Post-class edits

Advanced/Definable Sets/main.tex

@@ -5,10 +5,6 @@
 	singlenumbering
 ]{../../resources/ormc_handout}
 
-% Typewriter tabs
-\usepackage{tabto}
-\TabPositions{1cm, 2cm, 3cm, 4cm, 5cm, 6cm, 7cm, 8cm}
-
 % for \coloneqq, a centered :=
 \usepackage{mathtools}
 

Advanced/Definable Sets/parts/1 structures.tex

@@ -90,16 +90,16 @@ 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 with a universe $U$. \par
+Say $S$ is a structure 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{}
 Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bigr)$? \par
-\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$}
+\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$. Also, $2 \times 2 = 4$.}
 
 \begin{solution}
-	$-2 \notin \mathbb{Z}^+$, so $2$ can be defined by $[x \text{ where } x \times x = 4]$.
+	$2$ is the only element in $\mathbb{Z}^+$ that satisfies $[x \text{ where } x \times x = 4]$.
 \end{solution}
 
 \vfill
@@ -111,16 +111,23 @@ Try to define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr
 \begin{solution}
 	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.
+
+	\begin{instructornote}
+		Actually, it is. Bonus problem: how? \par
+		Do this after understanding quantifiers.
+	\end{instructornote}
 \end{solution}
 
 \vfill
 
 
 \problem{}
-What numbers are definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
+What numbers are definable in the structure $\Bigl( \mathbb{R}^+_0 ~\big|~ \{1, 2, \div \} \Bigr)$?
 
 \begin{solution}
-	With the tools we have so far, we can only define powers of two, positive and negative.
+	We can define powers of two, positive and negative.
+
+	If you're clever, you can define many more: $\sqrt{2}, \sqrt[3]{2}, ...$.
 \end{solution}
 
 \vfill

Advanced/Definable Sets/parts/2 quantifiers.tex

@@ -37,22 +37,22 @@ Which are true in $\mathbb{R}^+_0$? \par
 
 	\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.}
+	\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}
+%\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

Advanced/Definable Sets/parts/3 sets.tex

@@ -32,21 +32,28 @@ Only even numbers satisfy the formula $\varphi(x) = \exists y ~ (y + y = x)$, \p
 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{}
+Is the empty set definable in any structure?
+
+\vfill
+
+
+\problem{}
+Define $\{0, 1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$
+
+\begin{instructornote}
+	Here's an interesting fact:
+
+	A finite set of definable elements is always definable. Why? \par
+	An infinite set of definable elements may not be definable.
+\end{instructornote}
+
+\vfill
+
 \problem{}
 Define the set of prime 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
 
 
@@ -84,7 +91,9 @@ Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
 \vfill
 
 \problem{}
-Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
+Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$ \par
+\hint{We can't formally define a relation yet. Don't worry about that for now. \\
+You can repharase this question as \say{given $a,b \in \mathbb{Z}$, can you write a sentence that is true iff $a < b$?}}
 
 \vfill
 \pagebreak
@@ -94,7 +103,7 @@ Consider the structure $S = ( \mathbb{R} ~|~ \{0, \diamond \} )$ \par
 The relation $a \diamond b$ holds if $| a - b | = 1$
 
 \problempart{}
-Define the empty set in $S$.
+Define 0 in $S$.
 
 \problempart{}
 Define $\{-1, 1\}$ in $S$.
@@ -106,10 +115,12 @@ Define $\{-2, 2\}$ in $S$.
 
 \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
+Let $S$ be the stucture $( P ~|~ \{\subseteq\})$ \par
 
 \problempart{}
-Show that the empty set is definable in $S$.
+Show that the empty set is definable in $S$. \par
+\hint{Defining $\{\}$ with $\{x ~|~ x \neq x\}$ is \textbf{not} what we need here. \\
+We need $\varnothing \in P$, the \say{empty set} element in the power set of $\mathbb{Z}^+_0$.}
 
 \problempart{}
 Let $x \Bumpeq y$ be a relation on $P$. $x \Bumpeq y$ holds if $x \cap y \neq \{\}$. \par