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Merge branch 'master' of ssh://git.betalupi.com:33/Mark/ormc-handouts

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Mark 2023-05-13 17:15:22 -07:00
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6
Advanced/Definable Sets/parts/3 sets.tex
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@ -94,10 +94,10 @@ Consider the structure $S = ( \mathbb{R} ~|~ \{0, \diamond \} )$ \par
The relation $a \diamond b$ holds if $| a - b | = 1$ The relation $a \diamond b$ holds if $| a - b | = 1$
\problempart{} \problempart{}
Define $\{\}$ in $S$. Define the empty set in $S$.
\problempart{} \problempart{}
Define ${-1, 1}$ in $S$. Define $\{-1, 1\}$ in $S$.
\problempart{} \problempart{}
Define $\{-2, 2\}$ in $S$. Define $\{-2, 2\}$ in $S$.
@ -116,7 +116,7 @@ Let $x \Bumpeq y$ be a relation on $P$. $x \Bumpeq y$ holds if $x \cap y \neq \{
Show that $\Bumpeq$ is definable in $S$. Show that $\Bumpeq$ is definable in $S$.
\problempart{} \problempart{}
Let $f$ be a function on $P$ defined by $f(x) = \mathbb{Z}^+_0 - x$. This is called the \textit{compliment} of the set $x$. \par Let $f$ be a function on $P$ defined by $f(x) = \mathbb{Z}^+_0 - x$. This is called the \textit{complement} of the set $x$. \par
Show that $f$ is definable in $S$. Show that $f$ is definable in $S$.
\vfill \vfill