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66
Advanced/Definable Sets/parts/3 sets.tex
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@ -3,37 +3,40 @@
Armed with $(), \land, \lor, \lnot, \rightarrow, \forall,$ and $\exists$, we have enough tools to define sets.
\definition{Set-Builder Notation}
Say we have a condition $c$. \par
The set of all elements that satisfy that condition can be written as follows:
$$
\{ x ~|~ \text{$c$ is true} \}
$$
This is read \say{The set of $x$ where $c$ is true} or \say{The set of $x$ that satisfy $c$.}
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
The set of all even integers can then be written as
$$
\{ x ~|~ \varphi(x) \} = \{ x ~|~ \exists y ~ (y + y = x) \}
\{ x ~|~ \exists y ~ (y + y = x) \}
$$
\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 $x \in M$.
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 $\Bigl( \mathbb{Z} ~\big|~ \{+\} \Bigr)$ \par
\vspace{2mm}
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
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{}
Is the empty set definable in any structure?
When is the empty set definable?
\begin{solution}
Always: $\{ x ~|~ \lnot (x = x) \}$
\end{solution}
\vfill
@ -44,31 +47,22 @@ 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. Why? \par
An infinite set of definable elements may not be definable.
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
\vfill
\pagebreak
\problem{}
Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z)\} \Bigr)$ \par
\hint{$\text{real}(z)$ gives the real part of a complex number: $\text{real}(3 + 2i) = 3$}
\hint{$z$ is nonreal if $x \in \mathbb{C}$ and $x \notin \mathbb{R}$}
\begin{solution}
$\Bigl\{ x ~\bigl|~ \text{real}(x) \neq x \Bigr\}$
\end{solution}
\vfill
\problem{}
@ -77,7 +71,7 @@ Define $\mathbb{R}^+_0$ in $\Bigl( \mathbb{R} ~\big|~ \{\times\} \Bigr)$ \par
\vfill
\problem{}
Let $\bigtriangleup$ be a relational symbol. $a \bigtriangleup b$ holds iff $a$ divides $b$. \par
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
@ -93,7 +87,7 @@ Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
\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 $a,b \in \mathbb{Z}$, can you write a sentence that is true iff $a < b$?}}
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
@ -111,20 +105,24 @@ Define $\{-2, 2\}$ in $S$.
\vfill
\problem{}
Let $P$ be the set of all subsets of $\mathbb{Z}^+_0$. This is called a \textit{power set}. \par
Let $S$ be the structure $( P ~|~ \{\subseteq\})$ \par
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 $\varnothing \in P$, the \say{empty set} element in the power set of $\mathbb{Z}^+_0$.}
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 $P$. $x \Bumpeq y$ holds if $x \cap y \neq \{\}$. \par
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 a function on $P$ defined by $f(x) = \mathbb{Z}^+_0 - x$. This is called the \textit{complement} of the set $x$. \par
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