commit 4ca009f025a29351bcf1160cd68d513f04687713 Author: Sebastiano Tronto <sebastiano.tronto@gmail.com> Date: Sun, 6 Sep 2020 18:04:41 +0200 First commit Diffstat:
| A | .gitignore | | | 3 | +++ |
| A | exercises/preplogic-exercises.aux | | | 5 | +++++ |
| A | exercises/preplogic-exercises.fdb_latexmk | | | 62 | ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | exercises/preplogic-exercises.fls | | | 85 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | exercises/preplogic-exercises.log | | | 151 | ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | exercises/preplogic-exercises.pdf | | | 0 | |
| A | exercises/preplogic-exercises.synctex.gz | | | 0 | |
| A | exercises/preplogic-exercises.tex | | | 123 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | slides/preplogic-slides.aux | | | 114 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | slides/preplogic-slides.fdb_latexmk | | | 218 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
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| A | slides/preplogic-slides.nav | | | 86 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | slides/preplogic-slides.out | | | 10 | ++++++++++ |
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| A | slides/preplogic-slides.tex | | | 520 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | slides/preplogic-slides.toc | | | 10 | ++++++++++ |
| A | slides/unilu.jpg | | | 0 |
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within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 1 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/exercises/preplogic-exercises.pdf b/exercises/preplogic-exercises.pdf Binary files differ. diff --git a/exercises/preplogic-exercises.synctex.gz b/exercises/preplogic-exercises.synctex.gz Binary files differ. diff --git a/exercises/preplogic-exercises.tex b/exercises/preplogic-exercises.tex @@ -0,0 +1,123 @@ +\documentclass[a4paper,oneside]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amssymb} + +\theoremstyle{definition} \newtheorem{exercise}{Exercise}[section] + +\author{Sebastiano Tronto (uni.lu)} +\title{Elementary logic exercises (Prep Camp 2020)} + +\begin{document} +\maketitle + +\section{Logical operations} + +\begin{exercise} + Determine if the following statement are \textbf{true} or \textbf{false}: + \begin{enumerate} + \item ``Today is Tuesday or Germany has more inhabitants than Luxembourg'' + \item ``$7$ is odd and $2+2=5$'' + \item Every number of the form $2^{2^n}+1$, for $n=1,2,3...$, is prime. + \end{enumerate} +\end{exercise} + +\begin{exercise} + What is the negation of the sentence ``\emph{I payed attention in class and I + did not do my homework}'' ? +\end{exercise} + +\begin{exercise} + Simplify the following logical expressions using the properties of logical + operations (where $A,B$ and $C$ are statements): + \begin{enumerate} + \item $A\land(A\lor B)$ + \item $A\lor (B\land A)$ + \item $(A\lor B) \land \neg A$ + \item $A \lor (\neg A\land B)$ + \item $(\neg (A\lor \neg B))\land ((A\lor C) \land \neg C)$ + \end{enumerate} +\end{exercise} + +\section{Implication} + +\begin{exercise} + Fill in the following truth table: + \begin{align*} + \begin{array}{|c|c|c|c|c|} + \hline + A & B & C & \neg(A\implies B) & (A\implies B) \implies C \\ + \hline + 0 & 0 & 0 & & \\ + \hline + 0 & 0 & 1 & & \\ + \hline + 0 & 1 & 0 & & \\ + \hline + 0 & 1 & 1 & & \\ + \hline + 1 & 0 & 0 & & \\ + \hline + 1 & 0 & 1 & & \\ + \hline + 1 & 1 & 0 & & \\ + \hline + 1 & 1 & 1 & & \\ + \hline + \end{array} + \end{align*} +\end{exercise} + +\begin{exercise}[Transitivity] + Prove that the following statement is true for any statements $A,B$ and $C$: + \begin{align*} + ((A\implies B)\land (B\implies C))\implies (A\implies C) + \end{align*} +\end{exercise} + +\begin{exercise} +What is the contrapositive of ``\emph{If this table is not reserved, we sit +here}'' ? +\end{exercise} + +\section{Quantifiers} + +\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 ``$\exists!x\in \mathbb N,\,x+2=5$. + 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 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). + + \emph{Warning: your definition must depend on a set $S$ and on a ``variable + statement'' $A(x)$, as the existential and universal quantifiers.} +\end{exercise} + +\section{Proofs} + +\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 + \end{align*} +\end{exercise} + +\begin{exercise} + Do the last point of Exercise 1.1 again, but this time give a proof of your + answer. +\end{exercise} + + + +\end{document} diff --git a/slides/preplogic-slides.aux b/slides/preplogic-slides.aux @@ -0,0 +1,114 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} 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+\logo{\includegraphics[scale=0.065]{unilu.jpg}} +\institute{uni.lu} +\date{September 7-8, 2020} +%\subject{} + +\newtheorem {proposition}{Proposition} +\theoremstyle{definition} +\newtheorem {remark}{Remark} +\newtheorem {exercise}{Exercise} + + +\newcommand{\refgithub}{ + \begin{itemize} + \item \textbf{These slides:} + \item \textbf{Exercises:} + \item \textbf{Contact:} \texttt{sebastiano.tronto@uni.lu} + \end{itemize} +} + +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +\begin{frame} + \begin{columns} + \column{0.5\linewidth} + \tableofcontents + \column{0.5\linewidth} + \refgithub + \end{columns} +\end{frame} + + +\section{Statements} +\begin{frame}{Statements} +\begin{itemize} + \pause + \item Unambiguous + \pause +\begin{example}[A bad joke] +\textbf{Q:} How many months have 30 days? +\pause + +\textbf{A:} 11, some of them have even more! +\pause + +:-( +\end{example} +\pause + \item Objective + \pause +\begin{example} +\textbf{Good:} 3 is greater than 4 + +\textbf{Bad:} 3 is nicer than 4 +\end{example} +\end{itemize} +\end{frame} + +\begin{frame}{Statements} + \begin{itemize} + \item Mathematical: ``\emph{Three is greater than four}\,'' + (or ``$3 > 4$'') + \item ...or not: ``\emph{I am 26 years old}\,'' + \item \textbf{Key point:} staments can be \textbf{true} or + \textbf{false} +\end{itemize} +\end{frame} + +%\begin{frame}[plain] +%\begin{center} +%\includegraphics[scale=0.4]{xkcd169.png} +%https://xkcd.com/169/ +%\end{center} +%\end{frame} + + +\section{Logical operations} + +\begin{frame}{Logical operations} +\begin{itemize} +\item We can combine statements to make new ones +\item Negation (\textbf{not}), conjunction (\textbf{and}), disjunction + (\textbf{or}) +\end{itemize} +\end{frame} + +\begin{frame}{Negation (\textbf{not})} +\begin{center} +If $A$ is a statement, the statement ``not $A$'' (in symbols: $\neg A$) is +\textbf{true} when $A$ is \textbf{false}, and it is \textbf{false} when $A$ is +\textbf{true}. +\pause +\end{center} + +\begin{example} +\begin{center} +$\neg (3>4)$ is equivalent to $3\leq 4$ + +``\emph{$3$ is \textbf{not} greater than $4$}'' is equivalent to +``\emph{$3$ is less or equal than $4$}'' +\end{center} +\end{example} +\end{frame} + +\begin{frame}{Conjunction (\textbf{and})} +\begin{center} +The statement ``$A$ and $B$'' (in symbols: +$A\land B$) is \textbf{true} when both $A$ and $B$ are \textbf{true}, and it is +\textbf{false} if at \emph{at least} one of them is \textbf{false}. +\pause + +\begin{example} +``$(3<4)\land (5$ is an odd number$)$'' is \textbf{true} +\end{example} + +\begin{example} +``(Today is Monday) $\land$ (we are in France)'' is \textbf{false} +\end{example} +\end{center} + +\end{frame} + +\begin{frame}{Disjunction (\textbf{or})} +\begin{center} +The statement ``$A$ or $B$'' (in symbols: +$A\lor B$) is \textbf{true} when at least one of $A$ and $B$ is \textbf{true}, +and it is \textbf{false} if both of them are \textbf{false}. +\pause + +\begin{example} +``$(3=4)\lor (5$ is an even number$)$'' is \textbf{false} +\end{example} + +\begin{example} +``(Today is Monday) $\lor$ (we are in Luxembourg)'' is \textbf{true} +\end{example} +\end{center} + +\end{frame} + + +\begin{frame}{Logical operations} +\begin{itemize} + \item \textbf{Important:} $\lor$ is always \emph{inclusive}: + \pause + +\begin{center} +\begin{example}[Another bad joke] +Waiter: ``Would you like cheese or dessert?'' + +Mathematician: ``Yes.'' +\end{example} +\end{center} +\pause + \item $\neg$ has precedence over $\land$ and $\lor$: + \begin{align*} + \neg A\land B \text{ means } (\neg A)\land B,\qquad + \neg A\lor B \text{ means } (\neg A)\lor B + \end{align*} + (or just use parenthesis) +\end{itemize} + + +\end{frame} + +\begin{frame}{Properties} + \begin{center} + If $A$, $B$ and $C$ are statements: + \end{center} + {\fontsize{9}{17}\selectfont + \begin{align*} + \begin{array}{cc|c} + A\land B = B\land A & A \lor B = B\lor A &\textbf{commutativity}\\ + \hline + A\land (B\land C) = (A\land B)\land C \quad & + A\lor (B\lor C) = (A\lor B)\lor C & \textbf{associativity}\\ + \hline + A\land(B\lor C) = (A\land B)\lor(A\land C) & & \textbf{distributivity} \\ + A\lor(B\land C) = (A\lor B)\land(A\lor C) & &\textbf{distributivity*} \\ + \hline + \neg(\neg A) = A & & \textbf{double negation} \\ + \hline + A \land \textbf{true} = A & A \land \textbf{false} = \textbf{false} \\ + A\lor \textbf{true} = \textbf{true} & A \lor \textbf{false} = A \\ + (\neg A) \land A = \textbf{false} & (\neg A) \lor A = \textbf{true} \\ + \hline + \neg (A\land B) = (\neg A)\lor (\neg B) & + \neg (A\lor B) = (\neg A)\land (\neg B) & \textbf{De Morgan's laws} + \end{array} + \end{align*}} +\end{frame} + +\subsection{Boolean algebra} + +\begin{frame}{Boolean algebra} + \begin{itemize} + \item For simplicity: \textbf{true}\,$=1$, \textbf{false}\,$=0$ + \pause \item We have a set $\{0,1\}$ with some operations + $(\land,\lor,\neg)$ + \pause \item This is called a \textbf{Boolean algebra} + \end{itemize} +\end{frame} + +\subsection{Truth tables} + +\begin{frame}{Truth tables} + A compact way of describing an operator, or a composition of operators + \pause + \vspace{4pt} + Example: +\begin{align*} + \begin{array}{|c|c|c|c|c|c|} + \hline + A & B & \neg A & A\land B & A\lor B & (A\lor B)\land (\neg A) \\ + \hline + 0 & 0 & 1 & 0 & 0 & 0 \\ + 0 & 1 & 1 & 0 & 1 & 1 \\ + 1 & 0 & 0 & 0 & 1 & 0 \\ + 1 & 1 & 0 & 1 & 1 & 0 \\ + \hline + \end{array} +\end{align*} +\end{frame} + +\begin{frame}{Truth tables} + \begin{center} + We can check that two statements are equivalent with truth tables + \end{center} +\begin{align*} + \begin{array}{|c|c|c|c|} + \hline + A & B & \neg(A\land B) & (\neg A)\lor (\neg B)\\ + \hline + 0 & 0 & 1 & 1 \\ + 0 & 1 & 1 & 1 \\ + 1 & 0 & 1 & 1 \\ + 1 & 1 & 0 & 0 \\ + \hline + \end{array} +\end{align*} +\end{frame} + + +\section{Implication} + +\begin{frame}{Implication} + \begin{itemize} + \item ``$A\implies B$'' means \emph{``If $A$ (is true), then $B$ (is true)''} + \end{itemize} + \pause + \begin{example} + \emph{``If it rains, I will bring an umbrella''} + + (It rains)$\implies$(I will bring an umbrella) + \end{example} + \pause + \begin{example} + \emph{``If my grandpa had wheels, he would be a bike''} + + (My grandpa has wheels)$\implies$(My grandpa is a bike) + \end{example} +\end{frame} + + +\begin{frame}{Implication} + \begin{itemize} + \item It is a logical operation: ``$A\implies B$'' means ``$B\lor(\neg A)$'' + \pause + \end{itemize} + \begin{align*} + \begin{array}{|c|c|c|c|} + \hline + A & B & A\implies B & \\ + \hline + 0 & 0 & 1 & \text{No rain, I don't bring an umbrella} \\ + \hline + 0 & 1 & 1 & \text{No rain, I bring an umbrella anyway} \\ + \hline + 1 & 0 & 0 & \text{It rains, I don't bring an umbrella} \\ + \hline + 1 & 1 & 1 & \text{It rains, I bring an umbrella} \\ + \hline + \end{array} + \end{align*} + \pause + \begin{remark} + ``\textbf{false} $\implies A$'' is always \textbf{true}, whatever $A$ is + (\emph{ex falso quodlibet}) + ``$A\implies$\textbf{true}'' is always true, whatever $A$ is + \end{remark} +\end{frame} + + +\begin{frame}{Notation} + Sometimes we use the following symbols: + \begin{itemize} + \item ``$A\impliedby B$'' is the same as ``$B\implies A$'' + \item ``$A\iff B$ is the same as ``$(A\implies B)\land (B\implies A)$''.\\ + It is read ``$A$ is equivalent to $B$'' or ``$A$ if and only if $B$''. + \end{itemize} +\end{frame} + +\begin{frame}{Contrapositive} + \begin{itemize} + \item The statement $(\neg B)\implies (\neg A)$ is called + \emph{contrapositive} of $A\implies B$ + \pause + \item It is equivalent to ``$A\implies B$'' + \pause + \item Two proofs: + \begin{enumerate} + \item Properties of logical operations + \item Truth tables + \end{enumerate} + \end{itemize} +\end{frame} + +\begin{frame}{End of part 1} + See you tomorrow! + + \refgithub +\end{frame} + +\section{Quantifiers} + +\begin{frame}{Quantifiers} + 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 + + \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}. + \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} +\end{frame} + +\begin{frame}{Quantifiers - examples} + \begin{example} + $S=$``the set of all cars'', $A(x)$=``$x$ is red'' + + $\forall x\in S,\, A(X)$ is \textbf{false}. + + $\exists x\in S,\, A(X)$ is \textbf{true}. + \end{example} + \begin{example} + $S=\mathbb{N}$, $A(x)=x>5$ + + $\forall x\in S,\, A(x)$ is \textbf{false}. + + $\exists x\in S,\, A(X)$ is \textbf{true}. + \end{example} +\end{frame} + +\begin{frame}{Negation of quantifiers} + \begin{center} + \textbf{Today's most important fact:} + \end{center} + \begin{align*} + \neg(\forall x\in S,\, A(x))&=\only<1>{\quad?} + \onslide<2->{\exists x\in S,\,\neg A(x)}\\ + \neg(\exists x\in S,\, A(x))&=\only<1-2>{\quad?} + \onslide<3->{\forall x\in S,\,\neg A(x)}\\ + \end{align*} + \pause + \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) +\end{frame} + +\section{Proofs} + +\begin{frame}{Proofs} + \begin{itemize} + \item A proof is a sequence of statements, each one logically deriving from + the previous. + \pause + \item Proofs are used to derive new statements from statements that are + known to be true. + \pause + \item If $A$ is known to be true and the implication $A\implies B$ is + logically clear, then also $B$ must be true. + \pause + \item Every mathematical theorem must be justified with a proof. + \end{itemize} +\end{frame} + +\subsection{Direct proofs} +\begin{frame}{Example: direct proof} + \begin{theorem} + The sum of two even numbers is even. + \end{theorem} + \pause + \begin{proof} + \begin{enumerate} + \item Recall the definition: a natural number $x$ is called \emph{even} + if there is some natural number $n$ such that $x=2n$. + \pause + \item If $x$ and $y$ are even numbers, then there are natural numbers $n$ + and $m$ such that $x=2n$ and $y=2m$. + \pause + \item Then $x+y=2n+2m=2(n+m)$. + \pause + \item Then $x+y$ is even. + \end{enumerate} + \end{proof} +\end{frame} + + +\subsection{Proofs by contradiction} + +\begin{frame}{Proofs by contradiction} + Idea: I want to show $A=\textbf{true}$. I show that the implication + ``$(\neg A)\implies\textbf{false}$'' is \textbf{true}. + Then $\neg A=\textbf{false}$, so $A=\textbf{true}$. + \pause +\begin{definition} + A natural number is called \emph{prime} if it is different from $1$ and it + is only divisible by $1$ and itself. +\end{definition} + \begin{theorem} + There are infinitely many prime numbers. + \end{theorem} + +\end{frame} + +\begin{frame}{Proofs by contradiction} + \begin{theorem} + There are infinitely many prime numbers. + \end{theorem} + \pause + \begin{proof} + \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 $p_1,p_2,\dots,p_n$. + \pause + \item Let $u=p_1\times p_2\times \dots \times p_n +1$. + \pause + \item $u$ is not divisible by any of the prime numbers $p_1,\dots,p_n$. + \pause + \item Therefore $u$ is only divisible by $1$ and itself. So $u$ is prime. + \pause + \item So $p_1,\dots,p_n$ are not the only prime numbers. + \end{enumerate} + \end{proof} + +\end{frame} + + +\subsection{Proofs by induction} + +\begin{frame}{Proofs by induction} + If I want to prove $\forall n\in \mathbb N,\, A(n)$: + \begin{enumerate} + \item Prove $A(0)$ (\emph{base step}) + \item Prove $\forall n\in \mathbb N,\, (A(n)\implies A(n+1))$ + (\emph{inductive step}) + \end{enumerate} + \pause + \begin{theorem}[Sum of natural numbers] + $\forall n\in \mathbb N,\quad 0+1+\cdots + n=\frac{n(n+1)}{2}$ + \end{theorem} + +\end{frame} + + +\begin{frame} + \begin{proof} + \begin{enumerate} + \item Base case: $0=0$. + \pause + \item Let $n$ be any natural number. + \pause + + If $A(n)=\textbf{false}$, then $A(n)\implies A(n+1)$ is \textbf{true}. + \pause + + If $A(n)=\textbf{true}$, we have to show that $A(n+1)=\textbf{true}$. + \pause + + \begin{align*} + \begin{array}{rlr} + 0+\cdots +(n+1)& = (0+\cdots+ n)+(n+1)=\\ + & = \frac{n(n+1)}{2}+(n+1)= & \text{since }A(n)=\textbf{true}\\ + & = \frac{n^2+n+2n+2}{2}\\ + & = \frac{(n+1)(n+2)}{2} + \end{array} + \end{align*} + so $A(n+1)=\textbf{true}$ + + \end{enumerate} + \end{proof} +\end{frame} + +\end{document} diff --git a/slides/preplogic-slides.toc b/slides/preplogic-slides.toc @@ -0,0 +1,10 @@ +\beamer@sectionintoc {1}{Statements}{3}{0}{1} +\beamer@sectionintoc {2}{Logical operations}{11}{0}{2} +\beamer@subsectionintoc {2}{1}{Boolean algebra}{22}{0}{2} +\beamer@subsectionintoc {2}{2}{Truth tables}{25}{0}{2} +\beamer@sectionintoc {3}{Implication}{28}{0}{3} +\beamer@sectionintoc {4}{Quantifiers}{39}{0}{4} +\beamer@sectionintoc {5}{Proofs}{46}{0}{5} +\beamer@subsectionintoc {5}{1}{Direct proofs}{50}{0}{5} +\beamer@subsectionintoc {5}{2}{Proofs by contradiction}{55}{0}{5} +\beamer@subsectionintoc {5}{3}{Proofs by induction}{64}{0}{5} diff --git a/slides/unilu.jpg b/slides/unilu.jpg Binary files differ.
