129 lines
3.6 KiB
TeX
129 lines
3.6 KiB
TeX
\section{Definable Sets}
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Armed with $(), \land, \lor, \lnot, \rightarrow, \forall,$ and $\exists$, we have enough tools to define sets.
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\definition{Set-Builder Notation}
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Say we have a sentence $\varphi(x)$. \par
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The set of all elements that satisfy that sentence can be written as follows:
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\begin{equation*}
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\{ x ~|~ \varphi(x) \}
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\end{equation*}
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This is read \say{The set of $x$ where $\varphi$ is true} or \say{The set of $x$ that satisfy $\varphi$.}
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\vspace{2mm}
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For example, take the formula $\varphi(x) = \exists y ~ (y + y = x)$. \par
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The set of all even integers can then be written as
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$$
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\{ x ~|~ \exists y ~ (y + y = x) \}
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$$
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\definition{Definable Sets}
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Let $S$ be a structure with a universe $U$. \par
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We say a subset $M$ of $U$ is \textit{definable} if we
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can write a formula that is true for some $x$ iff $x \in M$.
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\vspace{4mm}
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For example, consider the structure $\big\langle~ \mathbb{Z} ~\big|~ \{+\} ~\big\rangle$ \par
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Only even numbers satisfy the formula $\varphi(x) = \exists y ~ (y + y = x)$, \par
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so we can define \say{the set of even numbers} as $\{ x ~|~ \exists y ~ (y + y = x) \}$. \par
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Remember---we can only use symbols that are available in our structure!
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\problem{}
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When is the empty set definable?
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\begin{solution}
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Always: $\{ x ~|~ \lnot (x = x) \}$
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\end{solution}
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\vfill
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\problem{}
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Define $\{0, 1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$
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\begin{instructornote}
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Here's an interesting fact:
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A finite set of definable elements is always definable. \note{(Why?)} \par
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An infinite set of definable elements might not be definable.
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\end{instructornote}
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\vfill
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\problem{}
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Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
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\vfill
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\pagebreak
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\problem{}
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Define $\mathbb{R}^+_0$ in $\Bigl( \mathbb{R} ~\big|~ \{\times\} \Bigr)$ \par
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\vfill
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\problem{}
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Let $\bigtriangleup$ be a relational symbol. $a \bigtriangleup b$ is true if and only if $a$ divides $b$. \par
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Define the set of prime numbers in $\Bigl( \mathbb{Z}^+ ~\big|~ \{ \bigtriangleup \} \Bigr)$ \par
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\vfill
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\theorem{Lagrange's Four Square Theorem}
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Every natural number may be written as a sum of four integer squares.
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\problem{}
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Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
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\vfill
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\problem{}
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Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$ \par
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\hint{We can't formally define a relation yet. Don't worry about that for now. \\
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You can repharase this question as \say{given $a,b \in \mathbb{Z}$,\\*/ write a sentence that is only true if $a < b$?}}
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\vfill
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\pagebreak
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\problem{}
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Consider the structure $S = ( \mathbb{R} ~|~ \{0, \diamond \} )$ \par
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The relation $a \diamond b$ holds if $| a - b | = 1$
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\problempart{}
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Define $\{-1, 1\}$ in $S$.
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\problempart{}
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Define $\{-2, 2\}$ in $S$.
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\vfill
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\problem{}
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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
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Let $S$ be the structure $( \mathcal{P} ~|~ \{\subseteq\})$ \par
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\problempart{}
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Show that the empty set is definable in $S$. \par
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\hint{Defining $\{\}$ with $\{x ~|~ x \neq x\}$ is \textbf{not} what we need here. \\
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We need $\varnothing \in \mathcal{P}$, the \say{empty set} element in the power set of $\mathbb{Z}^+_0$.}
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\vfill
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\problempart{}
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Let $x \Bumpeq y$ be a relation on $\mathcal{P}$. $x \Bumpeq y$ holds if $x \cap y \neq \{\}$. \par
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Show that $\Bumpeq$ is definable in $S$.
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\vfill
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\problempart{}
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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
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Show that $f$ is definable in $S$.
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\vfill
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\pagebreak |