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.
