Cleanup

Advanced/Definable Sets/parts/1 structures.tex

@@ -86,7 +86,7 @@ For the sake of time, I will not provide a formal definition. It isn't particula
 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
+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}
@@ -99,7 +99,7 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi
 \hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$. Also, $2 \times 2 = 4$.}
 
 \begin{solution}
-	$2$ is the only element in $\mathbb{Z}^+$ that satisfies $[x \text{ where } x \times x = 4]$.
+	$2$ is the only element in $\mathbb{Z}^+$ that satisfies $\varphi(x) = [x \times x = 4]$.
 \end{solution}
 
 \vfill
@@ -109,7 +109,7 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi
 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 $-2$. \\
+	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.
 
 	\begin{instructornote}