Post-class edits

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