Minor edits

Advanced/Definable Sets/parts/1 structures.tex

@@ -9,27 +9,31 @@ A \textit{universe} is a set of meaningless objects. Here are a few examples:
 \end{itemize}
 
 \definition{}
-A \textit{structure} consists of a universe $U$ and a set of symbols. \par
+A \textit{structure} consists of a universe and a set of \textit{symbols}. \par
 A structure's symbols give meaning to the objects in its universe.
 
 \vspace{2mm}
 
 Symbols come in three types:
 \begin{itemize}
-	\item Constant symbols, which let us specify specific elements of our universe. \par
+	\item \textit{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 universe. \par
-		Examples: $+, \times, \sin{x}, \sqrt{x}$
+	\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
+		The only difference between $a + b$ and $+(a, b)$ is notation.}
 		\vspace{2mm}
 
-	\item Relation symbols, which let us compare elements of our universe. \par
+	\item \textit{Relation symbols}, which let us compare elements of our universe. \par
 		Examples: $<, >, \leq, \geq$ \par
 		\vspace{2mm}
 \end{itemize}
 
-The equality check $=$ is \textbf{not} a relation symbol. It is included in every structure by default.
+The equality check $=$ is \textbf{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.
+
 
 \vspace{3mm}
 
@@ -42,7 +46,7 @@ $$
 
 \vspace{2mm}
 
-This is a structure with the universe $\mathbb{Z}$ that contains the following symbols:
+This is a structure over the universe $\mathbb{Z}$ that provides the following symbols:
 \begin{itemize}
 	\item Constants: \tab $\{0, 1\}$
 	\item Functions: \tab $\{+, -\}$
@@ -51,24 +55,37 @@ This is a structure with the universe $\mathbb{Z}$ that contains the following s
 
 \vspace{2mm}
 
-If we look at our set of constant symbols, we 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 this structure offers.
+If we look at our set of constant symbols, we 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 this structure offers.
 
-\vspace{1mm}
+\vspace{2mm}
 
-To \say{define} an element of a set, we need to write a sentence that is only true for that element. For example, if we want to define 2 in the structure above, we could use the sentence $\varphi(x) = [1 + 1 = x]$. \par
-Clearly, this is only true when $x = 2$.
+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 $[1 + 1 = x]$
+evaluates to \texttt{true}.
 
 
 \problem{}
 Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)$.
 
+\begin{solution}
+	The sentences \say{$x$ where $[x + 1 = 0]$} and \say{$x$ where $[0 - 1 = x]$} both work.
+\end{solution}
+
 \vfill
 \pagebreak
 
+
+
+
 Let us formalize what we found in the previous two problems. \par
 
-\definition{}
-A \textit{formula} in a structure $S$ is a well-formed string of constants, functions, and relations. \par
+\definition{Formulas}
+A \textit{formula} in a structure $S$ is a well-formed string
+of constants, functions, relations, \par and logical operators.
 
 \vspace{2mm}
 
@@ -77,15 +94,20 @@ For the sake of time, I will not provide a formal definition. It isn't particula
 
 \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.
+
+
+\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 $\varphi(x) = [x > 0]$. \par
-$\varphi(3)$ is true and $\varphi(-3)$ is false. \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
 
 \vspace{2mm}
 
-This \say{free variable} notation is much like the function notation you are used to: \par
+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 $x$.