Added equivalence problems

Advanced/Definable Sets/main.tex

@@ -21,5 +21,6 @@
 	\input{parts/1 structures.tex}
 	\input{parts/2 quantifiers.tex}
 	\input{parts/3 sets.tex}
+	\input{parts/4 equivalence.tex}
 
 \end{document}

Advanced/Definable Sets/parts/1 structures.tex

@@ -14,7 +14,7 @@ A structure's symbols give meaning to the objects in its universe.
 
 \vspace{2mm}
 
-Symbols generally come in three types:
+Symbols come in three types:
 \begin{itemize}
 	\item Constant symbols, which let us specify specific elements of our universe. \par
 		Examples: $0, 1, \frac{1}{2}, \pi$

Advanced/Definable Sets/parts/4 equivalence.tex

@@ -0,0 +1,55 @@
+\section{Equivalence (Bonus)}
+
+\generic{Notation:}
+Let $S$ be a structure and $\varphi$ a formula. \par
+If $\varphi$ is true in $S$, we write $S \models \varphi$.
+
+\definition{}
+Let $S$ and $T$ be structures. \par
+We say $S$ and $T$ are \textit{equivalent} and write $S \equiv T$ if for any formula $\varphi$, $S \models \varphi \Longleftrightarrow T \models \varphi$.
+
+\problem{}
+Show that $
+	\Bigl(\mathbb{Z} ~\big|~ \{ +, 0 \}\Bigr)
+	\not\equiv
+	\Bigl(\mathbb{R} ~\big|~ \{ +, 0 \}\Bigr)
+$
+\vfill
+
+\problem{}
+Show that $
+	\Bigl(\mathbb{Z} ~\big|~ \{ +, 0 \}\Bigr)
+	\not\equiv
+	\Bigl(\mathbb{N} ~\big|~ \{ +, 0 \}\Bigr)
+$
+\vfill
+
+\problem{}
+Show that $
+	\Bigl(\mathbb{R} ~\big|~ \{ +, 0 \}\Bigr)
+	\not\equiv
+	\Bigl(\mathbb{N} ~\big|~ \{ +, 0 \}\Bigr)
+$
+\vfill
+
+\problem{}
+Show that $
+	\Bigl(\mathbb{R} ~\big|~ \{ +, 0 \}\Bigr)
+	\not\equiv
+	\Bigl(\mathbb{Z}^2 ~\big|~ \{ +, 0 \}\Bigr)
+$
+\vfill
+
+\problem{}
+Show that $
+	\Bigl(\mathbb{Z} ~\big|~ \{ +, 0 \}\Bigr)
+	\not\equiv
+	\Bigl(\mathbb{Z}^2 ~\big|~ \{ +, 0 \}\Bigr)
+$
+
+\begin{solution}
+	All of the above are easy, but the last one can take a while. \par
+	The trick is to notice that $\mathbb{Z}$ has two equivalence classes mod 2, while $\mathbb{Z}^2$ has four.
+\end{solution}
+
+\vfill