Advanced handouts

Add missing file
Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>

src/Advanced/Definable Sets/parts/4 equivalence.tex

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+\section{Equivalence}
+
+\generic{Notation:}
+Let $S$ be a structure and $\varphi$ a formula. \par
+If $\varphi$ is true in $S$, we write $S \models \varphi$. \par
+This is read \say{$S$ satisfies $\varphi$}
+
+\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 $\varphi$, $S \models \varphi \Longleftrightarrow T \models \varphi$. \par
+If $S$ and $T$ are not equivalent, we write $S \not\equiv T$.
+
+\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