mathtools}
@@ -24,5 +24,7 @@
 
 	\input{parts/0 logic.tex}
 	\input{parts/1 structures.tex}
+	\input{parts/2 quantifiers.tex}
+	\input{parts/3 sets.tex}
 
 \end{document}

Advanced/Definable Sets/parts/1 structures.tex

@@ -84,7 +84,6 @@ Can we define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, +, -, <\} \Bigr)$? \par
 \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
@@ -99,15 +98,6 @@ For the sake of time, I will not provide a formal definition. It isn't particula
 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
@@ -119,6 +109,18 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi
 
 \vfill
 
+
+\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{}
 What is definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
 

Advanced/Definable Sets/parts/2 quantifiers.tex

@@ -0,0 +1,72 @@
+\section{Quantifiers}
+
+Recall the logical symbols we introduced earlier: $(), \land, \lor, \lnot, \rightarrow$ \par
+We will now add two more: $\forall$ (for all) and $\exists$ (exists).
+
+\definition{}
+$\forall$ and $\exists$ are \textit{quantifiers}. They allow us to make statements about arbitrary symbols.
+
+\vspace{2mm}
+
+Let's look at $\forall$ first. Let $\varphi$ be a formula. \par
+Then, the formula $\forall x ~ \varphi$ says \say{$\varphi$ is true for all possible $x$.}
+
+\vspace{1mm}
+
+For example, take the formula $\forall x ~ (0 < x)$. \par
+In english, this means \say{For any $x$, $x$ is bigger than zero,} or simply \say{Any $x$ is positive.}
+
+\vspace{3mm}
+
+$\exists$ is very similar: the formula $\exists x ~ \varphi$ states that there is at least one $x$ that makes $\varphi$ true. \par
+For example, $\exists ~ (0 < x)$ means \say{there is a positive number in our set}.
+
+\vspace{4mm}
+
+\problem{}
+Which of the following are true in $\mathbb{Z}$? \par
+Which are true in $\mathbb{R}^+_0$? \par
+\hint{$\mathbb{R}^+_0$ is the set of positive real numbers and zero.} \par
+
+\begin{itemize}[itemsep = 1mm]
+
+	\item $\forall x ~ (x \geq 0)$
+	\item $\lnot (\exists x ~ (x = 0))$
+
+	\item $\forall x ~ [\exists y ~ (y \times y = x)]$
+
+	\item $\forall xy ~ \exists z ~ (x < z < y)$ \tab \note{This is a compact way to write $\forall x ~ (\forall y ~ (\exists z ~ (x < z < y)))$}
+
+	\item $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \tab~\tab \note{Solution is below.}
+\end{itemize}
+
+\begin{examplesolution}
+	Here is a solution to the last part: $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \par
+
+	\vspace{4mm}
+
+	Reading this term-by-term, we get \tab \say{not exists $x$ where (for all $y$ ($x$ smaller than $y$))} \par
+	If we add some grammar, we get \tab \say{There isn't an $x$ where all $y$ are bigger than $x$} \par
+	which we can rephrase as \tab~\tab \say{There isn't a minimum value} \par
+
+	\vspace{4mm}
+
+	Which is true in $\mathbb{Z}$ and false in $\mathbb{R}^+_0$
+\end{examplesolution}
+
+
+\vfill
+\pagebreak
+
+\problem{}
+Define $\forall$ using logical symbols and $\exists$
+
+\vfill
+
+\problem{}
+Does the order of $\forall$ and $\exists$ in a formula matter? \par
+What's the difference between $\exists x ~ \forall y ~ (x < y)$ and $\forall y ~ \exists x ~ (x < y)$? \par
+\hint{In $\mathbb{R}^+$, the first is false and the second is true. $\mathbb{R}^+$ does not contain zero.}
+
+\vfill
+\pagebreak

Advanced/Definable Sets/parts/3 sets.tex

@@ -0,0 +1,83 @@
+\section{Definable Sets}
+
+Armed with $(), \land, \lor, \lnot, \rightarrow, \forall,$ and $\exists$, we have enough tools to define sets.
+
+\definition{Set-Builder Notation}
+Say we have a condition $c$. \par
+The set of all elements that satisfy that condition can be written as follows:
+$$
+	\{ x ~|~ \text{$c$ is true} \}
+$$
+This is read \say{The set of $x$ where $c$ is true} or \say{The set of $x$ that satisfy $c$.}
+
+
+\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 $x \in M$.
+
+\vspace{4mm}
+
+For example, consider the structure $\Bigl( \mathbb{Z} ~\big|~ \{+\} \Bigr)$ \par
+
+\vspace{2mm}
+
+Only even numbers satisfy the formula $\varphi(x) = \exists y ~ (y + y = x)$, \par
+So we can define \say{the set of even numbers} as $\{ x ~|~ \exists y ~ (y + y = x) \}$. \par
+Remember---we can only use symbols that are available in our structure!
+
+\problem{}
+Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
+
+\vfill
+
+\problem{}
+Define the set of rational numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
+
+\vfill
+
+\problem{}
+Define the set of irrational numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
+
+\vfill
+
+\problem{}
+Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
+
+\vfill
+
+\problem{}
+Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
+
+\vfill
+\pagebreak
+
+
+\problem{}
+Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ i, ~ \text{real}(z), \times\} \Bigr)$ \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 $x \notin \mathbb{R}$}
+
+\begin{solution}
+	$\Bigl\{ x ~\bigl|~ \lnot (\text{real}(i \times x) = 0) \Bigr\}$
+\end{solution}
+
+\vfill
+
+\problem{}
+Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ \text{real}(z), \times\} \Bigr)$ \par
+
+\begin{solution}
+	$\Biggl\{ x ~\Bigl|~ \forall y ~ \Bigl( \text{real}(y) = 0 \rightarrow \lnot \bigl[ \text{real}(x \times y) = 0 \bigr] \Bigr) \Biggr\}$
+\end{solution}
+
+\vfill
+
+\problem{}
+Define $\mathbb{R}$ in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z), \times\} \Bigr)$ \par
+
+\begin{solution}
+	$\Biggl\{ x ~\Bigl|~ \forall y ~ \Bigl( \text{real}(x) \times y = \text{real}(x) \Bigr) \Biggr\}$
+\end{solution}
+
+\vfill
+\pagebreak