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Mark
2023-05-09 21:23:09 -07:00
parent 3b197b0759
commit 90fd1e6ee1
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47
Advanced/Definable Sets/parts/3 sets.tex
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@ -10,6 +10,13 @@ $$
$$
This is read \say{The set of $x$ where $c$ is true} or \say{The set of $x$ that satisfy $c$.}
\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
$$
\{ x ~|~ \varphi(x) \} = \{ x ~|~ \exists y ~ (y + y = x) \}
$$
\definition{Definable Sets}
Let $S$ be a structure over a language $\mathcal{L}$. \par
@ -30,15 +37,6 @@ Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\
\vfill
\problem{}
Define the set of rational numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
\vfill
\problem{}
Define the set of irrational numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
\vfill
\problem{}
Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
@ -53,7 +51,22 @@ Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
\problem{}
Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ i, ~ \text{real}(z), \times\} \Bigr)$ \par
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}
$\Bigl\{ x ~\bigl|~ \lnot (\text{real}(i \times x) = 0) \Bigr\}$
\end{solution}
\vfill
\problem{}
Define the set of integers in $\Bigl( \mathbb{R} ~\big|~ \{\times\} \Bigr)$ \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 $x \notin \mathbb{R}$}
@ -63,21 +76,17 @@ Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ i, ~ \tex
\vfill
\problem{}
Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{0, ~ \text{real}(z), \times\} \Bigr)$ \par
\theorem{Lagrange's Four Square Theorem}
Every natural number may be written as a sum of four integer squares.
\begin{solution}
$\Biggl\{ x ~\Bigl|~ \forall y ~ \Bigl( \text{real}(y) = 0 \rightarrow \lnot \bigl[ \text{real}(x \times y) = 0 \bigr] \Bigr) \Biggr\}$
\end{solution}
\problem{}
Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
\vfill
\problem{}
Define $\mathbb{R}$ in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z), \times\} \Bigr)$ \par
Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
\begin{solution}
$\Biggl\{ x ~\Bigl|~ \forall y ~ \Bigl( \text{real}(x) \times y = \text{real}(x) \Bigr) \Biggr\}$
\end{solution}
\vfill
\pagebreak