diff --git a/Advanced/Definable Sets/parts/1 structures.tex b/Advanced/Definable Sets/parts/1 structures.tex
index 5345d39..87ec66d 100644
--- a/Advanced/Definable Sets/parts/1 structures.tex	
+++ b/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}
 
diff --git a/Advanced/Definable Sets/parts/2 quantifiers.tex b/Advanced/Definable Sets/parts/2 quantifiers.tex
index 4642870..a717741 100644
--- a/Advanced/Definable Sets/parts/2 quantifiers.tex	
+++ b/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}
 
diff --git a/Advanced/Definable Sets/parts/3 sets.tex b/Advanced/Definable Sets/parts/3 sets.tex
index d43d04c..7a5b354 100644
--- a/Advanced/Definable Sets/parts/3 sets.tex	
+++ b/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}
 
 
diff --git a/Advanced/Definable Sets/parts/4 equivalence.tex b/Advanced/Definable Sets/parts/4 equivalence.tex
index ba0fca8..ef86e17 100644
--- a/Advanced/Definable Sets/parts/4 equivalence.tex	
+++ b/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 $