56 lines
1.2 KiB
TeX
56 lines
1.2 KiB
TeX
\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 $\varphi$, $S \models \varphi \Longleftrightarrow T \models \varphi$.
|
|
|
|
\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 |