16 files changed, 267 insertions, 137 deletions

diff --git a/README.md b/README.md
index 3c40399..e4ef381 100644
--- a/README.md
+++ b/README.md
@@ -4,7 +4,7 @@ Here you can find the slides and the exercises for the course
 "Elementary Logic" (University of Luxembourg, PrepCamp 2020).
 
 For the slides, go in the ``slides`` folder and look for the file
-``preplogic-slides.pdf''.
+``preplogic-slides.pdf``.
 
-For the exercises, go in the ``exercises'' folder and look for the file
-``preplogic-exercises.pdf''.
+For the exercises, go in the ``exercises`` folder and look for the file
+``preplogic-exercises.pdf``.
diff --git a/exercises/preplogic-exercises.fdb_latexmk b/exercises/preplogic-exercises.fdb_latexmk
index c0a94f2..425b289 100644
--- a/exercises/preplogic-exercises.fdb_latexmk
+++ b/exercises/preplogic-exercises.fdb_latexmk
@@ -1,5 +1,5 @@
 # Fdb version 3
-["pdflatex"] 1599407243 "preplogic-exercises.tex" "preplogic-exercises.pdf" "preplogic-exercises" 1599407243
+["pdflatex"] 1599413298 "preplogic-exercises.tex" "preplogic-exercises.pdf" "preplogic-exercises" 1599413298
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-  "preplogic-exercises.tex" 1599407242 3546 94e18075fe4f9c11a865c64ac581a692 ""
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   (generated)
-  "preplogic-exercises.log"
-  "preplogic-exercises.pdf"
   "preplogic-exercises.aux"
+  "preplogic-exercises.pdf"
+  "preplogic-exercises.log"
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index 2dcc725..bcdb67c 100644
--- a/exercises/preplogic-exercises.fls
+++ b/exercises/preplogic-exercises.fls
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diff --git a/exercises/preplogic-exercises.log b/exercises/preplogic-exercises.log
index 340c2da..66c9478 100644
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+++ b/exercises/preplogic-exercises.log
@@ -1,4 +1,4 @@
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diff --git a/exercises/preplogic-exercises.pdf b/exercises/preplogic-exercises.pdf
index 7db8e9f..09381d6 100644
--- a/exercises/preplogic-exercises.pdf
+++ b/exercises/preplogic-exercises.pdf
diff --git a/exercises/preplogic-exercises.synctex.gz b/exercises/preplogic-exercises.synctex.gz
index 1a25b1d..7531587 100644
--- a/exercises/preplogic-exercises.synctex.gz
+++ b/exercises/preplogic-exercises.synctex.gz
diff --git a/exercises/preplogic-exercises.tex b/exercises/preplogic-exercises.tex
index cd37db7..1bf37b0 100644
--- a/exercises/preplogic-exercises.tex
+++ b/exercises/preplogic-exercises.tex
@@ -3,11 +3,12 @@
 \usepackage{amsmath}
 \usepackage{amsthm}
 \usepackage{amssymb}
+\usepackage[top=2cm]{geometry}
 
 \theoremstyle{definition} \newtheorem{exercise}{Exercise}[section]
 
-\author{Sebastiano Tronto (uni.lu)}
-\title{Elementary logic exercises (Prep Camp 2020)}
+\author{Sebastiano Tronto (\texttt{sebastiano.tronto@uni.lu})}
+\title{Elementary Logic exercises (Prep Camp 2020)}
 
 \begin{document}
 \maketitle
@@ -15,7 +16,7 @@
 \section{Logical operations}
 
 \begin{exercise}
-  Determine if the following statement are \textbf{true} or \textbf{false}:
+  Determine if the following statements are \textbf{true} or \textbf{false}:
   \begin{enumerate}
     \item ``Today is Tuesday or Germany has more inhabitants than Luxembourg''
     \item ``$7$ is odd and $2+2=5$''
@@ -84,21 +85,35 @@ here}'' ?
 \section{Quantifiers}
 
 \begin{exercise}
+  Write the negation of the following statements:
+  \begin{enumerate}
+    \item $\exists x\in \mathbb N,\, x^2-2=0$
+    \item ``Every prime number is odd''
+    \item ``Every person I have met likes pizza''
+    \item ``There is at least one number greater than $7$''
+    \item $\forall x\in \mathbb N,\,x\geq 0$
+    \item $\forall x\in \mathbb Z,\,(\exists y\in\mathbb Z,\,x+y=0)$
+  \end{enumerate}
+\end{exercise}
+
+\begin{exercise}
   There is another quantifier that we did not cover in the lecture, namely
   $\exists!$ (read ``there exists exactly one''). For example, the sentence
   ``\emph{there exists exactly one natural number x such that x+2=5}'' can be
-  written in symbols as ``$\exists!x\in \mathbb N,\,x+2=5$.
+  written in symbols as ``$\exists!x\in \mathbb N,\,x+2=5$''.
+
   In this exercise, your task is to give a formal definition of this quantifier
   using the logical symbols that we have defined in class. In particular, you
   will need the following:
   \begin{itemize}
-    \item the universal ($\forall$) and existential ($\exists$) quantifier
+    \item the universal ($\forall$) and existential ($\exists$) quantifiers
     \item the conjunction $\land$
     \item the implication $\implies$
   \end{itemize}
   Moreover, you will need the equality symbol $=$ between two elements of a set
   (if $a$ and $b$ are two elements of the same set, ``$a=b$'' is a mathematical
-  statement and it is \textbf{true} if and only if they are the same element).
+  statement and it is \textbf{true} if and only if $a$ and $b$ are the same
+  element).
   
   \emph{Warning: your definition must depend on a set $S$ and on a ``variable
   statement'' $A(x)$, as the existential and universal quantifiers.}
@@ -107,6 +122,27 @@ here}'' ?
 \section{Proofs}
 
 \begin{exercise}
+  Prove by induction that
+  \begin{align*}
+    \forall n\in\mathbb N,\quad \sum_{k=1}^n(2k-1)=n^2
+  \end{align*}
+    (here $\sum_{k=1}^n(2k-1)$ means $1+3+5+\cdots+ (2n-1)$).
+\end{exercise}
+
+\begin{exercise}
+  If $n\in \mathbb N$ the \emph{factorial} of $n$, denoted by $n!$ is defined
+  as follows:
+  \begin{align*}
+    n!=\begin{cases}
+      1&\text{if } n=0,\\
+      n\times (n-1)! & \text{if } n> 0.
+    \end{cases}
+  \end{align*}
+  Prove by induction that if $n\geq 4$ then $n!\geq 2^n$.
+\end{exercise}
+
+
+\begin{exercise}
   Is the following statement true or false? Give a proof of your answer.
   \begin{align*}
     \forall n\in \mathbb N,\, n^2 -4n +5>n
diff --git a/slides/preplogic-slides.aux b/slides/preplogic-slides.aux
index 70bc721..d51ee85 100644
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index 6819258..b01ff7a 100644
--- a/slides/preplogic-slides.fdb_latexmk
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+  "preplogic-slides.out"
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 \headcommand {\beamer@partpages {1}{71}}
-\headcommand {\beamer@subsectionpages {47}{71}}
-\headcommand {\beamer@sectionpages {47}{71}}
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 \headcommand {\beamer@documentpages {71}}
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diff --git a/slides/preplogic-slides.pdf b/slides/preplogic-slides.pdf
index 44c0113..e9bc6b7 100644
--- a/slides/preplogic-slides.pdf
+++ b/slides/preplogic-slides.pdf
diff --git a/slides/preplogic-slides.synctex.gz b/slides/preplogic-slides.synctex.gz
index dab34a5..7394213 100644
--- a/slides/preplogic-slides.synctex.gz
+++ b/slides/preplogic-slides.synctex.gz
diff --git a/slides/preplogic-slides.tex b/slides/preplogic-slides.tex
index 5194ebf..dc68c1f 100644
--- a/slides/preplogic-slides.tex
+++ b/slides/preplogic-slides.tex
@@ -350,7 +350,7 @@ Mathematician: ``Yes.''
 
   \begin{itemize}
     \item \textbf{Universal quantifier} ($\forall$ or ``for all''):
-          ``$\forall x\in S,\,A(x)$'' means that if we replace $x$ with any
+          ``$\forall x\in S,\,A(x)$'' means that if we replace ``$x$'' with any
           element of $S$, $A(x)$ is always \textbf{true}.
     \item \textbf{Existential quantifier} ($\exists$ or ``there exists''):
           ``$\exists x\in S,\, A(x)$'' means that $A(x)$ is \textbf{true} for
@@ -385,16 +385,21 @@ Mathematician: ``Yes.''
     \neg(\exists x\in S,\, A(x))&=\only<1-2>{\quad?}
     \onslide<3->{\forall x\in S,\,\neg A(x)}\\
   \end{align*}
-  \pause
+  \onslide<2->{
   \begin{example}
     \begin{center}
   $\neg$``every number is even'' = ``there is at least one odd number''
     \end{center}
   \end{example}
-\end{frame}
-
-\begin{frame}
-  (exercise)
+  }
+  \onslide<3->{
+  \begin{example}
+    \begin{center}
+    $\neg$``$\exists x\in \mathbb N,\, x+3=9$'' =
+    ``$\forall x\in \mathbb N,\, x+3\neq 9$''
+    \end{center}
+  \end{example}
+  }
 \end{frame}
 
 \section{Proofs}
@@ -523,4 +528,16 @@ Mathematician: ``Yes.''
   \end{proof}
 \end{frame}
 
+\begin{frame}{The end}
+  What you have learned in this course:
+  \begin{itemize}
+    \item Basic logic operations and logic implication
+    \item Quantifiers and \textbf{their negation}
+    \item Some basic mathematical proofs
+  \end{itemize}
+  \vspace{10pt}
+  \refgithub
+\end{frame}
+
+
 \end{document}
diff --git a/slides/preplogic-slides.toc b/slides/preplogic-slides.toc
index 1dc8678..80170ac 100644
--- a/slides/preplogic-slides.toc
+++ b/slides/preplogic-slides.toc
@@ -2,4 +2,4 @@
 \beamer@sectionintoc {2}{Logical operations}{11}{0}{2}
 \beamer@sectionintoc {3}{Implication}{28}{0}{3}
 \beamer@sectionintoc {4}{Quantifiers}{40}{0}{4}
-\beamer@sectionintoc {5}{Proofs}{47}{0}{5}
+\beamer@sectionintoc {5}{Proofs}{46}{0}{5}