Edits

Advanced/Definable Sets/parts/1 structures.tex

@@ -22,7 +22,7 @@ Symbols come in three types:
 
 	\item \textit{Function symbols}, which let us navigate between elements of our universe. \par
 		Examples: $+, \times, \sin{x}, \sqrt{x}$ \par
-		\note{In this handout, symbols we usually call \say{operators} are also called functions. \par
+		\note{Note that symbols we usually call \say{operators} are functions under this definition. \par
 		The only difference between $a + b$ and $+(a, b)$ is notation.}
 		\vspace{2mm}
 
@@ -31,7 +31,7 @@ Symbols come in three types:
 		\vspace{2mm}
 \end{itemize}
 
-The equality check $=$ is \textbf{not} a relation symbol. It is included in every structure by default. \par
+The equality check $=$ is \textit{not} a relation symbol. It is included in every structure by default. \par
 By definition, $a = b$ is true if and only if $a$ and $b$ are the same element of our universe.
 
 
@@ -61,10 +61,15 @@ others, we must define them using the tools this structure offers.
 
 \vspace{2mm}
 
+% NOTE: this is a great example for typesetting.
+% The line breaks here are ugly without a centered sentence.
 To \say{define} an element of a set, we need to write a sentence that is only true for that element. \par
-For example, if we want to define 2 in the structure above, \par
-we could use the sentence \say{$2$ is the $x$ that satisfies $[1 + 1 = x]$.} \par
-This is a valid definition because $2$ is the \textbf{only} element of $\mathbb{Z}$ for which $[1 + 1 = x]$
+If we want to define 2 in the structure above,
+we could use the following sentence:
+\begin{center}
+	\say{$2$ is the $x$ that satisfies $[1 + 1 = x]$.} \par
+\end{center}
+This is a valid definition because $2$ is the \textit{only} element of $\mathbb{Z}$ for which $[1 + 1 = x]$
 evaluates to \texttt{true}.
 
 
@@ -81,6 +86,9 @@ Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)$.
 
 
 
+
+
+
 Let us formalize what we found in the previous two problems. \par
 
 \definition{Formulas}
@@ -89,31 +97,51 @@ of constants, functions, relations, \par and logical operators.
 
 \vspace{2mm}
 
-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.
+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}
 
-As a quick example, the formula $\phi(x) = [1 + 1 = x]$ evaluates to \texttt{true} when $x$ is 2 \par
-and to \texttt{false} otherwise.
+As a quick example, the formula $\psi \coloneqq [\lnot (1 = 1)]$ is always false, \par
+and $\varphi(x) \coloneqq [1 + 1 = x]$ evaluates to \texttt{true} only when $x$ is 2.
+
+
+
 
 
 \definition{Free Variables}
 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 above. \par
-$\varphi(2)$ is \texttt{true} and $\varphi(-3)$ is \texttt{false}. \par
+Formulas with free variables let us define \say{properties} that certain objects have.
 
 \vspace{2mm}
 
-This \say{free variable} notation is much like the function notation we are used to: \par
-$\varphi(x) = [x > 0]$ is similar to $f(x) = x + 1$, since the values of $\varphi(x)$ and $f(x)$ depend on $x$.
+For example, consider the two formulas from the previous definition, $\psi$ and $\varphi$:
+\begin{itemize}
+	\item $\psi \coloneqq [\lnot (1 = 1)]$ \par
+	There are no free variables in this formula. \par
+	In any structure, $\psi$ is always either \texttt{true} or \texttt{false}.
+
+	\vspace{2mm}
+
+	\item $\varphi(x) \coloneqq [1 + 1 = x]$ \par
+	This formula has one free variable, labeled $x$. \par
+	The value of $\varphi(x)$ depends on the $x$ we're talking about: \par
+	$\varphi(72)$ is false, and $\varphi(2)$ is true.
+\end{itemize}
+
+\vspace{2mm}
+
+\note{
+	This \say{free variable} notation is very similar to the function notation we are used to: \par
+	The values of both $\varphi(x) \coloneqq [x > 0]$ and $f(x) = x + 1$ depend on $x$.
+}
+
+
 
 
 \definition{Definable Elements}
-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.
+Let $S$ be a structure over a universe $U$. \par
+We say an element $x \in U$ is \textit{definable in $S$} if we can write a formula $\varphi(x)$ that only $x$ satisfies.
 
 
 \problem{}
@@ -121,10 +149,16 @@ Define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bigr)$. \
 \hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$. Also, $2 \times 2 = 4$.}
 
 \begin{solution}
-	$2$ is the only element in $\mathbb{Z}^+$ that satisfies $\varphi(x) = [x \times x = 4]$.
+	$2$ is the only element in $\mathbb{Z}^+$ that satisfies $\varphi(x) \coloneqq [x \times x = 4]$.
 \end{solution}
 
+
 \vfill
+\pagebreak
+
+
+
+
 
 
 \problem{}
@@ -132,8 +166,10 @@ Try to define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr
 Why can't you do it?
 
 \begin{solution}
-	This isn't possible. We could try $\varphi(x) = [x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \par
-	We have no way to distinguish between negative and positive numbers.
+	We could try $\varphi(x) \coloneqq [x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \par
+	We have no way to distinguish between negative and positive numbers. \par
+
+	\note{This problem is intentionally hand-wavy. We don't have the tools to write a proper proof.}
 
 	\begin{instructornote}
 		Actually, it is. Bonus problem: how? \par
@@ -145,12 +181,21 @@ Why can't you do it?
 
 
 \problem{}
-What numbers are definable in the structure $\Bigl( \mathbb{R}^+_0 ~\big|~ \{1, 2, \div \} \Bigr)$?
+Consider the structure $\Bigl( \mathbb{R}^+_0 ~\big|~ \{1, 2, \div \} \Bigr)$
+
+\begin{itemize}
+	\item Define $2^2$
+	\item Define $2^n$ for all positive integers $n$
+	\item Define $2^{-n}$ for all positive integers $n$
+
+	\item What other numbers can we define in this structure? \par
+	\hint{There is at least one more \say{class} of numbers we can define.}
+\end{itemize}
+
 
 \begin{solution}
-	We can define powers of two, positive and negative.
-
-	If you're clever, you can define many more: $\sqrt{2}, \sqrt[3]{2}, ...$.
+	As far as I've seen, we can define any $2^{\nicefrac{a}{b}}$ for $a, b \in \mathbb{Z}$. \par
+	For example, $\phi(x) \coloneqq [2 = x \div (1 \div x)]$ defines $\sqrt{2}$.
 \end{solution}
 
 \vfill