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Buendía y José Dosal} +\materia{Conjuntos y lógica} +\alum{} + +%\xsimsetup{solution/print = true} + +\begin{document} +\HojaExamen{}{e} +\begin{exercise} + Sea $f: \mathcal{X} \to \mathcal{Y}$ una función. Demuestra que $F: \mathcal{P}(\mathcal{X}) \to \mathcal{P}(\mathcal{Y})$ + y $G: \mathcal{P}(\mathcal{Y}) \to \mathcal{P}(\mathcal{X})$ definidas como: $\begin{aligned} + F(A) = f(A)\; \text{y}\; G(A) = f^{-1}(A), + \end{aligned}$ +son funciones. +\end{exercise} + +\begin{solution} +Sean $(A, f(A))$ y $(B, f(B))$ en $F$. Por demostrar que $f(A)=f(B)$. Supongamos que $f(A) \neq f(B)$, entonces +$f(B) \menos f(A) \neq \emptyset$, es decir, existe $y \in f(B) \menos f(A)$. Por lo anterior, existe $x \in A$ tal que +$f(x) = y$, es decir $y \in f(A)$. Por lo tanto $f(A) = f(B)$ +\end{solution} + +\begin{exercise} +Sean $f: A \to C$ y $g:A \to B$ funciones. Demostrar que existe una función $h: B \to C$ tal que $f=h \circ g$ +si y solo si para cada $x,\; y \in A$ $g(x) = g(y)$ implica $f(x) = f(y)$. +\end{exercise} + +\begin{solution} +\textbf{($\Rightarrow$)} Supongamos que existe una función $h: B \to C$ tal que $f = h \circ g$. +Por demostrar que para cualesquiera $x, y \in A$, si $g(x) = g(y)$, entonces $f(x) = f(y)$. +Sean $x, y \in A$ tales que $g(x) = g(y)$, +$$f(x) = (h \circ g)(x) = h(g(x)) = h(g(y)) = (h \circ g)(y) = f(y)$$ +Dado que $g(x) = g(y)$ y $h$ es función, +$$h(g(x)) = h(g(y))$$ +Por lo tanto $f(x) = f(y)$. +\\ +\textbf{($\Leftarrow$)} Supongamos que para todo $x, y \in A$ tenemos que $g(x) = g(y) \implies f(x) = f(y)$ es cierta. +Construyamos una función $h: B \to C$ tal que $f = h \circ g$. Sea $h$ definida para cualquier $b \in B$: +\begin{enumerate} + \item Si $b \in \text{Im}(g)$, entonces existe al menos un $a \in A$ tal que $g(a) = b$. Para el cual definimos $h(b) = f(a)$. + Si existiera otro elemento $a' \in A$ tal que $g(a') = b$, entonces tendríamos $g(a) = g(a')$. Por hipótesis, esto implica + que $f(a) = f(a')$. Por lo tanto, el valor $h(b)$ es único. + + \item Si $b \notin \text{Im}(g)$, entonces no existe ningún $a \in A$ tal que $g(a)=b$. Así, $h(b) = c_0$. + Escogemos un elemento fijo $c_0 \in C$. +\end{enumerate} + +\end{solution} + +\begin{exercise} + Sean $A \neq \emptyset$ y $B \neq \emptyset$ conjuntos. Para cualquier conjunto $C$ y cualesquiera funciones + $f_1: C \to A$ y $f_2 : C \to B$ existe una única función $f: C \to A \times B$ tal que $f_1 = p_1 \circ f$ + y $f_2 = p_2 \circ f$. (Las funciones $f_1$ y $f_2$ se denominan funciones coordenadas) +\end{exercise} + +\begin{solution} + $\forall c \in C$ se tiene que $f_1(c) \in A$ y $f_2(c) \in B$. Definimos a $f(c) = (f_1(c), f_2(c))$. + Es claro que es función ya que la pareja $(f_1(c), f_2(c))$ es única. Por demostrar que f cumple las + dos propiedades. + \begin{enumerate} + \item $\forall c \in C$ se tiene que $$ (p_1 \circ f)(c) = p_1(f(c)) = p_1(f_1(c), f_2(c)) = f_1(c)$$ + Por lo tanto, $p_1 \circ f = f_1$. + \item $\forall c \in C$ se tiene que $$ (p_2 \circ f)(c) = p_2(f(c)) = p_2(f_1(c), f_2(c)) = f_2(c)$$ + Por lo tanto, $p_2 \circ f = f_2$. + \end{enumerate} +\end{solution} + +\begin{exercise} + Demuestra que si $I\neq \emptyset$ y algún $A_\alpha = \emptyset$ si y solo si $\prod_{\alpha \in I} A_{\alpha} = \emptyset$. +\end{exercise} + +\begin{exercise} + Sea $I \neq \emptyset$ un conjunto de índices. Considera dos familias indizadas $\set{A_{\alpha}}_{\alpha \in I}$ + y $\set{B_{\alpha}}_{\alpha \in I}$ . Demuestra lo siguiente: + \begin{enumerate} + \item Si $A_{\alpha} \subseteq B_{\alpha}$ para cada $\alpha \in I$, entonces + $$\prod_{\alpha \in I} A_{\alpha} \subseteq \prod_{\alpha \in I} B_{\alpha}$$. + \item El recíproco de 1 se cumple si $\prod_{\alpha \in I} A_{\alpha} \neq \emptyset$ + \end{enumerate} +\end{exercise} + +\begin{exercise} + Sea la relación $E$ en $\mathcal{R}^2$ definida por $E = \set{((x_1,y_1),(x_2,y_2)) |\; y_1 - x_1^2 = y_2 - x_2^2 }$. Demuestre que $E$ es de equivalencia y describa las clases de equivalencia. +\end{exercise} + +\begin{exercise} + Sea $R$ una relaci\'{o}n reflexiva y transitiva. Defina $\approx$ en $A$ por $a \approx b$ si y s\'{o}lo si $(a,b)\in R $ y $(b,a)\in R$. + +\begin{enumerate} + \item[(a)] Muestre que $\approx$ es una relaci\'{o}n de equivalencia en $A$. + + \item[(b)] Si $\preceq$ se define por $[a] \preceq [b]$ si y s\'{o}lo si $(a,b)\in R$; muestre que $(A/\approx, \preceq)$ es un conjunto ordenado. +\end{enumerate} +\end{exercise} + +\begin{exercise} + Sea $R$ un orden en $A$. Pruebe que $R^{-1}$ es tambi\'{e}n un orden en $A$ (denominado dual de $R$) y para $B \subseteq A$ se cumple que + +\begin{enumerate} + \item[(a)] $a$ es el m\'{i}nimo elemento de $B$ en $R^{-1}$ si y s\'{o}lo si $a$ es el m\'{a}ximo elemento de $B$ en $R$. + + \item[(b)] Similarmente para minimal y maximal, y supremo e \'{i}nfimo. +\end{enumerate} +\end{exercise} + +\begin{exercise} + Muestre que $h$ es un isomorfismo entre $(A, \leq)$ y $(B, \preceq)$ si y s\'{o}lo si $h$ y $h^{-1}$ preservan el orden. +\end{exercise} + +\begin{exercise} + Sea $A$ ordenado por $\leq$, y $B \subseteq A$. Demuestra lo siguiente: + +\begin{enumerate}[(a)] + \item $B$ tiene a lo m\'{a}s un elemento m\'{i}nimo. + + \item El elemento m\'{i}nimo de $B$ (si existe) es tambi\'{e}n minimal. + + \item Si $B$ es una cadena, entonces todo elemento minimal de $B$ es tambi\'{e}n un m\'{i}nimo. +\end{enumerate} +\end{exercise} + +\end{document} \ No newline at end of file diff --git a/2026-1/T2.xsim b/2026-1/T2.xsim new file mode 100644 index 0000000000..2975b0ab74 --- /dev/null +++ b/2026-1/T2.xsim @@ -0,0 +1,42 @@ +\providecommand\numberofexercises{} +\XSIM{solution-body}{exercise-1=={Sean $(A, f(A))$ y $(B, f(B))$ en $F$. Por demostrar que $f(A)=f(B)$. Supongamos que $f(A) \neq f(B)$, entonces $f(B) \menos f(A) \neq \emptyset $, es decir, existe $y \in f(B) \menos f(A)$. Por lo anterior, existe $x \in A$ tal que $f(x) = y$, es decir $y \in f(A)$. Por lo tanto $f(A) = f(B)$}||exercise-2=={\textbf {($\Rightarrow $)} Supongamos que existe una función $h: B \to C$ tal que $f = h \circ g$. Por demostrar que para cualesquiera $x, y \in A$, si $g(x) = g(y)$, entonces $f(x) = f(y)$. Sean $x, y \in A$ tales que $g(x) = g(y)$, $$f(x) = (h \circ g)(x) = h(g(x)) = h(g(y)) = (h \circ g)(y) = f(y)$$ Dado que $g(x) = g(y)$ y $h$ es función, $$h(g(x)) = h(g(y))$$ Por lo tanto $f(x) = f(y)$. \\ \textbf {($\Leftarrow $)} Supongamos que para todo $x, y \in A$ tenemos que $g(x) = g(y) \implies f(x) = f(y)$ es cierta. Construyamos una función $h: B \to C$ tal que $f = h \circ g$. Sea $h$ definida para cualquier $b \in B$: \begin {enumerate} \item Si $b \in \text {Im}(g)$, entonces existe al menos un $a \in A$ tal que $g(a) = b$. Para el cual definimos $h(b) = f(a)$. Si existiera otro elemento $a' \in A$ tal que $g(a') = b$, entonces tendríamos $g(a) = g(a')$. Por hipótesis, esto implica que $f(a) = f(a')$. Por lo tanto, el valor $h(b)$ es único. \par \item Si $b \notin \text {Im}(g)$, entonces no existe ningún $a \in A$ tal que $g(a)=b$. Así, $h(b) = c_0$. 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Defina $\approx $ en $A$ por $a \approx b$ si y s\'{o}lo si $(a,b)\in R $ y $(b,a)\in R$. \par \begin {enumerate} \item [(a)] Muestre que $\approx $ es una relaci\'{o}n de equivalencia en $A$. \par \item [(b)] Si $\preceq $ se define por $[a] \preceq [b]$ si y s\'{o}lo si $(a,b)\in R$; muestre que $(A/\approx , \preceq )$ es un conjunto ordenado. \end {enumerate}}||exercise-8=={Sea $R$ un orden en $A$. Pruebe que $R^{-1}$ es tambi\'{e}n un orden en $A$ (denominado dual de $R$) y para $B \subseteq A$ se cumple que \par \begin {enumerate} \item [(a)] $a$ es el m\'{i}nimo elemento de $B$ en $R^{-1}$ si y s\'{o}lo si $a$ es el m\'{a}ximo elemento de $B$ en $R$. \par \item [(b)] Similarmente para minimal y maximal, y supremo e \'{i}nfimo. \end {enumerate}}||exercise-9=={Muestre que $h$ es un isomorfismo entre $(A, \leq )$ y $(B, \preceq )$ si y s\'{o}lo si $h$ y $h^{-1}$ preservan el orden.}||exercise-10=={Sea $A$ ordenado por $\leq $, y $B \subseteq A$. 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+\author{Karina García Buendía} +\date{\today} + +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +\begin{frame}{Axiomas} + \begin{block}{Axioma 1: Existencia} + \textit {Existe un conjunto que no tiene elementos.} + \end{block} + + \begin{block}{Axioma 2: Extensión} + \textit {Si todo elemento de $X$ es elemento de $Y$ y todo elemento de $Y$ es elemento de $X$, entonces + $X =Y$.} + \end{block} + + \begin{block}{Axioma 3: Esquema de comprensión} + \textit {Sea una propiedad $P(x)$. Para cualquier conjunto $A$ hay un conjunto $B$ tal que $x \in B \Leftrightarrow$ + $x \in A$ y $P(x)$.} + \end{block} +\end{frame} + +\begin{frame} + Ejemplo 1. Si $P$ y $Q$ son conjuntos, entonces hay un conjunto $R$ tal que $x \in R$ + si y solo si $x \in P$ y $x \in Q$. +\pause + \begin{proof} + Consideremos la propiedad de que $x \in Q$, es decir $P(x,Q)$. Por el axioma esquema de comprensión se tiene + que para todo $Q$ y cualquier $P$ hay un conjunto $R$ tal que $x \in R$ si y solo si $x \in P$ y $x \in Q$. + \end{proof} +\pause + Ejemplo 2. El conjunto de todos los conjuntos no existen. + \begin{proof} + Supongamos lo contrario. + \end{proof} +\end{frame} + +\begin{frame} + \begin{lema} + Sea $P(x)$ una propiedad de $x$. Para todo conjunto $A$ hay un único conjunto $B$ tal que $x \in B$ + si y solo si $x \in A$ y $P(x)$ + \end{lema} +\end{frame} + +\begin{frame} + \begin{block}{Axioma 4: del Par} + \textit {Para cualesquiera $a$ y $b$ existe un conjunto $C$ tal que $x \in C$ si y solo si $x = a$ o $x = b$.} + \end{block} + +\pause + +\vspace{1em} + ¿Todo conjunto es un elemento de algún otro conjunto?, ¿dos conjuntos cuales quiera son simultáneamente elementos + de algún mismo conjunto? +\end{frame} + +\begin{frame}{Axiomas} + \begin{block}{Axioma 5: de Unión} + \textit {Para cualquier $S$, existe un conjunto $U$ tal que si $x \in U$ si y solo si $x \in X$ para algún $X \in S$.} + \end{block} + +\vspace{1em} + + \begin{itemize} + \item Ejemplo 3. Sea $S=\{\emptyset, \{\emptyset\}\}$ + \pause + ¿Quién es $\bigcup S$? + \pause + \vspace{1em} + + Entonces $x \in \bigcup S$ si y solo si $x \in A$ para algún conjunto $A$ en $S$. Es decir, si y solo si $x \in \emptyset$ + o $x \in \{\emptyset\}$ + \item $\bigcup \emptyset = \emptyset$ + \item Sean $A$ y $B$ conjuntos, existe $\bigcup \{A, B \}$ tal que $x \in \bigcup \{A, B \}$ si y solo si + $x \in A$ o $x \in B$. + \end{itemize} + +\end{frame} + +\end{document} diff --git a/2026-1/axiomas_con.xdv b/2026-1/axiomas_con.xdv new file mode 100644 index 0000000000..3b0a622fd0 Binary files /dev/null and b/2026-1/axiomas_con.xdv differ diff --git a/2026-1/axiomas_con.xsim b/2026-1/axiomas_con.xsim new file mode 100644 index 0000000000..f0bf4bb841 --- /dev/null +++ b/2026-1/axiomas_con.xsim @@ -0,0 +1,42 @@ +\providecommand\numberofexercises{} +\XSIM{solution-body}{} +\XSIM{exercise-body}{} +\XSIM{goal}{exercise}{points}{0} +\XSIM{totalgoal}{points}{0} +\XSIM{goal}{exercise}{bonus-points}{0} +\XSIM{totalgoal}{bonus-points}{0} +\XSIM{order}{} +\XSIM{use}{} +\XSIM{use!}{} +\XSIM{used}{} +\XSIM{print}{} +\XSIM{print!}{} +\XSIM{printed}{} +\XSIM{total-number}{0} +\XSIM{exercise}{0} +\XSIM{types}{exercise} +\XSIM{idtypes}{} +\XSIM{collections}{} +\XSIM{collection:all exercises}{} +\setcounter{totalexerciseinall exercises}{0} +\XSIM{id}{} +\XSIM{ID}{} +\XSIM{counter}{} +\XSIM{counter-value}{} +\XSIM{solution}{} +\XSIM{chapter-value}{} +\XSIM{chapter}{} +\XSIM{section-value}{} +\XSIM{section}{} +\XSIM{sectioning}{} +\XSIM{subtitle}{} +\XSIM{points}{} +\XSIM{bonus-points}{} +\XSIM{page-value}{} +\XSIM{page}{} +\XSIM{tags}{} +\XSIM{topics}{} +\XSIM{userpoints}{} +\XSIM{bodypoints}{} +\XSIM{userbonus-points}{} +\XSIM{bodybonus-points}{} diff --git a/2026-1/examen_extraordinario_2026-1.aux b/2026-1/examen_extraordinario_2026-1.aux new file mode 100644 index 0000000000..4067488a48 --- /dev/null +++ b/2026-1/examen_extraordinario_2026-1.aux @@ -0,0 +1,14 @@ +\relax +\providecommand*{\memsetcounter}[2]{} +\providecommand\babel@aux[2]{} +\@nameuse{bbl@beforestart} +\catcode `"\active +\providecommand\XSIM[1]{} +\XSIM{readaux} +\babel@aux{spanish}{} +\@writefile{toc}{\contentsline {paragraph}{Ejercicio\nobreakspace 1}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Ejercicio\nobreakspace 2}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Ejercicio\nobreakspace 3}{1}{}\protected@file@percent } +\memsetcounter{lastsheet}{1} +\memsetcounter{lastpage}{1} +\gdef \@abspage@last{1} diff --git a/2026-1/examen_extraordinario_2026-1.pdf b/2026-1/examen_extraordinario_2026-1.pdf new file mode 100644 index 0000000000..4483edd617 Binary files /dev/null and b/2026-1/examen_extraordinario_2026-1.pdf differ diff --git a/2026-1/examen_extraordinario_2026-1.tex b/2026-1/examen_extraordinario_2026-1.tex new file mode 100644 index 0000000000..925c117b87 --- /dev/null +++ b/2026-1/examen_extraordinario_2026-1.tex @@ -0,0 +1,107 @@ +\documentclass[fc]{tarea} + +\usepackage[style=mexican]{csquotes} + +\newcommand{\menos}{\backslash} +\newcommand{\al}{\symscr{A}} +\newcommand{\lan}{\symscr{L}} +\newcommand{\sis}{\symup{SF}} +\newcommand{\expresiones}{\symup{Exp}} +\NewDocumentCommand{\expr}{O{\al}}{\expresiones(#1)} + +%conjuntos +\providecommand\st{} +\newcommand\SetSymbol[1][]{% +\nonscript\:#1\vert +\allowbreak +\nonscript\: +\mathopen{}} +\DeclarePairedDelimiterX\set[1]{\{}{\}}{% +\renewcommand\st{\SetSymbol[\delimsize]}#1} +\DeclarePairedDelimiterX\bset[1]{\lbrack}{\rbrack}{#1} + +\examen{Tarea 2 (parte I)} +\prof{Karina G. Buendía y José Dosal} +\materia{Conjuntos y lógica} +\alum{} + +%\xsimsetup{solution/print = true} + + +\begin{document} +\HojaExamen{}{e} +\iffalse +\section*{1. Conjuntos} + +\begin{enumerate} + \item[1.1] Noci\'{o}n de conjunto. Noci\'{o}n de pertenencia a un conjunto. Notaci\'{o}n. + + \item[1.2] Relaciones entre conjuntos: $\cup$, $\cap$, $\setminus$, $\subset$. Uni\'{o}n e intersecci\'{o}n generalizadas. + + \item[1.3] Diagramas de Venn y diagramas de Euler. Representaci\'{o}n de operaciones. + + \item[1.4] Relaciones: pares ordenados y productos cartesianos, dominio y codominio, imagen o rango de una relaci\'{o}n. Operaciones con relaciones: inversa de una relaci\'{o}n, composici\'{o}n de relaciones. Relaciones de orden sobre un conjunto: Conjunto Parcialmente Ordenado (COPO), Conjunto Totalmente Ordenado (COTO), Conjunto Bien Ordenado (COBO), Conjunto Densamente Ordenado (CODO), Cotas (m\'{a}ximo, m\'{ı}nimo, maximales, \dots). ¿Qu\'{e} afirma el Lema de Zorn? + + \item[1.5] Relaciones de equivalencia y particiones. El conjunto cociente m\'{o}dulo una relaci\'{o}n de equivalencia. + + \item[1.6] Funciones. Dominio y Codominio, Rango o Imagen. Igualdad de funciones, funci\'{o}n constante, gr\'{a}fica de una funci\'{o}n. Funciones inyectivas, suprayectivas y biyectivas. Inversa de una funci\'{o}n, funciones invertibles, composici\'{o}n de funciones. + + \item[1.7] Cardinalidad. Equipotencia de dos conjuntos. Teorema de Cantor, conjuntos finitos y conjuntos infinitos. Conjuntos numerables y no numerables ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$). + + \item[1.8] Inducci\'{o}n y Recursi\'{o}n. Inducci\'{o}n finita y segundo principio de Inducci\'{o}n. Principio del no descenso infinito. Principio del Buen Orden. +\end{enumerate} +\fi + +\section*{Conjuntos} + +\begin{exercise}(Tema 4) + Demuestra lo siguiente: + \begin{tasks} + \task $\bigcup_\alpha$ distribuye sobre $\cap$ y $\bigcup_\alpha$ distribuye sobre $\cup$, + $$\bset{\bigcap_{\alpha \in I} A_\alpha} \cup \bset{\bigcap_{\beta \in J} B_\beta} = \bigcap \set{A_\alpha \cup B_\beta | (\alpha,\beta) \in I\times J}$$ + \task Si el complemento es tomado respecto a $X$, entonces + $$X\menos \bigcap \set{A_\alpha | \alpha \in I} = \bigcup \set{X\menos A_\alpha | \alpha \in I}$$ + \task $\bigcup_\alpha$ y $\bigcap_\alpha$ distribuyen sobre el producto cartesiano + $$\bset{\bigcap_{\alpha \in I} A_\alpha} \times \bset{\bigcap_{\beta \in J} B_\beta} = \bigcap \set{A_\alpha \times B_\beta | (\alpha, \beta)\in I \times J}$$ + \end{tasks} +\end{exercise} + +\begin{exercise}(Tema 5) +Demuestra que toda relación de equivalencia $E$ sobre un conjunto $A$ determina una partición del conjunto. Construye una biyección entre el conjunto de todas las relaciones de equivalencia sobre $A$ y el conjunto de todas las particiones de este. +\end{exercise} + +\begin{exercise}(Tema 6) + Demuestra que el único isomorfismo de un conjunto bien ordenado en sí mismos es la identidad. +\end{exercise} + +\section*{Lógica} +\iffalse +\begin{enumerate} + \item[2.1] Forma l\'{o}gica de un enunciado. + + \item[2.2] Simbolizaci\'{o}n de enunciados simples. Letras proposicionales y conectivos l\'{o}gicos. Sinonimia de conectivos: + \begin{enumerate} + \item A implica B; si A entonces B; B, si A; A s\'{o}lo si B; A es suficiente para B; B es necesaria para A. + \item A o B; A a menos que B; A o B o ambos. + \item A y B; A pero B; ambos: A y B. + \item no A; no es el caso que A; A, no. + \end{enumerate} + + \item[2.3] Simbolizaci\'{o}n de argumentos simples. + + \item[2.4] Simbolizaci\'{o}n de enunciados tomando en consideraci\'{o}n la estructura sujeto-predicado: predicados, constantes, variables y cuantificaciones: + \begin{enumerate} + \item Existe \dots / hay un \dots / para alg\'{u}n \dots / hay al menos un \dots / + \item Para todos \dots / para cada uno \dots / para cualquiera \dots /. + \end{enumerate} + Ejemplos y m\'{a}s ejemplos de traducciones de enunciados (de contenido matem\'{a}tico y de contenido no matem\'{a}tico). + + \item[2.5] Criterios de verdad: criterios de verdad de conectivos, cuantificadores e igualdad y analizar a partir de ellos la verdad de cualquier enunciado m\'{a}s complejo. + + \item[2.6] Equivalencias l\'{o}gicas elementales. Negaci\'{o}n de una conjunci\'{o}n, de una disyunci\'{o}n, de una implicaci\'{o}n, de un bicondicional. Negaci\'{o}n de cuantificadores universales y existenciales. Rec\'{i}proca y contrapuesta de una implicaci\'{o}n. Leyes distributivas. Optativo: uso de reglas de instanciaci\'{o}n y generalizaci\'{o}n, universal y existencial. +\end{enumerate} +\fi + + + +\end{document} \ No newline at end of file diff --git a/2026-1/examen_extraordinario_2026-1.xsim b/2026-1/examen_extraordinario_2026-1.xsim new file mode 100644 index 0000000000..fe29a6197e --- /dev/null +++ b/2026-1/examen_extraordinario_2026-1.xsim @@ -0,0 +1,42 @@ +\providecommand\numberofexercises{} +\XSIM{solution-body}{} +\XSIM{exercise-body}{exercise-1=={(Tema 4) Demuestra lo siguiente: \begin {tasks} \task $\bigcup _\alpha $ distribuye sobre $\cap $ y $\bigcup _\alpha $ distribuye sobre $\cup $, $$\bset {\bigcap _{\alpha \in I} A_\alpha } \cup \bset {\bigcap _{\beta \in J} B_\beta } = \bigcap \set {A_\alpha \cup B_\beta | (\alpha ,\beta ) \in I\times J}$$ \task Si el complemento es tomado respecto a $X$, entonces $$X\menos \bigcap \set {A_\alpha | \alpha \in I} = \bigcup \set {X\menos A_\alpha | \alpha \in I}$$ \task $\bigcup _\alpha $ y $\bigcap _\alpha $ distribuyen sobre el producto cartesiano $$\bset {\bigcap _{\alpha \in I} A_\alpha } \times \bset {\bigcap _{\beta \in J} B_\beta } = \bigcap \set {A_\alpha \times B_\beta | (\alpha , \beta )\in I \times J}$$ \end {tasks}}||exercise-2=={(Tema 5) Demuestra que toda relación de equivalencia $E$ sobre un conjunto $A$ determina una partición del conjunto. 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+\XSIM{id}{exercise-1=={1}||exercise-2=={2}||exercise-3=={3}} +\XSIM{ID}{exercise-1=={1}||exercise-2=={2}||exercise-3=={3}} +\XSIM{counter}{exercise-1=={1}||exercise-2=={2}||exercise-3=={3}} +\XSIM{counter-value}{exercise-1=={1}||exercise-2=={2}||exercise-3=={3}} +\XSIM{solution}{} +\XSIM{chapter-value}{exercise-1=={0}||exercise-2=={0}||exercise-3=={0}} +\XSIM{chapter}{exercise-1=={0}||exercise-2=={0}||exercise-3=={0}} +\XSIM{section-value}{exercise-1=={0}||exercise-2=={0}||exercise-3=={0}} +\XSIM{section}{exercise-1=={0.0}||exercise-2=={0.0}||exercise-3=={0.0}} +\XSIM{sectioning}{exercise-1=={{0}{0}{0}{0}{0}}||exercise-2=={{0}{0}{0}{0}{0}}||exercise-3=={{0}{0}{0}{0}{0}}} +\XSIM{subtitle}{} +\XSIM{points}{} +\XSIM{bonus-points}{} +\XSIM{page-value}{exercise-1=={1}||exercise-2=={1}||exercise-3=={1}} +\XSIM{page}{exercise-1=={1}||exercise-2=={1}||exercise-3=={1}} +\XSIM{tags}{} +\XSIM{topics}{} +\XSIM{userpoints}{} +\XSIM{bodypoints}{} +\XSIM{userbonus-points}{} +\XSIM{bodybonus-points}{} diff --git a/2026-1/pro_arbi.aux b/2026-1/pro_arbi.aux new file mode 100644 index 0000000000..8e217158fd --- /dev/null +++ b/2026-1/pro_arbi.aux @@ -0,0 +1,11 @@ +\relax +\providecommand*{\memsetcounter}[2]{} +\providecommand \babel@aux [2]{\global \let \babel@toc \@gobbletwo } +\@nameuse{bbl@beforestart} +\catcode `"\active +\providecommand\XSIM[1]{} +\babel@aux{spanish}{} +\XSIM{readaux} +\memsetcounter{lastsheet}{2} +\memsetcounter{lastpage}{2} +\gdef \@abspage@last{2} diff --git a/2026-1/pro_arbi.fdb_latexmk b/2026-1/pro_arbi.fdb_latexmk new file mode 100644 index 0000000000..97ce8f496e --- /dev/null +++ b/2026-1/pro_arbi.fdb_latexmk @@ -0,0 +1,160 @@ +# Fdb version 4 +["xdvipdfmx"] 1758911827.34546 "pro_arbi.xdv" "pro_arbi.pdf" "pro_arbi" 1758911827.60301 0 + "pro_arbi.xdv" 1758911827.28494 26072 d869d6fb1c82da023656414debadbdb6 "xelatex" + (generated) + "pro_arbi.pdf" + (rewritten before read) +["xelatex"] 1758911826.2839 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Buendía y José Dosal} +\materia{Conjuntos y lógica} +\alum{} + +\begin{document} +\HojaExamen{}{e} +\begin{tasks} + +\task El producto cartesiano de la familia de conjuntos $\al = \set{A_i}_{i \in \set{1,2,3}}$ + se denota como $\prod_{i=1}^{3} A_i$ donde $A_1=\set{a, b}$,\; + $A_2=\set{x,y,z}$ y $A_3=\set{Pepe}$. + + Es fácil ver que el producto es el conjunto: + \[ \prod_{i=1}^{3} A_i = + \set{ \begin{aligned} + &(a,x,\text{Pepe}),\; (a,y,\text{Pepe}),\; (a,z,\text{Pepe}), \\ + &(b,x,\text{Pepe}),\; (b,y,\text{Pepe}),\; (b,z,\text{Pepe}) + \end{aligned} } + \] + + Pero cada uno de estos elementos es una función. Dicho de otro modo, el producto calcula todas las funciones cuyo dominio es el conjunto de índices $I$ y cuyo codominio es la unión de los $A_i$. Por ejemplo, la tupla $(a, x, \text{Pepe})$ representa a la función $f_1$ tal que $f_1(1)=a$, $f_1(2)=x$ y $f_1(3)=\text{Pepe}$. + + El producto cartesiano es el conjunto que contiene exactamente estas seis funciones, que podemos describir formalmente como conjuntos de pares ordenados: + \begin{gather*} + f_1=\set{(1,a),(2,x),(3,\text{Pepe})}\\ + f_2=\set{(1,a),(2,y),(3,\text{Pepe})}\\ + f_3=\set{(1,a),(2,z),(3,\text{Pepe})}\\ + f_4=\set{(1,b),(2,x),(3,\text{Pepe})}\\ + f_5=\set{(1,b),(2,y),(3,\text{Pepe})}\\ + f_6=\set{(1,b),(2,z),(3,\text{Pepe})} + \end{gather*} + +\task Sea la familia de conjuntos $\mathcal{F} = \set{\set{1,2}, \set{a,b,c}, \set{\text{Pepe}}}$. + Si la función de indexación es la identidad $I: \mathcal{F} \to \mathcal{F}$ donde $I(A)=A$, + tenemos que nuestro dominio es la propia familia $\mathcal{F}$ y el codominio es la $\bigcup_{A \in \mathcal{F}}A$. + + Así, los elementos del producto $\prod_{A \in \mathcal{F}} A$ son funciones $g$ que deben cumplir lo siguiente: + \begin{itemize} + \item $g(\set{1,2})$ debe ser $1$ o $2$. + \item $g(\set{a,b,c})$ debe ser $a$, $b$ o $c$. + \item $g(\set{\text{Pepe}})$ debe ser $\text{Pepe}$. + \end{itemize} + + El producto cartesiano es, por tanto, el siguiente conjunto de $2 \times 3 \times 1 = 6$ funciones: + \begin{gather*} + g_1 = \set{(\set{1,2}, 1), (\set{a,b,c}, a), (\set{\text{Pepe}}, \text{Pepe})}\\ + g_2 = \set{(\set{1,2}, 1), (\set{a,b,c}, b), (\set{\text{Pepe}}, \text{Pepe})}\\ + g_3 = \set{(\set{1,2}, 1), (\set{a,b,c}, c), (\set{\text{Pepe}}, \text{Pepe})}\\ + g_4 = \set{(\set{1,2}, 2), (\set{a,b,c}, a), (\set{\text{Pepe}}, \text{Pepe})}\\ + g_5 = \set{(\set{1,2}, 2), (\set{a,b,c}, b), (\set{\text{Pepe}}, \text{Pepe})}\\ + g_6 = \set{(\set{1,2}, 2), (\set{a,b,c}, c), (\set{\text{Pepe}}, \text{Pepe})} + \end{gather*} + +\end{tasks} +\end{document} \ No newline at end of file diff --git a/2026-1/pro_arbi.xdv b/2026-1/pro_arbi.xdv new file mode 100644 index 0000000000..4f575555b7 Binary 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+\newcommand\SetSymbol[1][]{% +\nonscript\:#1\vert +\allowbreak +\nonscript\: +\mathopen{}} +\DeclarePairedDelimiterX\set[1]{\{}{\}}{% +\renewcommand\st{\SetSymbol[\delimsize]}#1} +\DeclarePairedDelimiterX\bset[1]{\lbrack}{\rbrack}{#1} + +\examen{Tarea 1} +\prof{Karina G. Buendía y José Dosal} +\materia{Conjuntos y lógica} +\alum{} + +\xsimsetup{solution/print = true} + + +\begin{document} +\HojaExamen{}{e} +\begin{exercise} + Demuestra que el conjunto de todos los $x$ tales que $x \in A$ y $x \notin B$ existe y que es único. + +\end{exercise} + +\begin{solution} +Por el axioma de comprensión tenemos que $P(x):= x \notin B$. Para cualquier conjunto $A$ hay un conjunto $B$, +tal que si $x\in B \Leftrightarrow$ $x \in A$ y $P(x)$. Construimos el conjunto $C = \{x \in A| x\notin B\}$. +Supongamos que existe un $C'=\{x \in A| x\notin B\}$. Por el axioma de extensión, debemos mostrar que +$C \subseteq C'$ y $C' \subseteq C$. Sea $x \in C$ si y solo si $x \in A$ y $x\notin B$ +pero esta propiedad es la que cumplen los elementos de $C'$. Por lo tanto $C = C'$. +\end{solution} + +\begin{exercise} + Demuestre que para cualquier conjunto $X$ hay algún $a\notin X$. +\end{exercise} + +\begin{solution} +La prueba es por contradicción. Supongamos que existe, para todo $a$ elemento $a \in X$ (lo contiene todo). +Por el axioma de comprensión, tenemos un subconjunto de $X$, digamos $A = \{a | P(x)\}$ donde $P(x):= x\notin x$ +Este conjunto es igual al de la paradoja de Russel. Cuando nos preguntamos si $A \in A$ nos lleva a una contradicción. +Por lo que suponer que existe tal conjunto no puede suceder. +\end{solution} + +\begin{exercise} + Demuestre que $A\subseteq \{A\}$ si y solo si $ A = \varnothing$. +\end{exercise} + +\begin{solution} + Para la implicación $A = \varnothing$, entonces $A\subseteq \{A\}$. Es trivial. Para la dirección contraria + tenemos que demostrarlo por contradicción. Supongamos que $A\subseteq \{A\}$ y $A \neq \varnothing$, entonces + existe $x \in A$ tal que $X \neq A$. Como $\{A\}$ tiene como único elemento el conjunto $A$ y $X \neq A$, entonces + como $x \in A$ y $x \notin \set{A}$. Por lo tanto $A \nsubseteq \set{A}$. +\end{solution} + +\begin{exercise} + Demuestre que si $A \subseteq B$, entonces $P(A) \subseteq P(B)$ +\end{exercise} + +\begin{solution} + Sea $X \in P(A)$, entonces $X \subseteq A$ y por hipótesis $A \subseteq B$ tenemos que $X \subseteq B$. Por lo tanto, + $X \in P(B)$. +\end{solution} + +\begin{exercise} + Demuestre que $A \subseteq C$ si y solo si $A \cup (B \cap C) = (A \cup B) \cap C$ +\end{exercise} + +\begin{solution} + Para la demostras la idea tenemos que demostral la doble contención. Primero demostramos que $A \cup (B \cap C) \subseteq (A \cup B) \cap C$. + Sea $x \in A \cup (B \cap C)$, entonces $x \in A$ o $x \in (B \cap C)$. Caso (1) supongamos que $x \in A$ y por hipótesis + $A \subseteq C$, por lo que $x \in C$. Ademmás, como $x \in A$, entonces $X \in A \cup B$. Por lo tanto $ x \in (A \cup B) + \cap C$. La otra contención se demuestra de manera similar. + Para el regreso. Sea $x \in A$, entonces $x \in A \cup (B \cap C) = (A \cup B) \cap C$. Así, $x \in (A \cup B)$ + y $x \in C$. +\end{solution} + +\begin{exercise} + Si $E$ es un conjunto que contiene a $A \cup B$, entonces: + \begin{tasks} + \task $E \menos (E \menos A) = A$ + \task $E \menos \varnothing = E, E \menos E= \varnothing$. + \end{tasks} +\end{exercise} + +\begin{solution} + \begin{tasks} + \task Demostramos la doble contención. Por demostrar que $E \menos (E \menos A) \subseteq A$. Sea + $x \in E \menos (E \menos A)$, entonces $x \in E$ y $x \notin (E \menos A)$. Lo anterior nos dice que + $x \in E$ y $x \notin E\; \text{o}\; x \in A$. Por tanto $x \in A$. + Para el regreso tenemos que demostrar que $x \in E \menos (E \menos A)$. Sea $x \in A$, entonces $x \notin E$ + o $x \in A$. Así $x \notin E\menos A$. Además, $A \subset A \cup B \subset E$ por hipótesis, entonces $x \in E$. + Por lo tanto $x \in E \menos (E\menos A)$. + \task + \begin{enumerate} + \item Por demostrar que $E \menos \varnothing = E$. Primero demostraremos que $E \menos \varnothing \subseteq E$, + entonces $x \in E$ y $x \notin \varnothing$. Por lo tanto $x \in E$. Tenemos que demostrar que $E \subseteq E\menos \varnothing$, + pero esta contención es trivial. + \item Por demostrar que $ E \menos E= \varnothing$. Sea $x \in E\menos E$, $x \in E$ y $x \notin E$. No hay elementos + que cumplan lo anterior. Por lo tanto es vacio. + \end{enumerate} + \end{tasks} + +\end{solution} + +\begin{exercise} + Para todo conjunto $A$, $B$ y $C$ se cumple lo siguiente: + \begin{tasks} + \task $A \triangle \varnothing = A$ + \task $A \triangle A = \varnothing$ + \task Si $A \triangle B = A \triangle C$, entonces $B = C$ + \end{tasks} +\end{exercise} + +\begin{exercise} + Sea $F$ una familia de conjuntos. Pruebe que $\bigcup F = \varnothing$ si y solo si $F = \varnothing$ + o $A \in F$ implica $A = \varnothing$. +\end{exercise} + +\begin{exercise} + Demuestre que la unión y la intersección generalizada satisface la siguiente forma de asociación: + + \begin{tasks} + \task $\bigcup \set{ A_\alpha | \alpha \in \bigcup I} = \bigcup_{I \in I} (\bigcup_{\alpha \in I}A_\alpha)$ + \task $\bigcap \set{ A_\alpha | \alpha \in \bigcap I} = \bigcap_{I \in I} (\bigcap_{\alpha \in I}A_\alpha)$ + \end{tasks} +\end{exercise} + +\begin{exercise} + Demuestra lo siguiente: + \begin{tasks} + \task $\bigcup_\alpha$ distribuye sobre $\cap$ y $\bigcup_\alpha$ distribuye sobre $\cup$, + $$\bset{\bigcap_{\alpha \in I} A_\alpha} \cup \bset{\bigcap_{\beta \in J} B_\beta} = \bigcap \set{A_\alpha \cup B_\beta | (\alpha,\beta) \in I\times J}$$ + \task Si el complemento es tomado respecto a $X$, entonces + $$X\menos \bigcap \set{A_\alpha | \alpha \in I} = \bigcup \set{X\menos A_\alpha | \alpha \in I}$$ + \task $\bigcup_\alpha$ y $\bigcap_\alpha$ distribuyen sobre el producto cartesiano + $$\bset{\bigcap_{\alpha \in I} A_\alpha} \times \bset{\bigcap_{\beta \in J} B_\beta} = \bigcap \set{A_\alpha \times B_\beta | (\alpha, \beta)\in I \times J}$$ + \end{tasks} +\end{exercise} + +\end{document} \ No newline at end of file diff --git a/2026-1/t1.xdv b/2026-1/t1.xdv new file mode 100644 index 0000000000..308ca9872f Binary files /dev/null and b/2026-1/t1.xdv differ diff --git a/2026-1/t1.xsim b/2026-1/t1.xsim new file mode 100644 index 0000000000..7683b9b97c --- /dev/null +++ b/2026-1/t1.xsim @@ -0,0 +1,42 @@ +\providecommand\numberofexercises{} +\XSIM{solution-body}{exercise-1=={Por el axioma de comprensión tenemos que $P(x):= x \notin B$. Para cualquier conjunto $A$ hay un conjunto $B$, tal que si $x\in B \Leftrightarrow $ $x \in A$ y $P(x)$. Construimos el conjunto $C = \{x \in A| x\notin B\}$. Supongamos que existe un $C'=\{x \in A| x\notin B\}$. Por el axioma de extensión, debemos mostrar que $C \subseteq C'$ y $C' \subseteq C$. Sea $x \in C$ si y solo si $x \in A$ y $x\notin B$ pero esta propiedad es la que cumplen los elementos de $C'$. Por lo tanto $C = C'$.}||exercise-2=={La prueba es por contradicción. Supongamos que existe, para todo $a$ elemento $a \in X$ (lo contiene todo). Por el axioma de comprensión, tenemos un subconjunto de $X$, digamos $A = \{a | P(x)\}$ donde $P(x):= x\notin x$ Este conjunto es igual al de la paradoja de Russel. Cuando nos preguntamos si $A \in A$ nos lleva a una contradicción. Por lo que suponer que existe tal conjunto no puede suceder.}||exercise-3=={Para la implicación $A = \varnothing $, entonces $A\subseteq \{A\}$. Es trivial. Para la dirección contraria tenemos que demostrarlo por contradicción. Supongamos que $A\subseteq \{A\}$ y $A \neq \varnothing $, entonces existe $x \in A$ tal que $X \neq A$. Como $\{A\}$ tiene como único elemento el conjunto $A$ y $X \neq A$, entonces como $x \in A$ y $x \notin \set {A}$. Por lo tanto $A \nsubseteq \set {A}$.}||exercise-4=={Sea $X \in P(A)$, entonces $X \subseteq A$ y por hipótesis $A \subseteq B$ tenemos que $X \subseteq B$. Por lo tanto, $X \in P(B)$.}||exercise-5=={Para la demostras la idea tenemos que demostral la doble contención. Primero demostramos que $A \cup (B \cap C) \subseteq (A \cup B) \cap C$. Sea $x \in A \cup (B \cap C)$, entonces $x \in A$ o $x \in (B \cap C)$. Caso (1) supongamos que $x \in A$ y por hipótesis $A \subseteq C$, por lo que $x \in C$. Ademmás, como $x \in A$, entonces $X \in A \cup B$. Por lo tanto $ x \in (A \cup B) \cap C$. La otra contención se demuestra de manera similar. Para el regreso. Sea $x \in A$, entonces $x \in A \cup (B \cap C) = (A \cup B) \cap C$. Así, $x \in (A \cup B)$ y $x \in C$.}||exercise-6=={\begin {tasks} \task Demostramos la doble contención. Por demostrar que $E \menos (E \menos A) \subseteq A$. Sea $x \in E \menos (E \menos A)$, entonces $x \in E$ y $x \notin (E \menos A)$. Lo anterior nos dice que $x \in E$ y $x \notin E\; \text {o}\; x \in A$. Por tanto $x \in A$. Para el regreso tenemos que demostrar que $x \in E \menos (E \menos A)$. Sea $x \in A$, entonces $x \notin E$ o $x \in A$. Así $x \notin E\menos A$. Además, $A \subset A \cup B \subset E$ por hipótesis, entonces $x \in E$. Por lo tanto $x \in E \menos (E\menos A)$. \task \begin {enumerate} \item Por demostrar que $E \menos \varnothing = E$. Primero demostraremos que $E \menos \varnothing \subseteq E$, entonces $x \in E$ y $x \notin \varnothing $. Por lo tanto $x \in E$. Tenemos que demostrar que $E \subseteq E\menos \varnothing $, pero esta contención es trivial. \item Por demostrar que $ E \menos E= \varnothing $. Sea $x \in E\menos E$, $x \in E$ y $x \notin E$. No hay elementos que cumplan lo anterior. Por lo tanto es vacio. \end {enumerate} \end {tasks} \par }} +\XSIM{exercise-body}{exercise-1=={Demuestra que el conjunto de todos los $x$ tales que $x \in A$ y $x \notin B$ existe y que es único. \par }||exercise-2=={Demuestre que para cualquier conjunto $X$ hay algún $a\notin X$.}||exercise-3=={Demuestre que $A\subseteq \{A\}$ si y solo si $ A = \varnothing $.}||exercise-4=={Demuestre que si $A \subseteq B$, entonces $P(A) \subseteq P(B)$}||exercise-5=={Demuestre que $A \subseteq C$ si y solo si $A \cup (B \cap C) = (A \cup B) \cap C$}||exercise-6=={Si $E$ es un conjunto que contiene a $A \cup B$, entonces: \begin {tasks} \task $E \menos (E \menos A) = A$ \task $E \menos \varnothing = E, E \menos E= \varnothing $. \end {tasks}}||exercise-7=={Para todo conjunto $A$, $B$ y $C$ se cumple lo siguiente: \begin {tasks} \task $A \triangle \varnothing = A$ \task $A \triangle A = \varnothing $ \task Si $A \triangle B = A \triangle C$, entonces $B = C$ \end {tasks}}||exercise-8=={Sea $F$ una familia de conjuntos. Pruebe que $\bigcup F = \varnothing $ si y solo si $F = \varnothing $ o $A \in F$ implica $A = \varnothing $.}||exercise-9=={Demuestre que la unión y la intersección generalizada satisface la siguiente forma de asociación: \par \begin {tasks} \task $\bigcup \set { A_\alpha | \alpha \in \bigcup I} = \bigcup _{I \in I} (\bigcup _{\alpha \in I}A_\alpha )$ \task $\bigcap \set { A_\alpha | \alpha \in \bigcap I} = \bigcap _{I \in I} (\bigcap _{\alpha \in I}A_\alpha )$ \end {tasks}}||exercise-10=={Demuestra lo siguiente: \begin {tasks} \task $\bigcup _\alpha $ distribuye sobre $\cap $ y $\bigcup _\alpha $ distribuye sobre $\cup $, $$\bset {\bigcap _{\alpha \in I} A_\alpha } \cup \bset {\bigcap _{\beta \in J} B_\beta } = \bigcap \set {A_\alpha \cup B_\beta | (\alpha ,\beta ) \in I\times J}$$ \task Si el complemento es tomado respecto a $X$, entonces $$X\menos \bigcap \set {A_\alpha | \alpha \in I} = \bigcup \set {X\menos A_\alpha | \alpha \in I}$$ \task $\bigcup _\alpha $ y $\bigcap _\alpha $ 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b/2026-1/t3.tex new file mode 100644 index 0000000000..70505f712f --- /dev/null +++ b/2026-1/t3.tex @@ -0,0 +1,244 @@ +\documentclass[fc]{tarea} + +\usepackage[style=mexican]{csquotes} + +\newcommand{\al}{\symscr{A}} +\newcommand{\lan}{\symscr{L}} +\newcommand{\sis}{\symup{SF}} +\newcommand{\expresiones}{\symup{Exp}} +\NewDocumentCommand{\expr}{O{\al}}{\expresiones(#1)} + +%conjuntos +\providecommand\st{} +\newcommand\SetSymbol[1][]{% +\nonscript\:#1\vert +\allowbreak +\nonscript\: +\mathopen{}} +\DeclarePairedDelimiterX\set[1]{\{}{\}}{% +\renewcommand\st{\SetSymbol[\delimsize]} +#1 +} + +\examen{Tarea-Examen 1} +\prof{Karina G. Buendía y José Dosal} +\materia{Conjuntos y lógica} +\alum{} + +%\xsimsetup{solution/print = true} + +\begin{document} +\HojaExamen{}{e} +\chapter{Sistemas formales} + +\begin{exercise} + Demuestra que si \(\Gamma\vdash\alpha\) entonces existe un subconjunto finito + \(\Delta\subseteq\Gamma\) tal que \(\Delta\vdash\alpha\). +\end{exercise} +\begin{solution} + Si \(\Gamma\vdash\alpha\) entonces existe una sucesión finita de fórmulas \(\alpha_1,\ldots,\alpha_n\) que satisfacen las condiciones de la definición~\ref{def:ded}. Consideramos + \(\Delta=\Gamma\cap\{\alpha_1,\ldots,\alpha_n\}\). Notamos que \(\Delta\) es un conjunto finito y que \(\Delta\vdash\alpha\). Lo último es porque la misma lista finita \(\alpha_1,\ldots,\alpha_n\) es una deducción de \(\alpha\) a partir de \(\Delta\). +\end{solution} + +\begin{exercise} + Demuestra que la cerradura deductiva es un operador de cerradura, es decir, + \begin{tasks} + \task \(\Gamma\subseteq\overline{\Gamma}\), + \task si \(\Delta\subseteq\Gamma\) entonces + \(\overline{\Delta}\subseteq\overline{\Gamma}\), + \task \(\overline{\overline{\Gamma}}=\overline{\Gamma}\) + \end{tasks} +\end{exercise} + +\begin{exercise} + Considera el alfabeto \(\al=\set{a,b,\circ}\), la definición de fórmula dada por \(\Phi=\set{\alpha\in\expr\st\alpha\text{ empieza con }a}\) y la regla de inferencia + \[ + R: + \begin{array}{c} + A,B\\ + \midrule + A\circ B + \end{array}. + \] + Con esto definimos el lenguaje \(\lan=(\al,\Phi)\) y al sistema formal \(\sis=(\lan,\set{R})\). Sea + \(T=\set{\alpha\st\alpha\text{ no tiene ocurrencias de }\circ}\). + \begin{tasks} + \task Demuestra que \(T\) es correcta y no completa respecto a \enquote{no tener \(b\) después de cada \(\circ\)}. + \task Muestra que \(T\) es completa y no correcta respecto a \enquote{tener \(aa\) después de cada \(\circ\)}. + \end{tasks} +\end{exercise} + +\begin{exercise} + Considera el sistema formal dado por lo siguiente: el alfabeto es + \(\al=\set{a,b,c}\), el conjunto de fórmulas es + \(\Phi=\set{\alpha\in\expr\st\alpha\text{ empieza con } a}\) con las reglas de inferencia siguientes + \[ + R_1: + \begin{array}{c} + axb\\ + \midrule + axbc + \end{array} + \quad + R_2: + \begin{array}{c} + ax\\ + \midrule + axx + \end{array} + \quad + R_3: + \begin{array}{c} + axbbby\\ + \midrule + axcy + \end{array} + \quad + R_4: + \begin{array}{c} + axccy\\ + \midrule + axy + \end{array}, + \] + donde \(x\) y \(y\) representan sucesiones finitas de símbolos de \(\al\). Considera \(\Gamma=\set{ab}\) y demuestra lo siguiente: + \begin{tasks}(2) + \task \(\Gamma\vdash abcc\), + \task \(\Gamma\vdash acbbc\), + \task ¿es posible obtener \(\Gamma\vdash ac\)? + \end{tasks} +\end{exercise} + +\iffalse +\begin{exercise} + Usa el hecho de que \(2 = \{{0, 1}\}\) para demostrar que si \(\alpha\) y \(\beta\) son proposiciones tales que \(e(\alpha)=1\) si y solo si \(e(\beta)=1\) para toda evaluación \(e: From \to 2\), entonces \(\alpha\equiv \beta\). +\end{exercise} + + +\begin{exercise} +Sea \(\mathbb{P}\) un conjunto de letras proposicionales. Consideramos el conjunto de todas las posibles evaluacione \(e: From \to 2\) una función. Demuestra que toda proposición \(e\) induce una partición en dos pedazos del conjunto \(2^]{\mathbb{P}}\) . +\end{exercise} +\fi + +\chapter{Lógica Proposicional} + +\begin{exercise} Demuestra que \(\{\neg,\;\land\}\) es un conjunto mínimo de conectivos, es decir, + que el resto de conectivos se pueden definir en términos de ellos dos. +\end{exercise} + +\begin{exercise} + Demuestra que \(\{\neg,\;\iff\}\) y \(\{\lor,\;\land\}\) no son conjuntos mínimos de conectivos, es decir, + hay al menos un conectivo que no se puede definir usando sólo los conectivos de cada conjunto. +\end{exercise} + +\begin{exercise} + Demuestra que las siguientes proposiciones son tautología. + \begin{tasks} + \task \(\alpha \rightarrow (\beta \rightarrow \alpha) \), + \task \(\alpha \rightarrow(\beta \rightarrow \gamma) \rightarrow [(\alpha \rightarrow \beta)\rightarrow(\alpha \rightarrow \gamma)] \), + \task \( (\neg \alpha \rightarrow \neg \beta) \rightarrow [(\neg \alpha \rightarrow \beta)\rightarrow \alpha] \) + \end{tasks} +\end{exercise} + + +\begin{exercise} + Usa el teorema de las \enquote{primas} para demostrar que de la deducción \(\set{\alpha,\;\neg \beta}\vdash (\neg \alpha \rightarrow \beta)\), con \(\gamma \equiv (\neg \alpha \rightarrow \beta)\), se puede demostrar que \(\set{\alpha',\;\beta'}\vdash \gamma'\). +\end{exercise} + +\begin{exercise} + Decimos que una teoría \(T\) es consistente si existe una fórmula \(\alpha\) tal que \(T\not \vdash \alpha\). Demuestra que el cálculo de proposiciones es consistente. +\end{exercise} + + +\begin{exercise} + Realiza las siguientes deducciones en el cálculo de proposiciones. + \begin{tasks}(1) + \task $\beta, \neg \beta \vdash \gamma$ + \task $\vdash \beta \rightarrow (\neg \gamma \rightarrow (\beta \rightarrow \gamma))$ + \task $\{\alpha \rightarrow \beta, \beta \rightarrow \gamma\} \vdash \alpha \rightarrow \gamma$ + \task \(\vdash \neg \alpha \rightarrow (\alpha \rightarrow \beta)\) + \end{tasks} +\end{exercise} + +\begin{solution} +Inciso a: +\smallskip + +\begin{tabular}{l|l|l} +No. & Fórmula & Justificación \\ +\hline +1 & $\neg \neg \beta$ & Premisa por el TD \\ +2 & $\neg \beta \rightarrow \neg \beta$ & Teorema ($\vdash P \rightarrow P$) \\ +3 & $(\neg \beta \rightarrow \neg \beta) \rightarrow ((\neg \beta \rightarrow \beta) \rightarrow \beta)$ & Axioma 3 \\ +4 & $(\neg \beta \rightarrow \beta) \rightarrow \beta$ & MP (2, 3) \\ +5 & $\neg \neg \beta \rightarrow (\neg \beta \rightarrow \neg \neg \beta)$ & Axioma 1 \\ +6 & $\neg \beta \rightarrow \neg \neg \beta$ & MP (1, 5) \\ +7 & $\beta$ & De (4) y (6) por MP \\ +\hline +8 & $\neg \neg \beta \rightarrow \beta$ & TD (1-7) +\end{tabular} + +\bigskip + +Inciso d (Ley de Contraposición): +\smallskip + +\begin{tabular}{l|l|l} +No. & Fórmula & Justificación \\ +\hline +1 & $\alpha \rightarrow \beta$ & Premisa \\ +2 & $\neg \beta$ & Premisa \\ +3 & $\neg \neg \alpha$ & Supuesto para Reducción \\ +4 & $\neg \neg \alpha \rightarrow \alpha$ & Elim. Doble Negación \\ +5 & $\alpha$ & MP (4, 3) \\ +6 & $\beta$ & MP (1, 5) \\ +7 & Contradicción & $\beta$ y $\neg \beta$ \\ +\hline +8 & $\neg \alpha$ & Reducción al Absurdo \\ +9 & $\neg \beta \rightarrow \neg \alpha$ & TD (2-8) \\ +10 & $(\alpha \rightarrow \beta) \rightarrow (\neg \beta \rightarrow \neg \alpha)$ & TD (1-9) +\end{tabular} + +\bigskip + +Inciso e (Silogismo Hipotético): +\smallskip + +\begin{tabular}{l|l|l} +No. & Fórmula & Justificación \\ +\hline +1 & $\alpha \rightarrow \beta$ & Premisa \\ +2 & $\beta \rightarrow \gamma$ & Premisa \\ +3 & $\alpha$ & Premisa adicional para TD \\ +4 & $\beta$ & MP (1, 3) \\ +5 & $\gamma$ & MP (2, 4) \\ +\hline +6 & $\alpha \rightarrow \gamma$ & TD (3-5) +\end{tabular} +\end{solution} + +\iffalse +\chapter{Ejercicios de conjuntos} + + \begin{exercise} + Demuestra las siguientes propiedades de los números naturales. + \begin{tasks} + \task Para todo $m,n,p \in \mathbb{N}$, + $$m={Classical TikZ Rightarrow[]}}} + +%Capítulos +\makechapterstyle{tareas}{% +\renewcommand{\chapterheadstart}{} +\renewcommand{\chapnamefont}{\Large\bfseries} +\renewcommand{\chapnumfont}{\Large\bfseries} +\renewcommand{\chaptitlefont}{\Large\bfseries} +\renewcommand{\printchaptername}{} +\renewcommand{\printchapternum}{\chapnumfont\thechapter} +\renewcommand{\afterchapternum}{ } +\renewcommand{\printchaptertitle}{\textcolor{gray}{\vrule}~\chaptitlefont} +\renewcommand{\afterchaptertitle}{\par\nobreak\vspace*{0.6\afterchapskip}} +} +\chapterstyle{tareas} + + + +%diseño IRC +\ifisirc +%tokens para hacer el diseño +\newtoks\materia +\newtoks\tarex +\newlength{\nombre} +\setlength{\nombre}{\textwidth-\widthof{Nombre del alumne:}} + +\newcommand{\info}{ + \begin{minipage}{0.4\linewidth} + \raggedright% + \includegraphics[width=0.9\linewidth]{logoIRC} + \end{minipage}% + \begin{minipage}{0.5\linewidth} + {\large\bfseries\sffamily \the\materia}\\[1mm] + {\scalebox{1.2}{\sffamily \the\tarex:\rule[-2pt]{1cm}{0.4pt}}}\\ + {\sffamily M. en F.C. 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b/CHANGELOG.md deleted file mode 100644 index 3fe8066156..0000000000 --- a/CHANGELOG.md +++ /dev/null @@ -1,1461 +0,0 @@ ---- -layout: plain -sitemap: false ---- - -# CHANGELOG -{:.no_toc} - -* this list will be replaced by the toc -{:toc .large-only} - -## v9.1.6 -Feb 07 2022 -{:.heading.post-date} - -* Removed unused clap button HTML tags -* Fixed an issue with using `featured` projects on the `welcome` layout -* Minor style adjustments -* Updated dependencies - -## v9.1.5 -Nov 30 2021 -{:.heading.post-date} - -* Now works with Ruby 3.0 -* Fixed theme color issue when using multiple theme colors -* Fixed UTF-8 characters not rendering correctly in breadcrumbs -* Fixed empty string warning in resume layout -* Removed clap button info box - -## v9.1.4 -Mar 30 2021 -{:.heading.post-date} - -* Fixed scroll position restoration in webkit. -* Fixed an issue that caused clap button to use canonical url incorrectly. - -## v9.1.3 -Mar 28 2021 -{:.heading.post-date} - -### Added -* Allow setting custom Google Fonts providers via `google_fonts_url` ([#264](https://github.com/hydecorp/hydejack/issues/264)). Defaults to `https://fonts.googleapis.com`. - -### Fixes -* Merged [#266](https://github.com/hydecorp/hydejack/pull/266) --- Fixed a bug that caused incorrect links to be rendered. -* Merged [#267](https://github.com/hydecorp/hydejack/pull/267) --- Fixed a bug that caused incorrect sub-titles when using multiple categories/tags. -* Fix [#262](https://github.com/hydecorp/hydejack/issues/262) --- Fixed a bug that causes warnings when not providing a skill level / fluency level in the resume. -* [PRO] Fixed a bug preventing scrolling when using sticky ToC with certain aspect ratios. - -### Design -* [PRO] Adjusted calculation to dark mode background color to prevent red tint of grey colors -* Hide underline in tooltips -* Show clap button on collection pages -* Show dingbat in `plain` layout - -## v9.1.2 -Feb 11 2021 -{:.heading.post-date} - -* Fix [#258](https://github.com/hydecorp/hydejack/issues/258) -* Fix [#259](https://github.com/hydecorp/hydejack/issues/259) - - -## v9.1.1 -Feb 9 2021 -{:.heading.post-date} - -* Added tooltips to post subtitles and last-modified-at lines. -* Added tooltips to abberavations (`` tags), e.g. IIAFE. See [Example Content](/blog/hyde/2012-02-07-example-content/#inline-html-elements){:.flip-title} on how to use them in markdown. -* Added tooltips to social media icons (inside main column only) -* Fixed KaTeX font rendering -* [PRO] Setting `clap_button: false` will hide the clap button preview, even in development -* [PRO] A overscrolling ToC will scroll itself to keep the active element in view. - -*[IIAFE]: Instantly Invoked Asynchronous Function Expression -*[ToC]: Table of Contents - -## v9.1.0 -Feb 5 2021 -{:.heading.post-date} - -Version 9.1 provides minor design changes, new features, and closes multiple issues: - -### Added - -* Code blocks now can have headers: - - ~~~js - // file: 'hello-world.js' - console.log('Hello World!'); - ~~~ - - To add the headers, simply make the first line a comment of the form `file: "dir/filename.ext"`. - - * Code blocks now have a copy-to-clipboard button - -* Resumes can now have download buttons. - - ![Download Buttons](/assets/img/blog/9.1.0-3.png){:.border.lead width="1776" height="258" loading="lazy"} - - Add the following to the front matter. Note that the PDF needs to be pre-generated. - See [the docs](docs/basics.md#downloads) for more. - - ```yml - # file: "resume.md" - buttons: - print: true - pdf: /assets/Resume.pdf - vcf: http://h2vx.com/vcf/ - json: /assets/resume.json - ``` - -* Added breadcrumbs above page title: - - ![Breadcrumbs](/assets/img/blog/9.1.0-2.png){:.border.lead width="1588" height="164" loading="lazy"} - - Note that this requires a [directory-like URL pattern](https://qwtel.com/posts/software/urls-are-directories/) like `/blog/:categories/:year-:month-:day-:title/` (default for Hydejack). - - Disable with `hydejack.no_breadcrumbs`. - -* Added "Last modified at" to post layout: - - ![Last modified at](/assets/img/blog/9.1.0-1.png){:.border.lead width="1254" height="218" loading="lazy"} - - To enable this feature, the post needs to have a `last_modified_at` property with a valid date. You can either set it manually in the frontmatter (not recommended), or use the [`jekyll-last-modified-at` plugin](https://github.com/gjtorikian/jekyll-last-modified-at) to set it for you (Not available on GitHub Pages!). - - You can remove this element by setting `hide_last_modified` in the front matter. You can disable it for all posts by setting `hydejack.hide_last_modified` in the config file. Setting `hydejack.hide_dates` (PRO version only) will also remove it, together with all other time-related UI elements. - - You can customize the hover text, icon, and date format in `_data/strings.yml` using the following keys: `last_modified_at` (hover text), `last_modified_at_icon` (icon name, default: `icon-history`) and `date_formats.last_modified_at` (date format, default: `%Y-%m-%d`). - -* Added option to "invert" / darken the font colors in the sidebar. This enables use of bright sidebar images. - Set `invert_sidebar: true` in the font matter to enable. Use `defaults` in the config file to enable this for all pages. - -* Added a demo of [Clap Button](https://getclaps.app/) during development. -* Added option to configure border radius -* Added dingbat to `page` layout -* Added `plain` layout that comes without a dingbat -* Added `smaller` and `larger` CSS classes that set the font size to the respective values. -* Added options to change the file paths to favicon and apple touch icon in the confog file. Use `favicon` and `apple_touch_icon` respectively. - -### Changed -* Added border radius to many elements -* Modernized table design -* [PRO] Setting `hydejack.advertise: false` will now remove the banner from the HTML and the JavaScript console. -* Changed the box shadow of cards (projects, posts) to reduce the amount of painting the browser has to do on when mouse hovering them. -* The layout when using the theme without the `no_break_layout` setting is now - - -### Fixes -* Allow transparent project and post images -* Removing/leaving out the `logo` key in the config file will now correctly remove the logo from the sidebar -* [PRO] Fixed a bug that caused blog posts to be included the the search even when set to `sitemap: false` in the front matter. - -## v9.0.5 -Sept 8 2020 -{:.heading.post-date} - -* Added GitHub Pages Starter Kit to PRO version -* Added chapter on how to deploy PRO on GitHub Pages -* JavaScript source files now included in PRO zip again -* Updated default config file -* Changed default code font -* Improved fallback image in dark mode - -## v9.0.4 -July 15 2020 -{:.heading.post-date} - -* Fixed image fade in animation for images with `srcset` -* Slightly increased size of post and project cards -* Added page margin to print layout -* Fixed KaTeX when JavaScript is disabled -* Fixed a layout bug in the `resume` layout when changing the content width in variable -* Fixed table of contents sticky breakpoint - -## v9.0.3 -July 9 2020 -{:.heading.post-date} - -* Updated print resume style -* Updated docs for GitHub Pages -* Slightly decreased size of dark mode icon -* Fixed a bug that caused a GitHub Pages build to fail with an empty configuration file -* Changed default icon so that it less resembles slashdot.org\~\~ - -## v9.0.2 -July 7 2020 -{:.heading.post-date} - -* Fixed a bug that prevented the search from updating when offline is enabled -* Fixed a bug that caused search terms to get lost during initialization -* Fixed a bug that prevented `site.legal` from getting stored for offline during service worker installation -* Added support to for `no-cache` param to service worker. - -## v9.0.1 -July 6 2020 -{:.heading.post-date} - -* Changing app icons has been revamped. See [this section](./docs/config.md#adding-custom-favicons-and-app-icons) in the docs. -* Changed default icons -* Changed default sidebar background -* Added `jekyll-compose` defaults to config file -* Slightly adjusted dark mode colors -* Fixed resume layout breakpoint - -## v9.0.5 -Sept 8 2020 -{:.heading.post-date} - -* Added GitHub Pages Starter Kit to PRO version -* Added chapter on how to deploy PRO on GitHub Pages -* JavaScript source files now included in PRO zip again -* Updated default config file -* Changed default code font -* Improved fallback image in dark mode - -## v9.0.4 -July 15 2020 -{:.heading.post-date} - -* Fixed image fade in animation for images with `srcset` -* Slightly increased size of post and project cards -* Added page margin to print layout -* Fixed KaTeX when JavaScript is disabled -* Fixed a layout bug in the `resume` layout when changing the content width in variable -* Fixed table of contents sticky breakpoint - -## v9.0.3 -July 9 2020 -{:.heading.post-date} - -* Updated print resume style -* Updated docs for GitHub Pages -* Slightly decreased size of dark mode icon -* Fixed a bug that caused a GitHub Pages build to fail with an empty configuration file -* Changed default icon so that it less resembles slashdot.org\~\~ - -## v9.0.2 -July 7 2020 -{:.heading.post-date} - -* Fixed a bug that prevented the search from updating when offline is enabled -* Fixed a bug that caused search terms to get lost during initialization -* Fixed a bug that prevented `site.legal` from getting stored for offline during service worker installation -* Added support to for `no-cache` param to service worker. - -## v9.0.1 -July 6 2020 -{:.heading.post-date} - -* Changing app icons has been revamped. See [this section](./docs/config.md#adding-custom-favicons-and-app-icons) in the docs. -* Changed default icons -* Changed default sidebar background -* Added `jekyll-compose` defaults to config file -* Slightly adjusted dark mode colors -* Fixed resume layout breakpoint - -## v9.0.0 -July 3 2020 -{:.heading.post-date} - -### Major -* Added Built-In Search Functionality - - Hydejack now has its own built-in search solution, that integrates well with the existing page style and the new navbar. - - The solution is entirely browser-based which means it even works while offline and doesn't depend on an 3rd party. - This works, because Hydejack is designed for personal sites that generally have less than 1000 pages. - In my testing, Jekyll build times have been a problem long before search query times. - - The results of the search are surprisingly good, but have only been tested in English and (somewhat) German. - For better language support, I might build an integration with Algolia at some point, which has the best results, - but requires stable internet connection and an API key. - I'd be interested to hear about problems with search in other languages to determine if this is necessary. - -* Added Table of Contents that is prettier, sticky, and dynamic. - - Adding a table of contents is part of kramdown and can be done in all versions of Hydejack. - However, v9 adds a dynamic version that will stick to the 3rd column on large screens and highlight the current section. - - Note that this will reduce the amount of space freed up by the `no_break_layout: false` setting (otherwise the ToC would overlap with code blocks, math blocks, etc). - -* Added a scroll-linked navbar that disappears when scrolling down and re-appears when scrolling up. - -* Math support has been revamped because the old solution stopped working with Jekyll 4. Hydejack now supports both KaTeX and MathJax. - - The MathJax implementation is more similar to the old solution. It comes with a client-side runtime (MathJax in this case) - and works on GitHub Pages. It is the more heavy-weight of the two and doesn't work without JavaScript enabled. - Due to the size of the complete MathJax package, it only works partially with offline support enabled. - - The KaTeX implementation has been changed to pre-render the KaTeX output during site building. - It _does not_ ship a client-side runtime, which is more lightweight and works _without_ JavaScript. - In my opinion, it is the more elegant solution, but it requires a JavaScript runtime on the machine that builds the site. - This means that this solution doesn't work on GitHub Pages. - - You can switch between the two implementations by changing the `kramdown.math_engine` key to either `katex` or `mathjax` in your config file. - The KaTeX implementation also requires the `kramdown-math-katex` gem in your `Gemfile`. - -* Drastically improved build times through the use of `jekyll-include-cache`. - Most of the previous tips on [improving page build speed](hydejack/_posts/2019-02-18-improving-site-build-speed.md) should now be obsolete. - -* Added `grid` layout in PRO version that mirrors the `projects` layout, but for posts. - If you've set `image`s for your posts, it will give your blog a more modern look. - -### Minor -* CSS variables are now configurable via `_data/variables.yml`. While there are other ways to change them, this has the broadest reach across HTML (`img[sizes]` attribute!), CSS and JS. -* Many JS content features (such as `#` heading links) now work even when `hydejack.no_push_state` is enabled. -* Linking to posts in the `home` and `post` layout now accept either paths or URLs. [Read more](docs/basics.md#adding-related-posts-to-a-post). -* The star icons in the resume layout can now be disabled via `no_skill_icons` and `no_language_icons`. -* Resume layout now supports the following keys: `born`, `citizenship`, and `maritalStatus`. -* PRO only: Added `hide_dates` option to disable showing blog post dates. -* The `blog` layout now works without pagination, simply showing all posts on a single page when the `jekyll-pagination` plugin is not found. -* The `projects`, `resume`, and `grid` layout now make better use of large screens, by letting content expand to the right of the screen. Restore the previous , set `no_break_columns` in front matter. -* Removed smooth scroll polyfill for Safari/Webkit as it has caused problems with sticky content. -* Figure captions can now be added to code blocks, math blocks, and tables in addition to just images using the `.figcaption` CSS class. -* The order of the comment section relative to the about and related posts sections can now be customized via the `hydejack.post_addons` and `hydejack.project_addons` keys. See the example `_config.yml` for more. -* Clicking the image in the `blog` layout will now navigate to the blog post. In the PRO version it will do so with the classic "move image in place" animation. -* Grouping projects by year can now be disabled. Use `no_groups: true` in the front matter. This also applies to the new `grid` layout and the old `list` layout. -* hy-img has been removed and replaced with browser's native `loading=lazy` attribute -* Now serving separate JavaScript files for old and new browsers, allowing it to take advantage of many new language features. -* JavaScript files are now chunked, so that only what is needed is loaded on demand. -* hy-drawer and hy-push-state have been rewritten in TypeScript and LitHTML, fixing many bugs in the process. -* Updated to Jekyll 4.1 -* Hydejack now has a dedicated offline page that will be shown when the client is offline and tries to open a page that hasn't been cached. - The content of the page can be customized by creating `offline.md` file in the root with `layout: offline`, similar to `404.md`. -* The code font can now be customized in the config file via the `font_code` key. - -### Design -* Many layouts (`projects`, `resume`, `home`) will now use more space on large screens (disable via `hydejack.no_third_column`) -* Headings now appear "oversized" on larger screens and extend to the right end of the screen (disable via `hydejack.no_large_headings`) -* The base font size is now smaller across screen sizes (can be configured in `_sass/_variables.scss`) -* The content width is now larger across screen sizes (can be configured in `_sass/_variables.scss`) -* The link style has been changed to make picking accent colors easier for dark mode. It's also possible to link images now without worrying about underlines. -* The amount of whitespace has been reduced. Previously it had been increased but it felt too much. -* Added a [`note` class](docs/writing.md#adding-notes) that succeeds the `message` for adding auxillary content. The `message` class still exists. -* The navbar buttons made to look more like buttons -* Dark mode colors have been toned down a bit -* Improved dark mode text rendering on macOS -* The permalink style has been changed to use "#" instead of an icon. -* Definition list now have a double colon after the definition term. -* Changed the style of footnote links on small screens to make them easier to click. - -### Fixes -* Horizontal scrolling on a code block, math black, or table can no longer accidentally open the drawer. -* The client-side scripts are now more robust to missing HTML elements. This should ease theme customization. -* Setting `sitemap: false` will now also add a `noindex` meta tag to prevent accidental search engine indexation. -* Fixed back button not showing when opening in standalone mode for the first time. -* The menu icon is now hidden when the drawer is disabled. -* Using `no_drawer: true` is now working properly on cover pages when `no_push_state: false` is set. -* Reduced the amount of pixels cached by the `will-change` optimization and fixed the corresponding warning in Firefox. - - -## v8.5.2 -August 31 2019 -{:.heading.post-date} - -* Fixed breaking builds due to name collision when upgrading to 8.5 - This is caused when using both `_plugins/jekyll-replace-imgs` and the new `jekyll-replace-img` ruby gem. - I recommend deleting `_plugins/jekyll-replace-imgs` when using the `jekyll-replace-img` gem. - Note that this is an optional plugin. -* Fixed `nomodule` script loading for Safari 10.1 -* Fixed [#176](https://github.com/hydecorp/hydejack/issues/176) - -## v8.5.1 -Aug 1 2019 -{:.heading.post-date} - -* Fixed minification bug - -## v8.5.0 -Aug 1 2019 -{:.heading.post-date} - -### Changed -* [PRO] The theme now matches the operating system's dark mode. -* Scrolling down on a page with an open drawer will now close the drawer -* Reloading a cover page after the drawer has been closed will now open the page with the drawer closed -* Removed JavaScript-based web font swapping in favor of using `font-display: swap` -* Darkened font color in dark mode to reduce contrast -* Updated KaTeX to the latest version -* Increased the durations of various animations slightly -* Now using [`jekyll-replace-img`](https://github.com/qwtel/jekyll-replace-img) instead of custom code in `_plugins` folder -* Added a separate, smaller JavaScript bundle for modern browsers - -### Added -* Added support for figure captions on code blocks -* Added `keybase` to social media icons -* [PRO] There's now a forward button when using the theme as a PWA - -### Fixed -* Fixed print layout when dark mode is enabled -* Using `CSSTransformValue` correctly -* Fixed a minor style bug for dates -* Fixed a bug where light mode would flash when loading a page in dark mode -* Minimal support for IE11 - -## v8.4.0 -Mar 9 2019 -{:.heading.post-date} - -* Added support for `noindex` property in the front matter -* Fixed ordering of selected projects/post in welcome layout -* Updated dependencies - -## v8.3.0 -Feb 18 2019 -{:.heading.post-date} - -This version adds new options to increase production build speed. Read [this post](hydejack/_posts/2019-02-18-improving-site-build-speed.md) for details. - -### Added -* Added `no_page_style` config option to increase site build speed. -* Added `menu` config option to increase site build speed. -* Copying math will now add the LaTeX source to the clipboard. - -### Changed -* Upgraded KaTeX to version 0.10.0. - -## v8.2.0 -Feb 1 2019 -{:.heading.post-date} - -* Added support for custom `related_posts` -* Removed footer from print layout -* Increased photo size in print resume -* Improved `welcome` layout generation performance -* Fixed a bug that prevented scrolling to headlines with non-ascii characters (Thanks [@ForelaxX](https://github.com/ForelaxX)) - - -## v8.1.1 -Sep 1 2018 -{:.heading.post-date} - -### Fixes -* Fixed an issue that prevented the drawer from working on iOS 10. -* Changing the page via push state will now also update the `link[rel=canonical]` tag. -* Changing the page via push state will now also update the `meta[name=description]` tag. -* Fixed an issue that prevented the JS from building on Netlify. - -## v8.1.0 -Aug 18 2018 -{:.heading.post-date} - -This release adds Dark Mode for Hydejack PRO customers. - -### Breaking -* Removed cookie banner from free version -* Removed offline support from free version - -In an attempt to make the PRO offering more appealing, I'm removing features that arguably should have never been included in the free version. -As software licenses go, nobody is stopping you from using the old code, but updates will no longer be included. - -### Changed -* The cookies banners is now showing at the bottom of the page and its background color is no longer transparent to increase visibility -* Changed the default syntax theme from "GitHub" to "Atom One Light" -* Adapted `figure` CSS class to accommodate different children, not just `img`s -* `video` tags now have a `max-width` of 100% -* Increased margin before headings to `5rem`, up from 4 -* Increased margin of `hr` elements. -* Cookies banner can now be enabled without using Google Analytics -* Clicking the cookie banner "Okay" button will now fire a `hy--cookies-ok` event on `document`, so that custom analytics solutions can plug in. -* All Google Analytics code has been removed from Hydejack's core and moved to `_includes/body/analytics.js`. -* All Disqus code has been removed from Hydejack's core and moved to `_includes/comments.html` and `_includes/my-comments.html`. -* Using CSS Custom Properties instead of SASS variables for certain properties to enable style customization using only CSS. -* Added shadow to sidebar -* Navbar is longer positioned `fixed` - -### Added -* [PRO] Added Dark Mode -* `border` CSS class - - -## v8.0.0 -Jul 16 2018 -{:.heading.post-date} - -So far Hydejack has been a decent Jekyll theme, but with v8 it really starts stand out among the competition: Beautiful and unique landing pages, lazy-loading images, and experimental offline support are just the most prominent new features. - -### Breaking -* The expected format for sidebar images has changed. - A sidebar image should now be a full-screen ~16:10 image. - - Comment: The sidebar can now be fully extended on desktop, which generally requires a large landscape image to fill the entire window. - To save bandwidth, you can blur the image on the left and right edges and save it as JPG. - -* The `about` and `welcome` layout no longer prepend the content with the author information. - Instead, the author info can be shown by adding the `` marker to the top of the file. You can also place it anywhere else. - - Comment: Showing the author description on the top of the `welcome` and `about` layouts felt like an imposition and was a left-over from when I was developing Hydejack primarily for myself. - -* [PRO] The `welcome` layout no longer adds recent posts and projects to the bottom of the page. Instead, they have to be explicitly set using the `` and `` markers. The `content_separator` front matter opton is now ignored. - - Comment: The old behavior felt arbitrary, and `` wasn't a good name to be replaced with recent projects ands posts. - -* Setting the accent color and sidebar image for an entire category/tag/author is no longer possible. - To achieve a similar effect, use [Front Matter defaults][ffd] instead. - - E.g. to set the accent color and image for every post in the `hydejack` folder, use: - - ~~~yml - defaults: - - scope: - path: hydejack - values: - accent_color: rgb(38,139,210) - accent_image: /assets/img/hydejack-bg.jpg - ~~~ - - Comment: The code to find the color for a given page was complicated and slow (potentially iterating all categories/tags to find the right one). - -### Changed - -* The drawer now responds to mouse inputs. -* The default heading font is now less bold. To restore the old behavior, edit (create if it's missing) `_sass/my-variables.scss` and add `$font-weight-heading: 700;`. -* Hydejack now uses lazy-loading hy-img tags instead of regular `img` tags. - To revert to using regular images, set `hydejack.no_img` in the config file to `true`. -* Cookie consent is now stored as a cookie (instead of `LocalStorage`) and expires after 1 year. -* Scrolling to a fragment link is now smooth. -* Font loading now works differently, and will be cancelled on slow connections. -* The sidebar content is now centered. -* The sidebar will now show the site's logo, which can be set in the config file under the `logo` key. -* [PRO] Updated embedded Bootstrap to v4. -* [PRO] Project cards now throw a shadow instead of having a border. - -### Added -* Pages can now have the `cover` key in the front matter. - When set to `true`, the sidebar will be opened when visiting the page directly. - E.g. {:.no-push-state} - -* Added a `_plugin` that automatically replaces `` tags with lazy-loading `` tags. If you don't want images to load lazily, delete or rename the `_plugins` folder. - Note that this plugin will never run when building the site on GitHub Pages. - - To get the most out of this plugin, it is recommended to provide the width and height of the image, e.g. - - ~~~md - ![Some image](assets/img/some-img.png){:width="800" height="600"} - ~~~ - - This will cause hy-img to render a placeholder of 800 by 600 `px`, preventing the document height from changing after the image has finished loading. - -* Added experimental offline support via Service Workers. Use with care! - For details, [read the docs](docs/advanced.md#enabling-offline-support). - -* Added the `figure` CSS class, which allows images to have nicer-looking captions. E.g. - - ~~~md - ![An image with a caption](https://via.placeholder.com/800x50){:.lead} - A caption to an image. - {:.figure} - ~~~ - -* Clicking on a footnote will give its corresponding text a subtle highlight. - -* [PRO] Projects can now have an optional `end_date` field in the front matter. - The `date` is treated as the start date in this case. - -### Fixes -* The back button now works in combination with fragment links. - -[ffd]: https://jekyllrb.com/docs/configuration/#front-matter-defaults - -## v7.5.1 -Apr 2 2018 -{:.heading.post-date} - -### Changed -* Moved from browserify to webpack -* Updated ruby dependencies -* Updated JS dependencies -* Updated hy-push-state and hy-drawer to latest versions - -## v7.5.0 -Dec 18 2017 -{:.heading.post-date} - -### Added -* Added secondary `legal` nav in footer: - - ```yml - # file: `_config.yml` - legal: - - title: Cookies Policy - href: /cookies-policy/ - - title: Foobar - href: https://foobar.com/ - ``` - -* The "heading permalink" can now be configured via `strings.yml`: - - ```yml - # file: `_data/strings.yml` - permalink: Permalink - permalink_icon: icon-link - ``` - -* Sections on resume layout can now be rearranged, e.g.: - - ```yml - # file: `resume.md` - left_column: - - work - - volunteer - - education - - awards - - publications - - references - right_column: - - languages - - skills - - interests - ``` - - -### Fixed -* Fixed a bug that caused `` tags to render as regular text ([#52](https://github.com/hydecorp/hydejack/pull/52)) -* Fixed a bug that caused Disqus to load the same thread on all pages ([#53](https://github.com/hydecorp/hydejack/pull/52)) -* Fixed a bug that prevented Disqus comments to be loaded on sites that didn't cause scroll events -* Fixed a bug that caused Disqus to be loaded over HTTP instead of HTTPS. -* Fixed a bug that caused an extra space in URLs ([#55](https://github.com/hydecorp/hydejack/pull/55)). -* Comments no longer show up in the print version of the page. - -### Other -* Set base font to `11pt` in print layout. -* Set resume print layout to use 2 columns (A4 sheet) - -## v7.4.2 -Dec 1 2017 -{:.heading.post-date} - -### Fixed -* Dramatically improved resume print layout. - It is now much less likely that there will be page breaks within logical units. -* Fixed a bug that cause the page to break when setting `no_drawer`. -* Fixed a bug that cased the "Random Posts" heading to appear, even when there are no posts to show - -## v7.4.1 -Nov 27 2017 -{:.heading.post-date} - -### Fixed -* Fixed storing user-related data before accepting cookies. -* Fixed tab order of cookie banner, so keyboard users can access it more easily. -* Accepting cookies no longer causes a page reload in some browsers. -* Fixed appearance of the okay button in the free version. -* Menu icon now useable while the cookies banner is active. -* Loading icon is now visible while the cookies banner is active. -* Removed cookies banner from print layout. -* Removed inline styles from cookie banner. - -## v7.4.0 -Nov 25 2017 -{:.heading.post-date} - -### Added -* Allow markdown in copyright string -* Added `theme_color` front-matter property to micro-manage the value of the the `theme-color` meta tag. - When not set, will use `accent_color`. -* Added `theme_color` site setting, to set the value of `themeColor` in the app manifest. - When not set, will use `accent_color`. -* Added `cookies_banner` setting: - - ~~~yml - # file: `_config.yml` - hydejack: - cookies_banner: true - ~~~ - - Enabling this setting will show a notice at the top of the page to new visitors. - You can change the wording of the notice in `_data/strings.yml` - with the `cookies_banner.text` and `cookies_banner.okay` keys: - - ~~~yml - # file: `_data/strings.yml` - cookies_banner: - text: This site uses cookies. - okay: Okay - ~~~ - -### Fixed -* Drawer no longer resizes/repaints in iOS Safari (iPhone) and Chrome for Android when the address bar autohides. -* Fixed a bug that caused the drawer flicker/open unexpectedly when scrolling in mobile browsers. -* Fixed how `image` works when using the `jekyll-seo-tag` plugin. - -### Design -* Changed how line breaks work in resume layout -* Changed margins of horizontals lines - -### Other -* Updated docs -* Updated posts - - -## v7.3.0 -Nov 17 2017 -{:.heading.post-date} - -### Added -* Allow markdown content on `projects` layout. -* Renamed `big_project` option on projects to `featured` (`big_project` still works) - -### Fixed -* Fixed default font weights -* Fixed hard-coded `/projects/` URL in project layout -* Link to feed.xml is only generated when using the `jekyll-feed` plugin - - -## v7.2.0 -Nov 13 2017 -{:.heading.post-date} - -### Added -* Added `_sass/my-variables.scss` file, which you can use to selectively override SCSS variables. -* Font weights can now be configured via SCSS variables: - * `$font-weight` for normal font. - * `$font-weight-bold` for `strong` tags and similar. - * `$font-weight-heading` for headings. - -### Design -* Message boxes will no longer span the full width, even with the break layout feature enabled. -* Increased space between project card rows, so they look less like a brick wall. - -### Fixes -* Reduced the draw range of the drawer on iOS, so that a larger portion of the screen is available for zooming (a11y). -* Default images are now optimized, so they are no longer flagged by Google PageSpeed Insights and similar tools. -* Query parameters are no longer used for cache busting. - Instead, the version number is no part of the file name for the CSS and JS resources. - - -## v7.1.1 -Nov 3 2017 -{:.heading.post-date} - -### Fixes -* Fix IE11 feature detection - -## v7.1.0 -Nov 2 2017 -{:.heading.post-date} - -### Changed -* Renamed `no_description` to `hide_description`. - Since this feature isn't yet documented outside of the change log, the old name *will not* continue to work. -* When providing images to `image`, `image.path`, `image.src`, `image.srcset` and `accent_image `, - it is no longer necessary to prepend the url with the `baseurl` of the site, - e.g. values like `accent_image: /assets/img/sidebar-bg.jpg` are now valid. -* Limited scope of `a` and `img` styles to content areas. -* Upgraded KaTeX to v0.8.3 -* Upgraded `jekyll-relative-links` to v5.0.1 - -### Fixes -* `font` and `font_heading` are now properly set when using the `no_inline_css` option [#47](https://github.com/hydecorp/hydejack/issues/47). -* Fixed default values for `image` and `logo` that were referring to non-existing images. -* Added missing JS dev dependencies. - -### Content -* Updated documentation -* Updated index, download, about and README pages. - -## v7.0.1 -Oct 27 2017 -{:.heading.post-date} - -### Fixes -* Removed readme files from `assets` that would show up as pages when building on GitHub Pages [#42](https://github.com/hydecorp/hydejack/issues/42). -* Disabled push state on Firefox for iOS -* Changed some default settings in `_config.yml` - -### Content -* Updated documentation - -### Removed -* Removed outdated example script in `my-scripts.html` - -## v7.0.0 -Oct 24 2017 -{:.heading.post-date} - -### License Change -The *free version* of Hydejack is now [GPL-3.0] licensed, which is a more restrictive license than MIT (but still *Open Source*). -This was necessary because the two major components that make up Hydejack, -[hy-push-state](https://hydecorp.github.io/hy-push-state/){:.external} and -[hy-drawer](https://hydecorp.github.io/hy-drawer/){:.external}, -are now GPL licensed in turn. - -How will this affect you? -* If you bought the *PRO version* you are not affected at all. -* You can continue to use previous versions of Hydejack according to their license (MIT). -* If you upgrade, keep the source code in a public repository and make sure you include the new `LICENSE.md` file. - DO NOT publish the *new code* with an *old license*. -* If you upgrade and make changes to the source code, you are required to make those changes available to the public - under a GPL-3.0 compatible license. - -The full license text is available [here][GPL-3.0]. -You can read a summary on [tl;drLegel](https://tldrlegal.com/license/gnu-general-public-license-v3-(gpl-3)). - -If this change is not acceptable to you, DO NOT upgrade or consider [buying][buy] a [PRO license][PRO-license]. - -Note that the above does not constitute legal advice. - -### Breaking -This is a major release, but almost all options and APIs that were described in the docs continue to work. -Some names have changed and are no longer mentioned in the docs, but they are still part of the code and continue to work. - -That being said, you should be aware of these (small) breaking changes: - -* The favicon is now located in `assets/icons`. To change the favicon of the page, edit `favicon.ico` (png) in the folder. - -* Changed the way tables work, so that they do the right thing more often. - Tables are now scrollable by default, but small tables are no longer stretched to span the full width. - Setting `scroll-table` on a larger table is sill recommended, as it will set `white-space: nowrap`. - -* Autogenerated ids for posts now look like `post-2017-01-01-my-title` instead of `post-2017/01/01/my-title`. - -* Event names described in the scripting chapter have changed from `y-push-state-*` to `hy-push-state-*`, - except `y-push-state-animationend`, which has been removed. See the [docs][pstate] for more. - -[pstate]: docs/scripts.md#registering-push-state-event-listeners - -### Changes -* `image` has been renamed to `accent_image`, but `image` continues to work unless you add the `jekyll-seo-tag` plugin. - This change was necessary because `jekyll-seo-tag` uses the `image` keyword to set the thumbnail image of a page. - While it *may* be desirable to use the same image for both the sidebar and the thumbnail, - the new preferred way to set sidebar images is by using the `accent_image` key. - -* `color` has been renamed to `accent_color` to be consistent with the new `accent_image` key, but `color` continues to work. - -* Various options that do not make sense outside the context of Hydejack (like `no_push_state` or `no_drawer`) - have been moved under a common `hydejack` key. However, the old options continue to work. - - ```yml - hydejack: - no_push_state: false - no_drawer: false - ``` - -* All plugins (gems) are now optional. - The gem-based version of the theme no longer uses any plugins by default, - while the download version follow a "batteries-included" approach and enables some by default. - -* Links to the `/assets/` folder are no longer intercepted by the push state features, - which means clickable images and download links should work fine now. - -* Reader views in Firefox and Safari have an easier time recognizing the main content. - -* [Internal] No more `