Typos

Advanced/Definable Sets/parts/1 structures.tex

@@ -51,7 +51,7 @@ This is a structure with the universe $\mathbb{Z}$ that contains the following s
 
 \vspace{2mm}
 
-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.
+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 this structure offers.
 
 \vspace{1mm}
 

Advanced/Definable Sets/parts/2 quantifiers.tex

@@ -14,7 +14,7 @@ Then, the formula $\forall x ~ \varphi(x)$ says \say{$\varphi$ is true for all p
 \vspace{1mm}
 
 For example, take the formula $\forall x ~ (0 < x)$. \par
-In english, this means \say{For any $x$, $x$ is bigger than zero,} or simply \say{Any $x$ is positive.}
+In English, this means \say{For any $x$, $x$ is bigger than zero,} or simply \say{Any $x$ is positive.}
 
 \vspace{3mm}
 

Advanced/Definable Sets/parts/3 sets.tex

@@ -64,7 +64,7 @@ Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z)
 \hint{$z$ is nonreal if $x \in \mathbb{C}$ and $x \notin \mathbb{R}$}
 
 \begin{solution}
-	$\Bigl\{ x ~\bigl|~ \text{real}(x) = x \Bigr\}$
+	$\Bigl\{ x ~\bigl|~ \text{real}(x) \neq x \Bigr\}$
 \end{solution}
 
 

Advanced/Definable Sets/parts/4 equivalence.tex

@@ -7,7 +7,8 @@ This is read \say{$S$ satisfies $\varphi$}
 
 \definition{}
 Let $S$ and $T$ be structures. \par
-We say $S$ and $T$ are \textit{equivalent} and write $S \equiv T$ if for any formula $\varphi$, $S \models \varphi \Longleftrightarrow T \models \varphi$.
+We say $S$ and $T$ are \textit{equivalent} and write $S \equiv T$ if for any formula $\varphi$, $S \models \varphi \Longleftrightarrow T \models \varphi$. \par
+If $S$ and $T$ are not equivalent, we write $S \not\equiv T$.
 
 \problem{}
 Show that $