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z6U@5%aiq!@;8~pAhB;PzYWByh!Hc=fWf1pG7PdX{$|)veTQi`|5l}ohXj_xZFXQ_t z9um7h*vogbZ7A3BdG**adEKw!pk##nTJwOvxoL^ zbGVQsux<7xV}Vy`U~zHp_qp4(`dW9Rz{0%hVmHoK<&b{btoH`1{P3Rs$z_YrUkq{l z|7=71`;A&t1!{uDA!B0K9b_dl$lMmlkCHLwM~U}hVT_@3!7o{I{n2O~m| zO|J(S8?pBzGXg}(4E4Vg(A&}UtRnsDNR{;-JK{x5;6xqgL_V!TSYY{8!mxBh98rb+ zvI@)6j=(j0#bd>^wL- zlA?h5l1!S}QkSQmutH}uwSHzAIZq7*i?aFc3Q_G4p@@0p7#U-&eYVqkO!e46fnszl z!#dUs7ta#3Y*tD2NPo=)SqnIt6+KF7iLSV;t-dy!U=~*3E2)X+SmG#rs)c2E*1qEh zyP5hj=CNp65`=Ty+Nwe2>&C|t@D7a 0$ e \textbf{Falsa} para $n \leq 0$. + \item Para todo o $n \in N$, $3n \geq 4n$. $\rightarrow$ Proposição \textbf{False}. +\end{enumerate} + +\subsection{Tipos de Proposições} + +\begin{itemize} + \item \textbf{Atómica}: Não pode ser decomposta em proposições mais simples. + \item \textbf{Composta}: É formada a partir da combinação de proposições atómicas usando conectivos lógicos. +\end{itemize} + +\section{Conectivos Lógicos} +\label{sec:1-interpretacao-conectivos} + +\subsection{Negação} + +\subsubsection{Símbolo} +O símbolo da negação é $\neg$. + +\subsubsection{Tabela de Verdade} +\begin{table}[H] + \begin{tabular}{|c|c|} + \hline + $p$ & $\neg p$ \\ + \hline + 1 & 0 \\ + 0 & 1 \\ + \hline + \end{tabular} +\end{table} + +\subsection{Conjunção} + +\subsubsection{Símbolo} +O símbolo da conjunção é $\land$. + +\subsubsection{Tabela de Verdade} +\begin{table}[H] + \begin{tabular}{|c|c|c|} + \hline + $p$ & $q$ & $p \land q$ \\ + \hline + 1 & 1 & 1 \\ + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 0 \\ + \hline + \end{tabular} +\end{table} + +\subsection{Disjunção} + +\subsubsection{Símbolo} +O símbolo da conjunção é $\lor$. + +\subsubsection{Tabela de Verdade} +\begin{table}[H] + \begin{tabular}{|c|c|c|} + \hline + $p$ & $q$ & $p \lor q$ \\ + \hline + 1 & 1 & 1 \\ + 1 & 0 & 1 \\ + 0 & 1 & 1 \\ + 0 & 0 & 0 \\ + \hline + \end{tabular} +\end{table} + +\subsection{Implicação} + +\subsubsection{Símbolo} +O símbolo da conjunção é $\rightarrow$. + +\subsubsection{Tabela de Verdade} +\begin{table}[H] + \begin{tabular}{|c|c|c|} + \hline + $p$ & $q$ & $p \rightarrow q$ \\ + \hline + 1 & 1 & 1 \\ + 1 & 0 & 1 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + \hline + \end{tabular} +\end{table} + +\subsection{Equivalência} + +\subsubsection{Símbolo} +O símbolo da conjunção é $\leftrightarrow$. + +\subsubsection{Tabela de Verdade} +\begin{table}[H] + \begin{tabular}{|c|c|c|} + \hline + $p$ & $q$ & $p \leftrightarrow q$ \\ + \hline + 1 & 1 & 1 \\ + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + \hline + \end{tabular} +\end{table} + +\subsection{Exemplos} + +\subsubsection{Exemplo 1} +Ou o José foi ao supermercado ou está sem ovos em casa. + +\begin{itemize} + \item $\phi$ = "O José foi ao supermercado" + \item $\psi$ = "O José está sem ovos em casa" +\end{itemize} + +\textbf{Resultado}: $\phi \lor \psi$ + +\subsubsection{Exemplo 2} +A Beatriz decidiu emigrar e não tenciona regressar. + +\begin{itemize} + \item $\phi$ = "A Beatriz decidiu emigrar" + \item $\psi$ = "A Beatriz não tenciona regressar" +\end{itemize} + +\textbf{Resultado}: $\phi \land \psi$ + +\subsubsection{Exemplo 3} + +Ou o meu pai está em casa e a minha mãe não ou o meu pai não está em casa mas a minha mão está. + +\begin{itemize} + \item $\phi$ = "O meu pai está em casa" + \item $\psi$ = "A minha mão não está em casa" +\end{itemize} + +\textbf{Resultado}: $(\psi \land \neg \phi) \lor (\neg \psi \land \phi)$ + +\subsubsection{Exemplo 4} + +Ficarei milionário se ganhar o euromilhões + +\begin{itemize} + \item $\phi$ = "Ficar milionário" + \item $\psi$ = "Ganhar o euromilhões" +\end{itemize} + +\textbf{Resultado}: $\psi \rightarrow \phi$ + +\section{Validade de Fórmulas} + +\subsection{Tautologia} +\subsubsection{Definição} +Uma fórmula diz-se \textbf{Tautologia} quando tem valor lógico \textbf{1} para todas as suas interpretações. + +Representa-se com $\top$. + +\subsubsection{Exemplos} +\begin{enumerate} + \item $\neg \psi \lor \psi$ + \item $(\psi \land \phi) \rightarrow \psi$ +\end{enumerate} + +\begin{table}[H] + \begin{tabular}{|c|c|c|c|} + \hline + $p$ & $q$ & $p \land q$ & $(p \land q) \rightarrow q$ \\ + \hline + 0 & 0 & 0 & 1 \\ + 0 & 1 & 0 & 1 \\ + 1 & 0 & 0 & 1 \\ + 1 & 1 & 1 & 1 \\ + \hline + \end{tabular} +\end{table} + +\subsection{Consistente} +\subsubsection{Definição} +Uma fórmula diz-se \textbf{Consistente} quando tem valor lógico \textbf{1} para alguma das suas interpretações. + +\subsection{Inconsistente ou Contradição} +\subsubsection{Definição} +Uma fórmula diz-se \textbf{Inconsistente} ou \textbf{Contradição} quando tem valor lógico \textbf{0} para todas as suas interpretações. + +Representa-se com $\bot$. + +\subsubsection{Exemplo} +\begin{enumerate} + \item $\neg \psi \land \psi$ +\end{enumerate} + +\section{Fórmulas Equivalentes} +\subsection{Definição} +As fórmulas $\phi$ e $\psi$ dizem-se equivalentes quando a fórmula $\phi \leftrightarrow \psi$ é uma tautologia. + +\subsubsection{Demonstação} +\begin{table}[H] + \begin{tabular}{|c|c|c||c|c|c|} + \hline + $p$ & $q$ & $p \rightarrow q$ & $\neg p$ & $\neg p \lor q$ & $(p \rightarrow q) \leftrightarrow (\neg p \lor q)$ \\ + \hline + 0 & 0 & 1 & 1 & 1 & 1 \\ + 0 & 1 & 1 & 1 & 1 & 1 \\ + 1 & 0 & 0 & 0 & 0 & 1 \\ + 1 & 1 & 1 & 0 & 1 & 1 \\ + \hline + \end{tabular} +\end{table} + +\subsection{Exemplos} +\noindent Equivalências: +\begin{enumerate} + \item $p \land q \equiv q \land p$ + \item $p \lor q \equiv q \lor p$ + \item $p \land (q \land r) \equiv (p \land q) \land r$ + \item $p \lor (q \lor r) \equiv (p \lor q) \lor r$ + \item $p \land p \equiv p$ + \item $p \lor p \equiv p$ + \item $p \land \top \equiv p$ + \item $p \lor \top \equiv \top$ + \item $p \land \bot \equiv \bot$ + \item $p \lor \bot \equiv p$ +\end{enumerate} +Distributividade: +\begin{enumerate} + \item $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$ + \item $p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$ +\end{enumerate} +Leis de Morgan: +\begin{enumerate} + \item $\neg (p \land q) \equiv \neg p \lor \neg q$ + \item $\neg (p \lor q) \equiv \neg p \land \neg q$ +\end{enumerate} +Contraposição e dupla negação: +\begin{enumerate} + \item $p \rightarrow q \equiv \neg q \rightarrow \neg p$ + \item $p \rightarrow q \equiv \neg p \lor q$ + \item $\neg \neg p \equiv p$ +\end{enumerate} + +\section{Formas Normais} +\label{sec:1-interpretacao-formas-normais} + +\subsection{Literais} +\subsubsection{Definição} +Um literal é uma proposição atómica ou a negação de uma proposição atómica. + +\subsubsection{Exemplos} +\begin{enumerate} + \item $p$, $q$, $\neg r$ são literais. + \item $\neg \neg p$, $p \rightarrow q$ não são literais. +\end{enumerate} + +\subsection{Forma Normal Conjuntiva (FNC)} +Uma fórmula está na \textbf{Forma Normal Conjuntiva} se é uma conjunção de disjunção de literais. + +\subsection{Forma Normal Disjuntiva (FND)} +Uma fórmula está na \textbf{Forma Normal Disjuntiva} se é uma disjunção de conjunções de literais. + +\subsection{Exemplos} +\begin{enumerate} + \item $p \land q \land \neg r$ está na FNC e na FND. + \item $(p \lor \neg q) \land (q \lor r)$ está na FNC. + \item $(p \land \neg q) \lor (q \land r)$ está na FND. + \item $(p \land q) \lor (p \land \neg q) \lor (q \land r)$ não está na FNC nem na FND. +\end{enumerate} + + diff --git a/1ano/2semestre/md/apontamentos/src/2pec.tex b/1ano/2semestre/md/apontamentos/src/2pec.tex new file mode 100644 index 0000000..79e3d57 --- /dev/null +++ b/1ano/2semestre/md/apontamentos/src/2pec.tex @@ -0,0 +1,3 @@ +\part{Princípios de Enumeração Combinatória} +\label{prt:2pec} + diff --git a/1ano/2semestre/md/apontamentos/src/3aic.tex b/1ano/2semestre/md/apontamentos/src/3aic.tex new file mode 100644 index 0000000..b830e6c --- /dev/null +++ b/1ano/2semestre/md/apontamentos/src/3aic.tex @@ -0,0 +1,3 @@ +\part{Agrupamentos e Identidades Combinatórias} +\label{prt:3aic} + diff --git a/1ano/2semestre/md/apontamentos/src/4rfg.tex b/1ano/2semestre/md/apontamentos/src/4rfg.tex new file mode 100644 index 0000000..3d243d1 --- /dev/null +++ b/1ano/2semestre/md/apontamentos/src/4rfg.tex @@ -0,0 +1,3 @@ +\part{Recorrência e Funções Geradoras} +\label{prt:4rfg} + diff --git a/1ano/2semestre/md/apontamentos/src/5etg.tex b/1ano/2semestre/md/apontamentos/src/5etg.tex new file mode 100644 index 0000000..ab62437 --- /dev/null +++ b/1ano/2semestre/md/apontamentos/src/5etg.tex @@ -0,0 +1,3 @@ +\part{Elementos de Teoria dos Grafos} +\label{prt:5etg} + diff --git a/1ano/2semestre/md/apontamentos/src/Makefile b/1ano/2semestre/md/apontamentos/src/Makefile new file mode 100644 index 0000000..c224436 --- /dev/null +++ b/1ano/2semestre/md/apontamentos/src/Makefile @@ -0,0 +1,11 @@ +.PHONY: all compile clean + +all: compile clean + +compile: main.tex + pdflatex main.tex + pdflatex main.tex + mv main.pdf ../md-tiagorg.pdf + +clean: + rm -f *.aux *.blg *.bbl *.toc *.log *.lof *.lot *.log.xml *.bcf *.out *.run.xml *.gz diff --git a/1ano/2semestre/md/apontamentos/src/images/TiagoRG_Logo.png 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Author = TiagoRG +%! GitHub = https://github.com/TiagoRG + +\documentclass[15pt]{report} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{csquotes} +\usepackage[portuguese]{babel} +\usepackage[printonlyused]{acronym} +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{indentfirst} +\usepackage{float} +\usepackage{fancyhdr} +\usepackage{geometry} +\usepackage{titlesec} +\usepackage{titletoc} +\usepackage{tocloft} +\usepackage{listings} +\usepackage{array} +\usepackage{tabularx} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{enumitem} +\usepackage{multirow} +\usepackage{multicol} +\usepackage{setspace} +\renewcommand{\familydefault}{\sfdefault} + +\geometry{ + paper=a4paper, + margin=45pt, + includefoot +} +\setcounter{tocdepth}{1} + +\begin{document} +%%%%%% CAPA %%%%%% +\begin{titlepage} +\begin{center} + \vspace*{60pt} + {\fontsize{10\textwidth}{0}\selectfont{\textbf{MATEMÁTICA\\\vspace*{10pt}DISCRETA\\}}} + \vspace*{40pt} + \Huge + \textbf{Tiago Rocha Garcia}\\ + \vspace*{20pt} + \Large + \textbf{\href{mailto:tiago.rgarcia@ua.pt}{tiago.rgarcia@ua.pt}}\\ + \vspace*{5pt} + \textbf{\href{https://tiagorg.pt}{https://tiagorg.pt}}\\ + \vspace*{30pt} + \huge + \textbf{2023/2024}\\ + \vspace*{200pt} + \includegraphics[width=0.2\textwidth]{images/ua}\\ + \vspace*{10pt} + \huge + \textbf{Universidade de Aveiro}\\ +\end{center} +\end{titlepage} + +\renewcommand{\headrulewidth}{0pt} +\renewcommand{\footrulewidth}{0.1pt} +\fancypagestyle{plain}{ + \fancyhf{}% Clear header/footer + \fancyfoot[L]{Matemática Discreta\\Universidade de Aveiro\\2023/2024} + \fancyfoot[C]{\\{\huge\thepage}} + \fancyfoot[R]{Tiago Garcia\\\href{https://tiagorg.pt}{https://tiagorg.pt}\\\href{mailto:tiago.rgarcia@ua.pt}{tiago.rgarcia@ua.pt}} +} +\pagestyle{plain} + +\renewcommand{\thepart}{Capítulo \arabic{part}} +\titleformat{\part}[display]{\bfseries\centering}{\Huge\thepart}{2em}{\Huge} + +\renewcommand{\thechapter}{Tópico \arabic{chapter}} +\titleformat{\chapter}[display]{\bfseries}{\LARGE\thechapter}{1em}{\huge} +\dottedcontents{chapter}[7.2em]{\bfseries\vspace{0.6em}}{5em}{0.65pc} + +\renewcommand{\thesection}{\arabic{chapter}.\arabic{section}} +\titleformat{\section}[block]{\bfseries}{\Large\thesection}{1em}{\Large} +\dottedcontents{section}[9.2em]{\vspace{0.6em}}{2em}{0.65pc} + +\renewcommand{\contentsname}{Índice} +\tableofcontents +%\listoftables % descomentar se necessário +%\listoffigures % descomentar se necessário + +\input{1lpo} +\input{2pec} +\input{3aic} +\input{4rfg} +\input{5etg} + +\chapter*{Acrónimos} +\begin{acronym} +\end{acronym} + +\end{document} diff --git a/1ano/2semestre/md/fichas/fichas/folha0_MD_2023.pdf b/1ano/2semestre/md/fichas/2022-2023/fichas/folha0_MD_2023.pdf similarity index 100% rename from 1ano/2semestre/md/fichas/fichas/folha0_MD_2023.pdf rename to 1ano/2semestre/md/fichas/2022-2023/fichas/folha0_MD_2023.pdf diff --git a/1ano/2semestre/md/fichas/fichas/folha1_MD_2023.pdf 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