\section{Definable Sets}

Armed with $(), \land, \lor, \lnot, \rightarrow, \forall,$ and $\exists$, we have enough tools to define sets.

\definition{Set-Builder Notation}
Say we have a sentence $\varphi(x)$. \par
The set of all elements that satisfy that sentence can be written as follows:
\begin{equation*}
	\{ x ~|~ \varphi(x) \}
\end{equation*}
This is read \say{The set of $x$ where $\varphi$ is true} or \say{The set of $x$ that satisfy $\varphi$.}

\vspace{2mm}

For example, take the formula $\varphi(x) = \exists y ~ (y + y = x)$. \par
The set of all even integers can then be written as
$$
	\{ x ~|~ \exists y ~ (y + y = x) \}
$$

\definition{Definable Sets}
Let $S$ be a structure with a universe $U$. \par
We say a subset $M$ of $U$ is \textit{definable} if we
can write a formula that is true for some $x$ iff $x \in M$.

\vspace{4mm}

For example, consider the structure $\big\langle~ \mathbb{Z} ~\big|~ \{+\} ~\big\rangle$ \par

Only even numbers satisfy the formula $\varphi(x) = \exists y ~ (y + y = x)$, \par
so we can define \say{the set of even numbers} as $\{ x ~|~ \exists y ~ (y + y = x) \}$. \par
Remember---we can only use symbols that are available in our structure!

\problem{}
When is the empty set definable?

\begin{solution}
	Always: $\{ x ~|~ \lnot (x = x) \}$
\end{solution}

\vfill


\problem{}
Define $\{0, 1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$

\begin{instructornote}
	Here's an interesting fact:

	A finite set of definable elements is always definable. \note{(Why?)} \par
	An infinite set of definable elements might not be definable.
\end{instructornote}

\vfill

\problem{}
Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$


\vfill
\pagebreak






\problem{}
Define $\mathbb{R}^+_0$ in $\Bigl( \mathbb{R} ~\big|~ \{\times\} \Bigr)$ \par

\vfill

\problem{}
Let $\bigtriangleup$ be a relational symbol. $a \bigtriangleup b$ is true if and only if $a$ divides $b$. \par
Define the set of prime numbers in $\Bigl( \mathbb{Z}^+ ~\big|~ \{ \bigtriangleup \} \Bigr)$ \par

\vfill

\theorem{Lagrange's Four Square Theorem}
Every natural number may be written as a sum of four integer squares.

\problem{}
Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$

\vfill

\problem{}
Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$ \par
\hint{We can't formally define a relation yet. Don't worry about that for now. \\
You can repharase this question as \say{given $a,b \in \mathbb{Z}$,\\*/ write a sentence that is only true if $a < b$?}}

\vfill
\pagebreak

\problem{}
Consider the structure $S = ( \mathbb{R} ~|~ \{0, \diamond \} )$ \par
The relation $a \diamond b$ holds if $| a - b | = 1$

\problempart{}
Define $\{-1, 1\}$ in $S$.

\problempart{}
Define $\{-2, 2\}$ in $S$.

\vfill

\problem{}
Let $\mathcal{P}$ be the set of all subsets of $\mathbb{Z}^+_0$. This is called the \textit{power set} of $\mathbb{Z}^+_0$. \par
Let $S$ be the structure $( \mathcal{P} ~|~ \{\subseteq\})$ \par

\problempart{}
Show that the empty set is definable in $S$. \par
\hint{Defining $\{\}$ with $\{x ~|~ x \neq x\}$ is \textbf{not} what we need here. \\
We need $\varnothing \in \mathcal{P}$, the \say{empty set} element in the power set of $\mathbb{Z}^+_0$.}

\vfill

\problempart{}
Let $x \Bumpeq y$ be a relation on $\mathcal{P}$. $x \Bumpeq y$ holds if $x \cap y \neq \{\}$. \par
Show that $\Bumpeq$ is definable in $S$.

\vfill

\problempart{}
Let $f$ be the function on $\mathcal{P}$ defined by $f(x) = \mathbb{Z}^+_0 - x$. This is called the \textit{complement} of $x$. \par
Show that $f$ is definable in $S$.

\vfill
\pagebreak