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