83 lines
2.4 KiB
TeX
83 lines
2.4 KiB
TeX
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\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 condition $c$. \par
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The set of all elements that satisfy that condition can be written as follows:
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$$
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\{ x ~|~ \text{$c$ is true} \}
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$$
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This is read \say{The set of $x$ where $c$ is true} or \say{The set of $x$ that satisfy $c$.}
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\definition{Definable Sets}
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Let $S$ be a structure over a language $\mathcal{L}$. \par
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We say a subset $M$ of $\mathcal{L}$ is \textit{definable} if we 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 $\Bigl( \mathbb{Z} ~\big|~ \{+\} \Bigr)$ \par
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\vspace{2mm}
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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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Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
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\vfill
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\problem{}
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Define the set of rational numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
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\vfill
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\problem{}
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Define the set of irrational numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
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\vfill
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\problem{}
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Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
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\vfill
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\problem{}
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Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
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\vfill
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\pagebreak
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\problem{}
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Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ i, ~ \text{real}(z), \times\} \Bigr)$ \par
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\hint{$\text{real}(z)$ gives the real part of a complex number: $\text{real}(3 + 2i) = 3$}
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\hint{$z$ is nonreal if $x \in \mathbb{C}$ and $x \notin \mathbb{R}$}
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\begin{solution}
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$\Bigl\{ x ~\bigl|~ \lnot (\text{real}(i \times x) = 0) \Bigr\}$
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\end{solution}
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\vfill
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\problem{}
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Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ \text{real}(z), \times\} \Bigr)$ \par
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\begin{solution}
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$\Biggl\{ x ~\Bigl|~ \forall y ~ \Bigl( \text{real}(y) = 0 \rightarrow \lnot \bigl[ \text{real}(x \times y) = 0 \bigr] \Bigr) \Biggr\}$
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\end{solution}
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\vfill
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\problem{}
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Define $\mathbb{R}$ in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z), \times\} \Bigr)$ \par
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\begin{solution}
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$\Biggl\{ x ~\Bigl|~ \forall y ~ \Bigl( \text{real}(x) \times y = \text{real}(x) \Bigr) \Biggr\}$
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\end{solution}
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\vfill
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\pagebreak
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