Minor edits

Advanced/Definable Sets/parts/2 quantifiers.tex

@@ -4,12 +4,14 @@ Recall the logical symbols we introduced earlier: $(), \land, \lor, \lnot, \righ
 We will now add two more: $\forall$ (for all) and $\exists$ (exists).
 
 \definition{}
-$\forall$ and $\exists$ are \textit{quantifiers}. They allow us to make statements about arbitrary symbols.
+$\forall$ and $\exists$ are \textit{quantifiers}. They allow us to make statements about arbitrary symbols. \par
+\hint{Quantifiers are aptly named: they tell us \textit{how many} symbols satisfy a certain sentence.}
+
 
 \vspace{2mm}
 
-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 $x$.}
+Let's look at $\forall$ first. If $\varphi(x)$ is a formula, \par
+the formula $\forall x ~ \varphi(x)$ is true only if $\varphi$ is true for all $x$ in our universe.
 
 \vspace{1mm}
 
@@ -18,8 +20,8 @@ In English, this means \say{For any $x$, $x$ is bigger than zero,} or simply \sa
 
 \vspace{3mm}
 
-$\exists$ is very similar: the formula $\exists x ~ \varphi(x)$ 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}.
+$\exists$ is very similar: the formula $\exists x ~ \varphi(x)$ is true if ther is at least one $x$ for which $\varphi(x)$ is true. \par
+For example, $\exists ~ (0 < x)$ means \say{there is a positive number in our set.}
 
 \vspace{4mm}
 
@@ -64,7 +66,7 @@ What's the difference between $\exists x ~ \forall y ~ (x \leq y)$ and $\forall
 \hint{In $\mathbb{R}^+$, the first is false and the second is true. $\mathbb{R}^+$ does not contain zero.}
 
 \begin{solution}
-	If $\exists x$ is inside $\forall y$, $x$ depends on $y$. We can have a different value of $x$ for every $y$. \par
+	If $\exists x$ is inside $\forall y$, $x$ depends on $y$. We may pick a different value of $x$ for every $y$. \par
 	If $\exists x$ is outside, $x$ is fixed \textit{before} we check all $y$.
 \end{solution}
 
@@ -98,10 +100,10 @@ Define $-1$ in $\Bigl( \mathbb{Z} ~\big|~ \{0, <\} \Bigr)$
 
 \problem{}
 Let $\varphi(x)$ be a formula. \par
-Define $(\forall x ~ \varphi(x))$ using logical symbols and $\exists$.
+Write a formula equivalent to $[~ \forall x ~ \varphi(x) ~]$ using only logical symbols and $\exists$.
 
 \begin{solution}
-	$\Bigl(\forall x ~ \varphi(x)\Bigr)$ is true iff $\lnot \Bigl(\exists x ~ \lnot \varphi(x) \Bigr)$ is true.
+	$[~ \forall x ~ \varphi(x) ~]$ is true if and only if $[~ \lnot \exists x ~ \lnot \varphi(x) ~]$ is true.
 \end{solution}
 
 \vfill