From 2e380a3f3b45d6d57be24ce0277132e6a34142d0 Mon Sep 17 00:00:00 2001 From: Mark Date: Wed, 26 Jul 2023 17:55:07 -0700 Subject: [PATCH] Cleanup --- Advanced/Definable Sets/parts/0 logic.tex | 13 +++++++++---- Advanced/Definable Sets/parts/1 structures.tex | 6 +++--- Advanced/Definable Sets/parts/4 equivalence.tex | 5 +++-- 3 files changed, 15 insertions(+), 9 deletions(-) diff --git a/Advanced/Definable Sets/parts/0 logic.tex b/Advanced/Definable Sets/parts/0 logic.tex index b516254..fd38943 100644 --- a/Advanced/Definable Sets/parts/0 logic.tex +++ b/Advanced/Definable Sets/parts/0 logic.tex @@ -1,8 +1,9 @@ \section{Logical Algebra} \definition{} -Odds are, you are familiar with \textit{logical symbols}. \par -In this handout, we'll use the following: +\textit{Logical operators} operate on the values $\{\text{True}, \text{False}\}$, \par +just like algebraic operators operate on numbers. \par +In this handout, we'll use the following operators: \begin{itemize} \item $\lnot$: not \item $\land$: and @@ -66,11 +67,15 @@ $\lnot A$ is the opposite of $A$, which is why it looks like a \say{negative} si \vspace{2mm} $A \rightarrow B$ is a bit harder to understand. Read aloud, this is \say{$A$ implies $B$.} \par -The only time $\rightarrow$ is false is when $T \rightarrow F$. This may seem counterintuitive, but it makes sense. Think about it. \par +The only time $\rightarrow$ is false is when $T \rightarrow F$. This may seem counterintuitive, but it will make more sense as we progress through this handout. \problem{} Evaluate the following. \begin{itemize} + \item $\lnot T$ + \item $F \lor T$ + \item $T \land T$ + \item $(T \land F) \lor T$ \item $(T \land F) \lor T$ \item $(\lnot (F \lor \lnot T) ) \rightarrow T$ \item $(F \rightarrow T) \rightarrow (\lnot F \lor \lnot T)$ @@ -110,7 +115,7 @@ Evaluate the following. \problem{} Show that $\lnot (A \rightarrow \lnot B)$ is equivalent to $A \land B$. \par -That is, show that these give the same result for the same $A$ and $B$. +That is, show that these give the same result for the same $A$ and $B$. \par \hint{Use a truth table} \vfill diff --git a/Advanced/Definable Sets/parts/1 structures.tex b/Advanced/Definable Sets/parts/1 structures.tex index b9e6958..5345d39 100644 --- a/Advanced/Definable Sets/parts/1 structures.tex +++ b/Advanced/Definable Sets/parts/1 structures.tex @@ -86,7 +86,7 @@ For the sake of time, I will not provide a formal definition. It isn't particula A formula can contain one or more \textit{free variables.} These are denoted $\varphi{(a, b, ...)}$. \par Formulas with free variables let us define \say{properties} that certain objects have. \par -For example, $x$ is a free variable in the formula $\varphi(x) = x > 0$. \par +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} @@ -99,7 +99,7 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi \hint{$\mathbb{Z}^+ = \{1, 2, 3, ...\}$. Also, $2 \times 2 = 4$.} \begin{solution} - $2$ is the only element in $\mathbb{Z}^+$ that satisfies $[x \text{ where } x \times x = 4]$. + $2$ is the only element in $\mathbb{Z}^+$ that satisfies $\varphi(x) = [x \times x = 4]$. \end{solution} \vfill @@ -109,7 +109,7 @@ Can we define 2 in the structure $\Bigl( \mathbb{Z^+} ~\big|~ \{4, \times \} \Bi 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 $-2$. \\ + This isn't possible. We could try $\varphi(x) = [x \times x = 4]$, but this is satisfied by both $2$ and $-2$. \par We have no way to distinguish between negative and positive numbers. \begin{instructornote} diff --git a/Advanced/Definable Sets/parts/4 equivalence.tex b/Advanced/Definable Sets/parts/4 equivalence.tex index 06f31ec..ba0fca8 100644 --- a/Advanced/Definable Sets/parts/4 equivalence.tex +++ b/Advanced/Definable Sets/parts/4 equivalence.tex @@ -1,8 +1,9 @@ -\section{Equivalence (Bonus)} +\section{Equivalence} \generic{Notation:} Let $S$ be a structure and $\varphi$ a formula. \par -If $\varphi$ is true in $S$, we write $S \models \varphi$. +If $\varphi$ is true in $S$, we write $S \models \varphi$. \par +This is read \say{$S$ satisfies $\varphi$} \definition{} Let $S$ and $T$ be structures. \par