Edits

Advanced/Definable Sets/parts/0 logic.tex

@@ -73,13 +73,23 @@ 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 $(F \rightarrow T) \rightarrow (\lnot F \lor \lnot T)$
-	\item $(\lnot (A \rightarrow B)) \rightarrow A$ for any $A,B$
 \end{itemize}
 
 \vfill
 \pagebreak
 
+
+\problem{}
+Evaluate the following.
+\begin{itemize}
+	\item $A \rightarrow T$ for any $A$
+	\item $(\lnot (A \rightarrow B)) \rightarrow A$ for any $A, B$
+	\item $(A \rightarrow B) \rightarrow (\lnot B \rightarrow \lnot A)$ for any $A, B$
+\end{itemize}
+
+\vfill
+
 \problem{}
 Show that $\lnot (A \rightarrow \lnot B)$ is equivalent to $A \land B$. \par
 \hint{Use a truth table}

Advanced/Definable Sets/parts/1 structures.tex

@@ -1,7 +1,7 @@
 \section{Structures}
 
 \definition{}<def:language>
-A \textit{language} is a set of meaningless symbols. Here are a few examples:
+A \textit{language} is a set of meaningless objects. Here are a few examples:
 \begin{itemize}
 	\item $\{a, b, ..., z\}$
 	\item $\{0, 1\}$
@@ -12,30 +12,27 @@ Every language comes with the equality check $=$, which checks if two elements a
 
 
 \definition{}
-A \textit{structure} over a language $\mathcal{L}$ consists of three sets:
+A \textit{structure} over a language $\mathcal{L}$ consists of a set of symbols. \par
-\begin{itemize}
+The purpose of a structure is to give a language meaning.
-	\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.
+Symbols generally come in three types:
+\begin{itemize}
+	\item Constant symbols, which let us specify specific elements of our language. \par
+		Examples: $0, 1, \frac{1}{2}, \pi$
+		\vspace{2mm}
+
+	\item Function symbols, which let us navigate between elements of our language. \par
+		Examples: $+, \times, \sin{x}, \sqrt{x}$
+		\vspace{2mm}
+
+	\item Relation symbols, which let us compare elements of our language. \par
+		Examples: $<, >, \leq, \geq$ \par
+		The symbol $=$ is \textbf{not} a relation. Why? \hint{See \ref{def:language}}
+		\vspace{2mm}
+
+\end{itemize}
 
 \vspace{3mm}
 
@@ -45,25 +42,21 @@ The purpose of a structure is to give a language meaning. This is best explained
 
 The first structure we'll look at is the following:
 $$
-	\Bigl(
+	\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)
-		\mathcal{L} ~\big|~ \{\mathcal{C}, ~ \mathcal{F}, ~ \mathcal{R}\}
-	\Bigr)
-	=
-	\Bigl( \mathbb{Z} ~\big|~ \{0, 1, ~ +, -, ~ <\} \Bigr)
 $$
 
 \vspace{2mm}
 
 This is a structure over $\mathbb{Z}$ with the following symbols:
 \begin{itemize}
-	\item $\mathcal{C} = \{0, 1\}$   \tab \note{(constants)}
+	\item Constants: \tab $\{0, 1\}$
-	\item $\mathcal{F} = \{+, -\}$   \tab \note{(functions)}
+	\item Functions: \tab $\{+, -\}$
-	\item $\mathcal{R} = \{<\}$      \tab \note{(relations)}
+	\item Relations: \tab $\{<\}$
 \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.
+If you look at our set of constant symbols, you'll see that 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}
 
@@ -104,7 +97,7 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi
 \hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$}
 
 \begin{solution}
-	Yes! $-2$ no longer exists, so $2$ can be defined by $[x \text{ where } x \times x = 4]$.
+	$-2 \notin \mathbb{Z}^+$, so $2$ can be defined by $[x \text{ where } x \times x = 4]$.
 \end{solution}
 
 \vfill
@@ -122,10 +115,10 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr
 
 
 \problem{}
-What is definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
+What numbers are definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
 
 \begin{solution}
-	All powers of two, positive and negative.
+	With the tools we have so far, we can only define powers of two, positive and negative.
 \end{solution}
 
 \vfill

Advanced/Definable Sets/parts/2 quantifiers.tex

@@ -58,8 +58,41 @@ Which are true in $\mathbb{R}^+_0$? \par
 \vfill
 \pagebreak
 
+
 \problem{}
-Define $\forall$ using logical symbols and $\exists$
+Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
+
+
+\vfill
+
+
+\problem{}
+Define 1 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
+
+
+\vfill
+
+
+\problem{}
+Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
+
+\vfill
+
+\problem{}
+Define $3$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
+
+
+
+\vfill
+\pagebreak
+
+\problem{}
+Let $\varphi(x)$ be a formula. \par
+Define $(\forall x ~ \varphi(x))$ using logical symbols and $\exists$.
+
+\begin{solution}
+	$\Bigl(\forall x ~ \varphi(x)\Bigr)$ is true iff $\lnot \Bigl(\exists x ~ \lnot \varphi(x) \Bigr)$ is true.
+\end{solution}
 
 \vfill
 

Advanced/Definable Sets/parts/3 sets.tex

@@ -10,6 +10,13 @@ $$
 $$
 This is read \say{The set of $x$ where $c$ is true} or \say{The set of $x$ that satisfy $c$.}
 
+\vspace{2mm}
+
+For example, take the formula $\varphi(x) = \exists y ~ (y + y = x)$. \par
+The set of all even integers can then be written
+$$
+	\{ x ~|~ \varphi(x) \} = \{ x ~|~ \exists y ~ (y + y = x) \}
+$$
 
 \definition{Definable Sets}
 Let $S$ be a structure over a language $\mathcal{L}$. \par
@@ -30,15 +37,6 @@ Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\
 
 \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 $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
@@ -53,7 +51,22 @@ Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
 
 
 \problem{}
-Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ i, ~ \text{real}(z), \times\} \Bigr)$ \par
+Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z)\} \Bigr)$ \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 integers 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 $x \notin \mathbb{R}$}
 
@@ -63,21 +76,17 @@ Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ i, ~ \tex
 
 \vfill
 
-\problem{}
+\theorem{Lagrange's Four Square Theorem}
-Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ \text{real}(z), \times\} \Bigr)$ \par
+Every natural number may be written as a sum of four integer squares.
 
-\begin{solution}
+\problem{}
-	$\Biggl\{ x ~\Bigl|~ \forall y ~ \Bigl( \text{real}(y) = 0 \rightarrow \lnot \bigl[ \text{real}(x \times y) = 0 \bigr] \Bigr) \Biggr\}$
+Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
-\end{solution}
 
 \vfill
 
 \problem{}
-Define $\mathbb{R}$ in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z), \times\} \Bigr)$ \par
+Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
 
-\begin{solution}
-	$\Biggl\{ x ~\Bigl|~ \forall y ~ \Bigl( \text{real}(x) \times y = \text{real}(x) \Bigr) \Biggr\}$
-\end{solution}
 
 \vfill
 \pagebreak