Edits

Advanced/Definable Sets/parts/0 logic.tex

@@ -60,7 +60,7 @@ The function of these is defined by \textit{truth tables}:
 
 \vspace{2mm}
 
-$A \land B$ is only true if both $A$ and $B$ are true. $A \lor B$ is only true if $A$ or $B$ (or both) are true. \par
+$A \land B$ is only true if both $A$ and $B$ are true. $A \lor B$ is true when $A$ or $B$ (or both) are true. \par
 $\lnot A$ is the opposite of $A$, which is why it looks like a \say{negative} sign. \par
 
 \vspace{2mm}
@@ -92,6 +92,7 @@ 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 $B$.
 \hint{Use a truth table}
 
 \vfill

Advanced/Definable Sets/parts/1 structures.tex

@@ -9,7 +9,7 @@ A \textit{universe} is a set of meaningless objects. Here are a few examples:
 \end{itemize}
 
 \definition{}
-A \textit{structure} consists of a universe $U$ and set of symbols. \par
+A \textit{structure} consists of a universe $U$ and a set of symbols. \par
 A structure's symbols give meaning to the objects in its universe.
 
 \vspace{2mm}
@@ -35,8 +35,6 @@ The equality check $=$ is \textbf{not} a relation symbol. It is included in ever
 
 
 \example{}
-\def\structgeneric{\ensuremath{}}
-
 The first structure we'll look at is the following:
 $$
 	\Bigl( \mathbb{Z} ~\big|~ \{0, 1, +, -, <\} \Bigr)

Advanced/Definable Sets/parts/2 quantifiers.tex

@@ -8,8 +8,8 @@ $\forall$ and $\exists$ are \textit{quantifiers}. They allow us to make statemen
 
 \vspace{2mm}
 
-Let's look at $\forall$ first. Let $\varphi$ be a formula. \par
-Then, the formula $\forall x ~ \varphi$ says \say{$\varphi$ is true for all possible $x$.}
+Let's look at $\forall$ first. Let $\varphi(x)$ be a formula. \par
+Then, the formula $\forall x ~ \varphi(x)$ says \say{$\varphi$ is true for all possible $x$.}
 
 \vspace{1mm}
 
@@ -18,7 +18,7 @@ In english, this means \say{For any $x$, $x$ is bigger than zero,} or simply \sa
 
 \vspace{3mm}
 
-$\exists$ is very similar: the formula $\exists x ~ \varphi$ states that there is at least one $x$ that makes $\varphi$ true. \par
+$\exists$ is very similar: the formula $\exists x ~ \varphi(x)$ states that there is at least one $x$ that makes $\varphi$ true. \par
 For example, $\exists ~ (0 < x)$ means \say{there is a positive number in our set}.
 
 \vspace{4mm}
@@ -58,6 +58,12 @@ Which are true in $\mathbb{R}^+_0$? \par
 \vfill
 \pagebreak
 
+\problem{}
+Does the order of $\forall$ and $\exists$ in a formula matter? \par
+What's the difference between $\exists x ~ \forall y ~ (x < y)$ and $\forall y ~ \exists x ~ (x < y)$? \par
+\hint{In $\mathbb{R}^+$, the first is false and the second is true. $\mathbb{R}^+$ does not contain zero.}
+
+\vfill
 
 \problem{}
 Define 0 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
@@ -71,6 +77,7 @@ Define 1 in $\Bigl( \mathbb{Z} ~\big|~ \{\times\} \Bigr)$
 
 
 \vfill
+\pagebreak
 
 
 \problem{}
@@ -78,13 +85,10 @@ Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
 
 \vfill
 
-\problem{}
-Define $3$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
+%\problem{}
+%Define $2$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
 
-
-
-\vfill
-\pagebreak
+%\vfill
 
 \problem{}
 Let $\varphi(x)$ be a formula. \par
@@ -94,12 +98,5 @@ Define $(\forall x ~ \varphi(x))$ using logical symbols and $\exists$.
 	$\Bigl(\forall x ~ \varphi(x)\Bigr)$ is true iff $\lnot \Bigl(\exists x ~ \lnot \varphi(x) \Bigr)$ is true.
 \end{solution}
 
-\vfill
-
-\problem{}
-Does the order of $\forall$ and $\exists$ in a formula matter? \par
-What's the difference between $\exists x ~ \forall y ~ (x < y)$ and $\forall y ~ \exists x ~ (x < y)$? \par
-\hint{In $\mathbb{R}^+$, the first is false and the second is true. $\mathbb{R}^+$ does not contain zero.}
-
 \vfill
 \pagebreak

Advanced/Definable Sets/parts/3 sets.tex

@@ -39,12 +39,12 @@ Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\
 
 
 \problem{}
-Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
+Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$
 
 \vfill
 
 \problem{}
-Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
+Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{<\} \Bigr)$
 
 \vfill
 \pagebreak