From 230fa50257bd2db4edee02a5b9779bfe2f2fa8c7 Mon Sep 17 00:00:00 2001 From: TiagoRG <35657250+TiagoRG@users.noreply.github.com> Date: Tue, 4 Apr 2023 18:34:15 +0100 Subject: [PATCH] [MD] re-re-removed this :skull: --- 1ano/2semestre/md/apontamentos/README.md | 10 - .../classes/logica-1a-ordem/src/main.bib | 3 - .../classes/logica-1a-ordem/src/main.tex | 291 ------------------ .../classes/logica-1a-ordem/src/ua.pdf | Bin 2082 -> 0 bytes .../md/apontamentos/pdf/logica-1a-ordem.pdf | Bin 171848 -> 0 bytes 1ano/2semestre/md/apontamentos/setup.sh | 34 -- .../md/apontamentos/template/src/main.bib | 3 - .../md/apontamentos/template/src/main.tex | 63 ---- .../md/apontamentos/template/src/ua.pdf | Bin 2082 -> 0 bytes 9 files changed, 404 deletions(-) delete mode 100644 1ano/2semestre/md/apontamentos/README.md delete mode 100644 1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/main.bib delete mode 100644 1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/main.tex delete mode 100755 1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/ua.pdf delete mode 100644 1ano/2semestre/md/apontamentos/pdf/logica-1a-ordem.pdf delete mode 100755 1ano/2semestre/md/apontamentos/setup.sh delete mode 100644 1ano/2semestre/md/apontamentos/template/src/main.bib delete mode 100644 1ano/2semestre/md/apontamentos/template/src/main.tex delete mode 100755 1ano/2semestre/md/apontamentos/template/src/ua.pdf diff --git a/1ano/2semestre/md/apontamentos/README.md b/1ano/2semestre/md/apontamentos/README.md deleted file mode 100644 index d484638..0000000 --- a/1ano/2semestre/md/apontamentos/README.md +++ /dev/null @@ -1,10 +0,0 @@ -# Matemática Discreta - Apontamentos - -### Conteúdos: -* Diretório com todo o código latex das aulas: [classes](https://github.com/TiagoRG/uaveiro-leci/tree/master/1ano/2semestre/md/apontamentos/classes) -* Diretório com todos os PDF com os apontamentos: [pdf](https://github.com/TiagoRG/uaveiro-leci/tree/master/1ano/2semestre/md/apontamentos/pdf) -* Diretório com o template usado: [template](https://github.com/TiagoRG/uaveiro-leci/tree/master/1ano/2semestre/md/apontamentos/template) -* Script em shell para a automação do sistema: [setup.sh](https://github.com/TiagoRG/uaveiro-leci/tree/master/1ano/2semestre/md/apontamentos/setup.sh) - ---- -*Pode conter erros, caso encontre algum, crie um* [*ticket*](https://github.com/TiagoRG/uaveiro-leci/issues/new) diff --git a/1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/main.bib b/1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/main.bib deleted file mode 100644 index 330cf1a..0000000 --- a/1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/main.bib +++ /dev/null @@ -1,3 +0,0 @@ -%! Author = tiagorg -%! Date = 31/01/2023 - diff --git a/1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/main.tex b/1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/main.tex deleted file mode 100644 index 7efc4f7..0000000 --- a/1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/main.tex +++ /dev/null @@ -1,291 +0,0 @@ -%! Author = tiagorg -%! Date = 31/01/2023 -\documentclass[11pt]{report} - -\usepackage{amsmath} -\usepackage[T1]{fontenc} % Fontes T1 -\usepackage[utf8]{inputenc} % Input UTF8 -\usepackage[backend=biber, style=ieee]{biblatex} % para usar bibliografia -\usepackage{csquotes} -\usepackage[portuguese]{babel} %Usar língua portuguesa -\usepackage{blindtext} % Gerar texto automaticamente -\usepackage{hyperref} % para autoref -\usepackage{graphicx} -\usepackage{indentfirst} -\usepackage[printonlyused]{acronym} -\usepackage{color} - -\begin{document} -\def\titulo{Matemática Discreta} -\def\autores{Tiago Garcia} -\def\autorescontactos{tiago.rgarcia@ua.pt} -\def\empresa{Universidade de Aveiro} -\def\logotipo{ua.pdf} - -% -\def\tema{Lógica de 1ª Ordem} -% - -\begin{titlepage} -\begin{center} -\vspace*{50mm} -{\Huge\textbf{\titulo}}\\ -\vspace{10mm} -{\Large \empresa}\\ -\vspace{10mm} -{\LARGE \autores}\\ -\vspace{30mm} -\begin{figure}[h] - \center - \includegraphics{ua}\label{fig:ua-title} -\end{figure} -\vspace{30mm} -\end{center} -\end{titlepage} -\title{ -{\LARGE\textbf{\titulo} }\\\\ -{\Large \aula\\ \empresa} -} -\author{ - \href{https://github.com/TiagoRG}{\autores} \\ - \href{mailto:tiago.rgarcia@ua.pt}{\autorescontactos} -} -\date{\today} -\maketitle -\pagenumbering{arabic} -\clearpage - - % Content - - \chapter*{Consequências Semânticas} - \section*{Teorema} - Uma fórmula $\Psi$ é consequência lógica (ou semântica) das fórmulas $\psi_1, \psi_2, \ldots, \psi_n$ se e só se $(\psi_1 \wedge \psi_2 \wedge \ldots \wedge \psi_n) \rightarrow \Psi$ é uma tautologia (fórmula válida). - \subsection*{Notação} - $ \psi_1, \ldots, \psi_n \models \Psi $\\ - $\Psi$ é consequência lógica (ou semântica) de $\psi_1, \ldots, \psi_n$\\ - \par $\psi_1, \ldots, \psi_n \vdash \Psi$ existe uma prova de $\Psi$ a partir de $\psi_1, \ldots, \psi_n$\\ - A prova recorre a regras de dedução designadas por regras de inferência, e a tautologias conhecidas. - - \section*{Teorema} - $\psi_1, \ldots, \psi_n \models \Psi$\\ - ($\Psi$ é consequẽncia lógica de $\psi_1, \ldots, \psi_n$) se e só se o conjunto ${{\psi_1, \ldots, \psi_n, \neg\Psi}}$ é inconsistente, isto é, não existe uma interpretação para a qual todas as fórmulas do conjunto tomam valor 1. - \par Para verificar se este conjunto de fórmulas é inconsistente usamos uma nova regra designada por resolução:\\\\ - $ \frac{\psi \rightarrow \theta~~~\Psi\vee\psi}{\theta\vee\psi} res $\\Indicam que aplicámos a regra/método da resolução. - \subsection*{Casos particulares} - \begin{enumerate} - \item{Se $ \theta \equiv \bot $ obtemos\\ - $\frac{\Psi \rightarrow \bot~~~\Psi\vee\psi}{\bot\vee\psi}$\\ - simplificando como: $\bot\vee\psi\equiv\psi~~$ e $~~\Psi\rightarrow\bot\equiv\ned\Psi\vee\bot\equiv\ned\Psi$ - \par Para este caso particular a regra da resolução é:\\ - $\frac{\neg\Psi~~~\Psi\vee\psi}{\psi} res ~~ \rightarrow \neg\Psi, \Psi $ são lineares complementares. - } - \item { - Se $ \theta\equiv\bot~~~e~~~\psi\equiv\bot $ (este é um caso particular do caso 1.) - \par Se $\psi\equiv\bot$ então $\Psi\vee\psi\equiv\Psi\vee\bot\equiv\Psi$\\ - Substituindo no caso particular da regra de resolução obtida em 1. tem-se\\ - $ \frac{\neg\Psi~~~\Psi}{\bot} res $ - } - \end{enumerate} - - \chapter*{Lógica Proposicional} - \section*{Definição} - \subsection*{Simbolos} - Variáveis proposicionais: $p, q, \Psi, \psi, \ldots$\\ - Constantes: $\bot e \top$ - Conetivos lógicos: $\wedge, \vee, \rightarrow, \leftrightarrow, \neg, \equiv$ - \subsection*{Regras de construção} - \begin{enumerate} - \item{Se $\psi$ é uma fórmula proposicional então $\neg\neg\psi$ é uma fórmula proposicional.} - \item{Se $\psi$ e $\theta$ são fórmulas proposicionais então $\psi\wedge\theta$ é uma fórmula proposicional.} - \item{Se $\psi$ e $\theta$ são fórmulas proposicionais então $\psi\vee\theta$ é uma fórmula proposicional.} - \item{Se $\psi$ e $\theta$ são fórmulas proposicionais então $\psi\rightarrow\theta$ é uma fórmula proposicional.} - \item{Se $\psi$ e $\theta$ são fórmulas proposicionais então $\psi\leftrightarrow\theta$ é uma fórmula proposicional.} - \end{enumerate} - - \section*{Dedução na lógica proposicional} - \begin{itemize} - \item {Verificar se uma fórmula é consequência lógica de um conjunto finito de fórmulas.\\ - $\psi_1, \ldots, \psi_n \models \Psi$ - } - \item {Vimos que a consequência lógica é válida se e só se a implicação\\ - $\psi_1 \wedge \psi_2 \wedge \ldots \wedge \psi_n \rightarrow \Psi$ é uma tautologia. - } - \end{itemize} - - \subsection*{Para verificar se uma consequência lógica é válida:} - \begin{enumerate} - \item Verificar se a implicação associada é uma tautologia. - \item Verificar se é possível obter (também são usados os termos deduzir, derivar, entre outros) $\Psi$ a partir de $\psi_1, \ldots, \psi_n$, recorrendo a regras de inferência e tautologias conhecidas (propriedades dos conetivos lógicos).\\ (através de uma sequência de deduções em que aplicamos as regras de inferências e tautologias), diz-se que existe uma prova de $\Psi$ a partir de $\psi_1, \ldots, \psi_n$ e escreve-se $\psi_1, \ldots, \psi_n \vdash \Psi$. - \item Aplicar a regra de resolução - Método de resolução. - \end{enumerate} - \subsubsection*{Método de resolução} - A consequência lógica $\psi_1, \ldots, \psi_n \models, \Psi$ é válida se e só se o conjunto de fórmulas {$\psi_1, \ldots, \psi_n, \neg\Psi$} é inconsistente, ou seja, este conjunto contém $\bot$ ou é possível deduzir $\bot$ a partir deste conjunto de fórmulas, isto é, existe uma prova de $\bot$ a partir de $\psi_1,\ldots,\psi_n,\neg\Psi$. - - \chapter*{Lógica de 1ª ordem} - \section*{Definição} { - Exemplo de uma fórmula da lógica proposicional:\\ - $(p \wedge q) \rightarrow r $\\ - Para traduzir frases do tipo:\\ - i) \color{red} todos \color{black} os gatos têm garras.\\ - ii) \color{red} alguns \color{black} alunos de MD têm 20.\\ - \par Passamos da lógica proposicional para a lógica de 1ª ordem (esta última engloba a outra). - } - \section*{Linguagem da lógica de 1ª ordem} { - \subsection*{Alfabeto} { - \begin{enumerate} - \item Variáveis: x, y, z, \ldots; - \item Conetivos lógicos da lógica proposicional: $\wedge, \vee, \rightarrow, \leftrightarrow, \neg, \equiv$; - \item Constantes da lógica proposicional: $\bot e \top$; - \item Os quantificadores $\forall~e~\exists$; - \item O símbolo de igualdade: =; - \item Símbolos de constantes; - \item Símbolos de funções com aridade $n \in N$ (isto é, com $n$ argumentos); - \item Símbolos de predicados. - \end{enumerate} - } - \subsection*{Termo} { - \begin{enumerate} - \item Cada variável e cada símbolo de constante é um termo; - \item { Se f é símbolo de função com aridade $n$ e $t_1, \ldots, t_n$ são termos então $f(t_1, \ldots, t_n)$ é um termo.\\\\ - Exemplo: { - \begin{itemize} - \item Variáveis: $x, y, z$; - \item Constantes: $a = 1$, $b = $ Maria, $c = $ Gato tareco; - \item Funções: pai(Maria), onde\\ Pai: $P\rightarrow P$, onde $P$ é o conjunto das pessoas. - \item Predicado: $par(x)="x$ é par\("\), $D=N$\\ $par(2)=1,~~par(3)=0$, etc. - \end{itemize} - Como é que se constroem as fórmulas da lógica de 1.ª ordem?\\ - Definição (recursiva) de fórmula: - \begin{itemize} - \item $P(t_1, \ldots, t_n)$ é uma fórmla, considerando $P$ um simbolo de predicado e $t_1,\ldots,t_n$ termos. - \item Se $\psi$ e $\Psi$ sao fórmulas então:\\ $\psi \wedge \Psi$, $\psi \vee \Psi$, $\psi \rightarrow \Psi$, $\psi \leftrightarrow \Psi$, $\neg\psi$, $\bot$ e $\top$ são fórmulas. - \item Se $\psi$ é uma fórmula e $x$ é uma variável então $\forall x \psi$ e $\exists x \psi$ também são fórmulas. - \item Se $t_1$ e $t_2$ são termos então $t_1 = t_2$ é uma fórmula. - \end{itemize} - } - } - \end{enumerate} - } - \subsection*{Átomo} { - Na lógica proposicional, os átomos são as proposições atómicas (ex: $p =$ "chove", $q = $ "vou à aula de MD")\\ - \par Os átomos da lógica de 1ª ordem são: - \begin{enumerate} - \item $\bot, \top$ - \item $t_1=t_2$, com $t_1$ e $t_2$ termos - \item $P(t_1,\ldots,t_n)$, onde $t_1,\ldots,t_n$ são termos e $P$ é um simbolo de predicado. - \end{enumerate} - \subsubsection*{Exemplo} { - Consideremos os espaços vetoriais estudados na ALGA.\\ - O alfabeto inclui: - \begin{itemize} - \item O símbolo de constante o que representa o elemento nulo dos espaço vetorial - \item Símbolos de funções - \begin{enumerate} - \item Para cada $\alpha \in R$, o símbolo de funções\\ $\alpha \cdot \_$\\ que tem aridade 1 correspondente à multiplicação escalar. - \item O símbolo de função + com aridade 2, que corresponde à adição de elementos do espaço vetorial. - \end{enumerate} - \end{itemize} - } - \subsubsection*{Exemplos} { - Converta as seguintes afirmações para linguagem simbólica da lógica de 1ª ordem: - \begin{enumerate} - \item{ \color{red}Todos \color{black} os gatos têm garras.\\ - \color{red} $\forall x$ \color{black} [$g(x) \rightarrow t(x)$]\\ - \color{red} Universo: $U$ = conjunto dos animais. - } - \item{ \color{red} Alguns \color{black} alunos de MD têm 20.\\ - \color{red} $\exists x$ \color{black} ($MD(x) \wedge V(x)$)\\ - MD(x) = "x é aluno de MD"\\ - V(x) = "x tem 20" - \color{red} Universo: $U$ = alunos da UA em 22/23 - } - \end{enumerate} - } - } - } - - \section*{Folha 1} - \subsection*{Exercício 2.} - \subsubsection*{c)} - \color{red} Todos \color{black} os insetos são mais leves do que \color{red} algum \color{black} mamífero.~~~~~\color{red} $\forall$ $\exists$\\ - \color{black} Predicados:\\ - $I(x)$ = ``x é um inseto``\\ - $L(y,z)$ = ``y é mais leve do que z``\\ - $M(w)$ = ``w é um mamífero``\\ - \par $\forall x \left(I(x) \rightarrow \exists y \left( M(y) \wedge L(x, y) \right) \right)$ - \par Obs: Alcance de cada quantificador:\\ - \begin{itemize} - \item Ocorrência de x ligada: $I(x)$ - \item Alcance de $\forall x$: $\left(I(x) \rightarrow \exists y \left( M(y) \wedge L(x, y \right) \right)$ - \item Ocorrências de y ligadas: $M(y)$ e $L(x, y)$ - \item Alcance de $\exists y$: $\left( M(y) \wedge L(x, y \right)$ - \end{itemize} - - - \chapter*{Fórmula fechada} { - \section*{Definição} { - Fórmula que não tem variáveis com occorrências livres. - \subsection*{Exemplo} { - $\forall x~\exists y~(P(x)~\rightarrow~R(x,y))$ é uma fórmula fechada. - \par $\exists y~((\forall x~P(x))~\wedge~R(x,y))$, esta fórmula não é uma fórmula fechada. - } - \subsection*{Negação de fórmula com quantificadores} { - \begin{enumerate} - \item $\neg (\forall x~\psi)~\equiv~\exists x~\neg \psi$. - \item $\neg (\exists x~\psi)~\equiv~\forall x~\neg \psi$. - \end{enumerate} - $\psi$ - parte da fórmula que está sob o quantificador. - } - } - \section*{Introdução das fórmulas da lógica de 1ª ordem} { - \subsection*{Definição} { - \begin{itemize} - \item Estrututa; - \item Valoração,~~~V:$var~\rightarrow~D$, onde $D$ é o conjunto das variáveis. - \end{itemize} - O conceito de valoração pode ser entendido por forma a podermos considerar a valoração de um termo.\\ - $V(a) = a$, se $a$ é uma constante $V(f(t_1,\ldots,t_n)) = f^M(V(t_1),\ldots,V(t_n))$. - \textbf{Obs:} Frequentemente denotamos o símbolo de função $f$ e a função correspondente na estrutura $f^M$, pela mesma letra. - } - \subsection*{Exemplo dos slides} { - $V(M(A, x)) = M^M(V(A), V(x)) = M(A^M, 2) = M(1,2) = |1-2| = |-1| = 1$,~~~~~~$V(A) = A$ porque $A$ é uma constante. - } - } - \section*{Interpretação de fórmulas} { - \subsection*{Exemplo de interpretação de fórmulas (ver slides)} { - \subsubsection*{i)} { - Mostre que $R(x, A)$ não é válida na interpretação $(M,V)$\\ - \par Note-se que $\neg (M,V)\models R(x,A)$ se e só se $(M,V) \models \neg R(x,A)$ ($\neg R(x,A)$ é válida na interpretação $(M,V)$)\\ - \par $V(\neg R(x,A))\equiv\neg R(V(x),V(A))\equiv\neg R(2, A^M)\equiv\neg R(2, 1)\\\equiv\neg(2 < 1)\equiv\neg\bot\equiv\top $\\Logo, $\neg R(x, A)$ é valida na interpretação $(M,V)$, isto é, $(M,V)\models\neg R(x, A)$\\ - Isto é equivalente a afirmar que $R(x,A)$ não é válida nesta interpretação. - } - } - } - } - \chapter*{Forma normal de Skolem} { - \section*{Definição} { - Uma fórmula $\phi$ é dita em forma normal de Skolem se $\phi$ é uma fórmula na forma normal conjuntiva e não contém nenhum quantificador universal. - } - \section*{Exemplo} { - \subsection*{1)} { - $\forall x~P(x, f(x))\wedge\neg R(x) $, onde $f$ é uma função e $R$ e $P$ são predicados.\\ - } - \subsection*{2)} { - $\forall x~\forall y~(P(x, f(x)) \wedge (\neg R(x)~\vee~P(x,y)))$ - \subsubsection*{Ideia} { - \begin{enumerate} - \item Convertemos $F$ numa fórmula $G$ que está na FNC prenex.\\ Note-se que $F \equiv G$ - \item A partir de $G$ obtemos uma fórmula $H$ que está na forma normal de Skolem. - \end{enumerate} - \textbf{Para tal:}\\ - \begin{itemize} - \item Se no início da fórmula temos um quantificador do tipo $\exists x$, substituimos todas as ocorrências de $x$ por um símbolo $a$ que represente uma constante e eliminamos o quantificador $\equiv x$. - \item Se na fórmula existe um quantificador existencial $\exists x_k$ com os quantificadores universais $\forall x_1~\forall x_2~\dots~\forall x_{k-1}$, à sua esquerda, substituimos todas as ocorrências de $x_k$ por um símbolo de função que ainda não esteja na fórmula, por exemplo $f$, que tem nos seus argumentos as variáveis $x_1, x_2, \dots, x_{k-1}$, isto é, $x_k$ é substituido por $f(x_1,\ldots,x_{k-1})$.\\\textbf{Atenção:} A fórmula $H$ que obtemos na forma normal de Skolen pode não ser (logicamente) equivalente à fórmula $G$ escrita na FNC prenex ou à fórmula $F$ original. - \end{itemize} - } - } - } - } - - -\end{document} diff --git a/1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/ua.pdf b/1ano/2semestre/md/apontamentos/classes/logica-1a-ordem/src/ua.pdf deleted file mode 100755 index 460cb187713ae1e843bf36ce5ff0a6cb1d1ea3b6..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 2082 zcmah~Yfuwc6!tNQE>L`ol&Pg$P$YmR`$`B(QSw6MkqiWjVyH{9Ks01ScF`08TSu9W z53u8VltGT6Ba=?xvgMrjsYH?N}@+HOsh;4!lvm& 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z<(oH{S_>#(=BO1EYc#+=4y=56d|MgdPz@_xiBjoLZSp}9EHaaIJB^HgZ-q+0Luw@c z&oXA7hm?nUhU_vbX-=UeXDWi1Q5FKk(-$y=g`pPjm3$4<-p)O%`g{;Lv=vJFpC=<% bH&YikulH3EGCMCDI~N}^4UM#l4D$a1G1lmz diff --git a/1ano/2semestre/md/apontamentos/setup.sh b/1ano/2semestre/md/apontamentos/setup.sh deleted file mode 100755 index 332f690..0000000 --- a/1ano/2semestre/md/apontamentos/setup.sh +++ /dev/null @@ -1,34 +0,0 @@ -# Confirma que o script está a ser executado no diretório correto -# shellcheck disable=SC2164 -parent_path=$( cd "$(dirname "${BASH_SOURCE[0]}")" ; pwd -P ) -cd "$parent_path" - -if [ $# != 1 ] -then - # Isto executa se o comando tiver mais do que 1 argumento - echo "Correct usage: md [class name (may not include spaces!)]" - -else - if [ "$1" == "reset" ] - then - # Reinicializa o template usando o git restore - git checkout 657720aba1fa3bde17d30f2021e001a972c18795 -- "$parent_path/template/" - - # Termina o script - exit 0 - fi - - # Cria o diretório da determinada aula - mkdir "classes/$1" - - # Copia o conteúdo da aula para o diretório respetivo - cp -a "template/out" "classes/$1" - cp -a "template/src" "classes/$1" - - # Reinicializa o template usando o git restore - git checkout 657720aba1fa3bde17d30f2021e001a972c18795 -- "$parent_path/template/" - - # Copia o pdf da aula para a pasta que contém todos os pdf - cp "classes/$1/out/main.pdf" "pdf" - mv "pdf/main.pdf" "pdf/$1.pdf" -fi diff --git a/1ano/2semestre/md/apontamentos/template/src/main.bib b/1ano/2semestre/md/apontamentos/template/src/main.bib deleted file mode 100644 index 330cf1a..0000000 --- a/1ano/2semestre/md/apontamentos/template/src/main.bib +++ /dev/null @@ -1,3 +0,0 @@ -%! Author = tiagorg -%! Date = 31/01/2023 - diff --git a/1ano/2semestre/md/apontamentos/template/src/main.tex b/1ano/2semestre/md/apontamentos/template/src/main.tex deleted file mode 100644 index 65a1cd1..0000000 --- a/1ano/2semestre/md/apontamentos/template/src/main.tex +++ /dev/null @@ -1,63 +0,0 @@ -%! Author = tiagorg -%! Date = 31/01/2023 -\documentclass[11pt]{report} - -\usepackage{amsmath} -\usepackage[T1]{fontenc} % Fontes T1 -\usepackage[utf8]{inputenc} % Input UTF8 -\usepackage[backend=biber, style=ieee]{biblatex} % para usar bibliografia -\usepackage{csquotes} -\usepackage[portuguese]{babel} %Usar língua portuguesa -\usepackage{blindtext} % Gerar texto automaticamente -\usepackage{hyperref} % para autoref -\usepackage{graphicx} -\usepackage{indentfirst} -\usepackage[printonlyused]{acronym} -\usepackage{color} - -\begin{document} -\def\titulo{Matemática Discreta} -\def\autores{Tiago Garcia} -\def\autorescontactos{tiago.rgarcia@ua.pt} -\def\empresa{Universidade de Aveiro} -\def\logotipo{ua.pdf} - -% -\def\tema{} -% - -\begin{titlepage} -\begin{center} -\vspace*{50mm} -{\Huge\textbf{\titulo}}\\ -\vspace{10mm} -{\Large \empresa}\\ -\vspace{10mm} -{\LARGE \autores}\\ -\vspace{30mm} -\begin{figure}[h] - \center - \includegraphics{ua}\label{fig:ua-title} -\end{figure} -\vspace{30mm} -\end{center} -\end{titlepage} -\title{ -{\LARGE\textbf{\titulo} }\\\\ -{\Large \aula\\ \empresa} -} -\author{ - \href{https://github.com/TiagoRG}{\autores} \\ - \href{mailto:tiago.rgarcia@ua.pt}{\autorescontactos} -} -\date{\today} -\maketitle -\pagenumbering{arabic} -\clearpage - - % Content - - - - -\end{document} diff --git a/1ano/2semestre/md/apontamentos/template/src/ua.pdf b/1ano/2semestre/md/apontamentos/template/src/ua.pdf deleted file mode 100755 index 460cb187713ae1e843bf36ce5ff0a6cb1d1ea3b6..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 2082 zcmah~Yfuwc6!tNQE>L`ol&Pg$P$YmR`$`B(QSw6MkqiWjVyH{9Ks01ScF`08TSu9W z53u8VltGT6Ba=?xvgMrjsYH?N}@+HOsh;4!lvm& zDu)IFp(u|MS`3*X&K17U#Jrpg2#L!%)||;GjCu}828xuuD^P-haK6CzeefV_tc*UH z=0F;3TcHAQU1{TDk~xTP{}7VGLAbLO2()V^5>oQfVeHmvqpNNK&p+68hwAq@#+XX% z=>bV?%D#OM-|R2(+g(t2`dmx=FAie#s?r62XZN+ey&u!tmt1=L?fltApJJ6~yL^+E zJ?`oeoTm0aZRGaN_X^umNsdbY$!k_IrK`9%!|!X89jiaT-Q)5hy&tPdY4oW-e#Kr{ zJu)q&5QI7ow%&eRnbI;j)gA2p*-df1#;9*JHh7TBULl`whhvP9jYr6)!dZ!T$xvT+$&=hi1fJdt)Or^ zD&D)cI<&F!dL)-`k8Z5^*q^o}~*{X^?>P0!BM&MyLc=Gg;C`!Yk*?CuFbB|C7e z_0f&YJ%~_QE^6#~)msxdwc)PKNWYl)K>GaV%<3jtdGW0iFM%Oi@LSC|#dk4hvnCY< zG-ztk|IR$Izd3cp^NhQFg7?K2cO3hoQtc9_K~vTbmw`&)82Xb4S9OV9zl#I9>iobEXV4tMhgd|LJ0<0CWpGh z85y&Jr-MK-3rEoiRE$Y6DJn&=Fcb?$(O~G8z!*N_0?Ff40-jE;0tS@e9pey@T9W{& z{N9C0Np7fpUtYOiUI`$L}yq;UWeFz=E8>LbcM;sl39YLX2=h%0S4W+4cDW!WfRB z1PYOsV3I{cu^qaowNe%fZGbldWn!Ud4+v(S$&^8v&HP}s*+v$?-#{HRzyjLJ!gPQT zVk9Pxz$G(rA?YfwzbFuITuuWV#qbcY%tCS5EKPys&M=7(6QUu5V6s{1OauiOilFZv zAQ6iRF~|fi8&1GE@((b(Y#4#T`F&_ZF#_HsLu@DpNBE%)Bf{Y+9%6&WVK!Vc+>a#S zsE5Ro(n0lcRtjD?R!9&aN;c-xd~c#8hJmEu&j=#ouNc5jn}1h;v`AbbkrJdrOyJ?D sgp?6Vg&3ES7#6J}%T%TZRYA5knEmm)zyO0^34AN3r>kpKVy