From a911f938427ec88618b6b1507ac3929616c770f7 Mon Sep 17 00:00:00 2001
From: Sebastiano Tronto <sebastiano.tronto@gmail.com>
Date: Tue, 8 Sep 2020 08:41:13 +0200
Subject: Minor fixes

---
 slides/preplogic-slides.tex | 10 +++++++---
 1 file changed, 7 insertions(+), 3 deletions(-)

(limited to 'slides/preplogic-slides.tex')

diff --git a/slides/preplogic-slides.tex b/slides/preplogic-slides.tex
index dc68c1f..e0c4886 100644
--- a/slides/preplogic-slides.tex
+++ b/slides/preplogic-slides.tex
@@ -347,15 +347,19 @@ Mathematician: ``Yes.''
   Let $S$ be a set and let $A(x)$ be a ``variable statement'' that depends on
   $x\in S$ (for example $S=\mathbb{N}$ and $A(x)=$``x is an even number'').
   \pause
-
+  \vspace{12pt}
   \begin{itemize}
     \item \textbf{Universal quantifier} ($\forall$ or ``for all''):
           ``$\forall x\in S,\,A(x)$'' means that if we replace ``$x$'' with any
           element of $S$, $A(x)$ is always \textbf{true}.
+  \vspace{9pt}
     \item \textbf{Existential quantifier} ($\exists$ or ``there exists''):
           ``$\exists x\in S,\, A(x)$'' means that $A(x)$ is \textbf{true} for
           at least one value of $x$ is $S$.
   \end{itemize}
+  \vspace{12pt}
+  \pause
+  \textbf{You always need a set $S$}
 \end{frame}
 
 \begin{frame}{Quantifiers - examples}
@@ -467,8 +471,8 @@ Mathematician: ``Yes.''
     \begin{enumerate}
     \item Assume that there are only finitely many prime numbers.
     \pause
-    \item So there are $n$ prime numbers, for some number $n$.
-          Call them $p_1,p_2,\dots,p_n$.
+    \item $\exists n\in \mathbb N$, there are $n$ prime numbers. Call them
+          $p_1,p_2,\dots,p_n$.
     \pause
     \item Let $u=p_1\times p_2\times \dots \times p_n +1$.
     \pause
-- 
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