From 90fd1e6ee133ede20f6271226ceff2db5fd59c95 Mon Sep 17 00:00:00 2001 From: Mark Date: Tue, 9 May 2023 21:23:09 -0700 Subject: [PATCH] Edits --- Advanced/Definable Sets/parts/0 logic.tex | 14 ++++- .../Definable Sets/parts/1 structures.tex | 61 ++++++++----------- .../Definable Sets/parts/2 quantifiers.tex | 35 ++++++++++- Advanced/Definable Sets/parts/3 sets.tex | 47 ++++++++------ 4 files changed, 101 insertions(+), 56 deletions(-) diff --git a/Advanced/Definable Sets/parts/0 logic.tex b/Advanced/Definable Sets/parts/0 logic.tex index 2d76a30..6180aca 100644 --- a/Advanced/Definable Sets/parts/0 logic.tex +++ b/Advanced/Definable Sets/parts/0 logic.tex @@ -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} diff --git a/Advanced/Definable Sets/parts/1 structures.tex b/Advanced/Definable Sets/parts/1 structures.tex index 5cc22c8..e161337 100644 --- a/Advanced/Definable Sets/parts/1 structures.tex +++ b/Advanced/Definable Sets/parts/1 structures.tex @@ -1,7 +1,7 @@ \section{Structures} \definition{} -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 diff --git a/Advanced/Definable Sets/parts/2 quantifiers.tex b/Advanced/Definable Sets/parts/2 quantifiers.tex index d63d8b7..f934cdc 100644 --- a/Advanced/Definable Sets/parts/2 quantifiers.tex +++ b/Advanced/Definable Sets/parts/2 quantifiers.tex @@ -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 diff --git a/Advanced/Definable Sets/parts/3 sets.tex b/Advanced/Definable Sets/parts/3 sets.tex index bc8e747..c65cbe5 100644 --- a/Advanced/Definable Sets/parts/3 sets.tex +++ b/Advanced/Definable Sets/parts/3 sets.tex @@ -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 \ No newline at end of file