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14
Advanced/Definable Sets/parts/0 logic.tex
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@ -73,13 +73,23 @@ Evaluate the following.
\begin{itemize}
\item $(T \land F) \lor T$
\item $(\lnot (F \lor \lnot T) ) \rightarrow T$
\item $A \rightarrow T$ for any $A$
\item $(\lnot (A \rightarrow B)) \rightarrow A$ for any $A,B$
\item $(F \rightarrow T) \rightarrow (\lnot F \lor \lnot T)$
\end{itemize}
\vfill
\pagebreak
\problem{}
Evaluate the following.
\begin{itemize}
\item $A \rightarrow T$ for any $A$
\item $(\lnot (A \rightarrow B)) \rightarrow A$ for any $A, B$
\item $(A \rightarrow B) \rightarrow (\lnot B \rightarrow \lnot A)$ for any $A, B$
\end{itemize}
\vfill
\problem{}
Show that $\lnot (A \rightarrow \lnot B)$ is equivalent to $A \land B$. \par
\hint{Use a truth table}

61
Advanced/Definable Sets/parts/1 structures.tex
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@ -1,7 +1,7 @@
\section{Structures}
\definition{}<def:language>
A \textit{language} is a set of meaningless symbols. Here are a few examples:
A \textit{language} is a set of meaningless objects. Here are a few examples:
\begin{itemize}
\item $\{a, b, ..., z\}$
\item $\{0, 1\}$
@ -12,30 +12,27 @@ Every language comes with the equality check $=$, which checks if two elements a
\definition{}
A \textit{structure} over a language $\mathcal{L}$ consists of three sets:
\begin{itemize}
\item A set of \textit{constant symbols} $\mathcal{C}$ \par
Constant symbols let us specify specific elements of our language. \par
$\mathcal{C}$ must thus be a subset of $\mathcal{L}$.
\vspace{3mm}
\item A set of \textit{function symbols} $\mathcal{F}$ \par
Function symbols let us navigate between elements of our language. \par
$+$, $-$ are functions, as are $\sin{x}$, $\cos{x}$, and $\sqrt{x}$ \par
Functions take inputs in $\mathcal{L}$ and produce outputs in $\mathcal{L}$.
\vspace{3mm}
\item A set of \textit{relation symbols} $\mathcal{R}$ \par
Relation symbols let us compare elements of our language. \par
You are already familiar with this concept: $>$, $<$, and $\leq$ are relation symbols. \par
$=$ is \textbf{not} a relational symbol. Why? \hint{See \ref{def:language}}
\end{itemize}
A \textit{structure} over a language $\mathcal{L}$ consists of a set of symbols. \par
The purpose of a structure is to give a language meaning.
\vspace{2mm}
The purpose of a structure is to give a language meaning. This is best explained by example.
Symbols generally come in three types:
\begin{itemize}
\item Constant symbols, which let us specify specific elements of our language. \par
Examples: $0, 1, \frac{1}{2}, \pi$
\vspace{2mm}
\item Function symbols, which let us navigate between elements of our language. \par
Examples: $+, \times, \sin{x}, \sqrt{x}$
\vspace{2mm}
\item Relation symbols, which let us compare elements of our language. \par
Examples: $<, >, \leq, \geq$ \par
The symbol $=$ is \textbf{not} a relation. Why? \hint{See \ref{def:language}}
\vspace{2mm}
\end{itemize}
\vspace{3mm}
@ -45,25 +42,21 @@ The purpose of a structure is to give a language meaning. This is best explained
The first structure we'll look at is the following:
$$
\Bigl(
\mathcal{L} ~\big|~ \{\mathcal{C}, ~ \mathcal{F}, ~ \mathcal{R}\}
\Bigr)
=
\Bigl( \mathbb{Z} ~\big|~ \{0, 1, ~ +, -, ~ <\} \Bigr)
\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)
$$
\vspace{2mm}
This is a structure over $\mathbb{Z}$ with the following symbols:
\begin{itemize}
\item $\mathcal{C} = \{0, 1\}$ \tab \note{(constants)}
\item $\mathcal{F} = \{+, -\}$ \tab \note{(functions)}
\item $\mathcal{R} = \{<\}$ \tab \note{(relations)}
\item Constants: \tab $\{0, 1\}$
\item Functions: \tab $\{+, -\}$
\item Relations: \tab $\{<\}$
\end{itemize}
\vspace{2mm}
Let's look at $\mathcal{C}$, our set of constant symbols. The only integers we can directly refer to in this structure are 0 and 1. If we want any others, we must define them using the tools the structure offers.
If you look at our set of constant symbols, you'll see that the only integers we can directly refer to in this structure are 0 and 1. If we want any others, we must define them using the tools the structure offers.
\vspace{1mm}
@ -104,7 +97,7 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi
\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$}
\begin{solution}
Yes! $-2$ no longer exists, so $2$ can be defined by $[x \text{ where } x \times x = 4]$.
$-2 \notin \mathbb{Z}^+$, so $2$ can be defined by $[x \text{ where } x \times x = 4]$.
\end{solution}
\vfill
@ -122,10 +115,10 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr
\problem{}
What is definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
What numbers are definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$?
\begin{solution}
All powers of two, positive and negative.
With the tools we have so far, we can only define powers of two, positive and negative.
\end{solution}
\vfill

35
Advanced/Definable Sets/parts/2 quantifiers.tex
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@ -58,8 +58,41 @@ Which are true in $\mathbb{R}^+_0$? \par
\vfill
\pagebreak
\problem{}
Define $\forall$ using logical symbols and $\exists$
Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
\vfill
\problem{}
Define 1 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
\vfill
\problem{}
Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
\vfill
\problem{}
Define $3$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
\vfill
\pagebreak
\problem{}
Let $\varphi(x)$ be a formula. \par
Define $(\forall x ~ \varphi(x))$ using logical symbols and $\exists$.
\begin{solution}
$\Bigl(\forall x ~ \varphi(x)\Bigr)$ is true iff $\lnot \Bigl(\exists x ~ \lnot \varphi(x) \Bigr)$ is true.
\end{solution}
\vfill

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