diff --git a/Advanced/Definable Sets/main.tex b/Advanced/Definable Sets/main.tex index 0e5f7bb..6929190 100755 --- a/Advanced/Definable Sets/main.tex +++ b/Advanced/Definable Sets/main.tex @@ -5,10 +5,6 @@ singlenumbering ]{../../resources/ormc_handout} -% Typewriter tabs -\usepackage{tabto} -\TabPositions{1cm, 2cm, 3cm, 4cm, 5cm, 6cm, 7cm, 8cm} - % for \coloneqq, a centered := \usepackage{mathtools} diff --git a/Advanced/Definable Sets/parts/1 structures.tex b/Advanced/Definable Sets/parts/1 structures.tex index 1c84f35..6a5e103 100644 --- a/Advanced/Definable Sets/parts/1 structures.tex +++ b/Advanced/Definable Sets/parts/1 structures.tex @@ -90,16 +90,16 @@ For example, $x$ is a free variable in the formula $\varphi(x) = x > 0$. \par $\varphi(3)$ is true and $\varphi(-3)$ is false. \definition{Definable Elements} -Say $S$ is a with a universe $U$. \par +Say $S$ is a structure with a universe $U$. \par We say an element $e \in U$ is \textit{definable in $S$} if we can write a formula that only $e$ satisfies. \problem{} Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bigr)$? \par -\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$} +\hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$. Also, $2 \times 2 = 4$.} \begin{solution} - $-2 \notin \mathbb{Z}^+$, so $2$ can be defined by $[x \text{ where } x \times x = 4]$. + $2$ is the only element in $\mathbb{Z}^+$ that satisfies $[x \text{ where } x \times x = 4]$. \end{solution} \vfill @@ -111,16 +111,23 @@ Try to define 2 in the structure $\Bigl( \mathbb{Z} ~\big|~ \{4, \times \} \Bigr \begin{solution} This isn't possible. We could try $[x \text{ where } x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \\ We have no way to distinguish between negative and positive numbers. + + \begin{instructornote} + Actually, it is. Bonus problem: how? \par + Do this after understanding quantifiers. + \end{instructornote} \end{solution} \vfill \problem{} -What numbers are definable in the structure $\Bigl( \mathbb{R} ~\big|~ \{1, 2, \div \} \Bigr)$? +What numbers are definable in the structure $\Bigl( \mathbb{R}^+_0 ~\big|~ \{1, 2, \div \} \Bigr)$? \begin{solution} - With the tools we have so far, we can only define powers of two, positive and negative. + We can define powers of two, positive and negative. + + If you're clever, you can define many more: $\sqrt{2}, \sqrt[3]{2}, ...$. \end{solution} \vfill diff --git a/Advanced/Definable Sets/parts/2 quantifiers.tex b/Advanced/Definable Sets/parts/2 quantifiers.tex index 0bee529..4642870 100644 --- a/Advanced/Definable Sets/parts/2 quantifiers.tex +++ b/Advanced/Definable Sets/parts/2 quantifiers.tex @@ -37,22 +37,22 @@ Which are true in $\mathbb{R}^+_0$? \par \item $\forall xy ~ \exists z ~ (x < z < y)$ \tab \note{This is a compact way to write $\forall x ~ (\forall y ~ (\exists z ~ (x < z < y)))$} - \item $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \tab~\tab \note{Solution is below.} + \item $\lnot \exists x ~ ( \forall y ~ (x < y) )$ %\tab~\tab \note{Solution is below.} \end{itemize} -\begin{examplesolution} - Here is a solution to the last part: $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \par - - \vspace{4mm} - - Reading this term-by-term, we get \tab \say{not exists $x$ where (for all $y$ ($x$ smaller than $y$))} \par - If we add some grammar, we get \tab \say{There isn't an $x$ where all $y$ are bigger than $x$} \par - which we can rephrase as \tab~\tab \say{There isn't a minimum value} \par - - \vspace{4mm} - - Which is true in $\mathbb{Z}$ and false in $\mathbb{R}^+_0$ -\end{examplesolution} +%\begin{examplesolution} +% Here is a solution to the last part: $\lnot \exists x ~ ( \forall y ~ (x < y) )$ \par +% +% \vspace{4mm} +% +% Reading this term-by-term, we get \tab \say{not exists $x$ where (for all $y$ ($x$ smaller than $y$))} \par +% If we add some grammar, we get \tab \say{There isn't an $x$ where all $y$ are bigger than $x$} \par +% which we can rephrase as \tab~\tab \say{There isn't a minimum value} \par +% +% \vspace{4mm} +% +% Which is true in $\mathbb{Z}$ and false in $\mathbb{R}^+_0$ +%\end{examplesolution} \vfill diff --git a/Advanced/Definable Sets/parts/3 sets.tex b/Advanced/Definable Sets/parts/3 sets.tex index d6400b0..d43d04c 100644 --- a/Advanced/Definable Sets/parts/3 sets.tex +++ b/Advanced/Definable Sets/parts/3 sets.tex @@ -32,21 +32,28 @@ Only even numbers satisfy the formula $\varphi(x) = \exists y ~ (y + y = x)$, \p So we can define \say{the set of even numbers} as $\{ x ~|~ \exists y ~ (y + y = x) \}$. \par Remember---we can only use symbols that are available in our structure! +\problem{} +Is the empty set definable in any structure? + +\vfill + + +\problem{} +Define $\{0, 1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$ + +\begin{instructornote} + Here's an interesting fact: + + A finite set of definable elements is always definable. Why? \par + An infinite set of definable elements may not be definable. +\end{instructornote} + +\vfill + \problem{} Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$ - \vfill - -\problem{} -Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$ - -\vfill - -\problem{} -Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$ - -\vfill \pagebreak @@ -84,7 +91,9 @@ Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$ \vfill \problem{} -Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$ +Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$ \par +\hint{We can't formally define a relation yet. Don't worry about that for now. \\ +You can repharase this question as \say{given $a,b \in \mathbb{Z}$, can you write a sentence that is true iff $a < b$?}} \vfill \pagebreak @@ -94,7 +103,7 @@ Consider the structure $S = ( \mathbb{R} ~|~ \{0, \diamond \} )$ \par The relation $a \diamond b$ holds if $| a - b | = 1$ \problempart{} -Define the empty set in $S$. +Define 0 in $S$. \problempart{} Define $\{-1, 1\}$ in $S$. @@ -106,10 +115,12 @@ Define $\{-2, 2\}$ in $S$. \problem{} Let $P$ be the set of all subsets of $\mathbb{Z}^+_0$. This is called a \textit{power set}. \par -Let $S$ be the stucture $( P ~|~ \{\subseteq\} )$ \par +Let $S$ be the stucture $( P ~|~ \{\subseteq\})$ \par \problempart{} -Show that the empty set is definable in $S$. +Show that the empty set is definable in $S$. \par +\hint{Defining $\{\}$ with $\{x ~|~ x \neq x\}$ is \textbf{not} what we need here. \\ +We need $\varnothing \in P$, the \say{empty set} element in the power set of $\mathbb{Z}^+_0$.} \problempart{} Let $x \Bumpeq y$ be a relation on $P$. $x \Bumpeq y$ holds if $x \cap y \neq \{\}$. \par