Latex

Online Resources

Books

LaTeX for authors

  • 1 Introduction
  • 2 Classes and packages
  • 3 Commands
  • 4 LaTeX 2.09 documents
  • 5 Local modifications
  • 6 Problems
  • 7 Enjoy!

The Not-so-short introduction to Latex 2e

  • Source
  • 1 Things you need to know
    • A bit of history
    • Basics
    • LaTeX input files
    • Input file structure
    • A typical command line session
    • The layout of the document
    • Files you may encounter
    • Big projects
  • 2 Typesetting text
    • The structure of text and language
    • Line breaking and page breaking
    • Ready-made strings
    • Special characters and symbols
    • International language support
    • The space between words
    • Titles, chapters, and sections
    • Cross references
    • Footnotes
    • Emphasized words
    • Environments
    • Including graphics and images
    • Floating bodies
  • 3 Typesetting mathematical formulae
    • The AMS-LaTeX bundle
    • Single equations
    • Building blocks of a mathematical formula
    • Single equations that are too long: multiline
    • Multiple equations
    • Arrays and matrices
    • Spacing in math mode
    • Fiddling with the math fonts
    • Theorems, lemmas, ...
    • List of mathematical symbols
  • 4 Specialties
    • Bibliography
    • Indexing
    • Fancy headers
    • The verbatim package
    • Installing extra packages
    • LaTeX and PDF
    • Working with XeLaTeX and PDF
    • Creating Presentations
  • 5 Producting mathematical graphics
    • Overview
    • The picture environment
    • The PGF and TikZ graphics packages
  • 6 Customising Latex
  • New commands, environments, and packages
  • Fonts and sizes
  • Spacing
  • Page layout
  • More fun with lengths
  • Boxes
  • Rules

Latex. A document preparation system (2nd Ed)

  • 1 Getting acquainted
  • 2 Getting started
    • Preparing an input file
    • The input
    • Running LaTeX
    • Helpful hints
  • 3 Carrying on
    • Changing the type style
    • Symbols from other languages
    • Mathematical formulas
    • Defining commands and environments
    • Figures and other floating bodies
    • Lining it up in columns
    • Simulating typed text
  • 4 Moving information around
    • The table of contents
    • Cross-references
    • Bibliography and citation
    • Splitting your input
    • Making an index or glossary
    • Keyboard input and screen output
    • Sending your document
  • 5 Other document classes
    • Books
    • Slides
    • Letters
  • 6 Designing it yourself
    • Document and page styles
    • Line and page breaking
    • Numbering
    • Length, spaces and boxes
    • Centering and flushing
    • List-making environments
    • Fonts
  • 7 Pictures and colors
    • Pictures
    • The graphics package
    • Color
  • 8 Errors
    • Finding the error
    • LaTeX's error messages
    • TeX's error messages
    • LaTeX warnings
    • TeX warnings
  • A Using MakeIndex
  • B The bibliography database
  • C Reference manual
  • D What's new
  • E Using plain Tex commands

LaTeX in 24 Hours

A practical guide for scientific writing

  • 1 Introduction
    • What is LaTeX?
    • Why LaTeX over other word processors?
    • How to prepare a LaTeX input file?
    • How to compile a LaTeX input file?
    • LaTeX syntax
    • Keyboard characters in LaTeX
    • How to read this book?
  • 2 Fonts Selection
    • Text-mode fonts
    • Math-mode fonts
    • Emphasized fonts
    • Colored fonts
  • 3 Formatting Texts I
    • Sectional units
    • Labeling and referring numbered items
    • Texts alignment
    • New lines and paragraphs
    • Creating and filling blank space
    • Producing dashes within texts
    • Preventing line break
    • Adjusting blank space after a period mark
    • Hyphenating a word
  • 4 Formatting Texts II
    • Increasing depth of sectional units
    • Changing titles and counters of sectional units
    • Multiple columns
    • Mini pages
    • Foot notes
    • Marginal notes
  • 5 Page Layout and Style
    • Page layout
    • Page style
    • Running header and footer
    • Page breaking and adjustment
    • Page numbering
  • 6 Listing and Tabbing Texts
    • Listing texts
    • Tabbing texts through the tabbing environment
  • 7 Table Preparation I
    • Table through the tabular environment
    • Table through the tabularx environment
    • Vertical positioning of tables
    • Sideways (rotated) texts in tables
    • Adjusting provisions for customizing columns of tables
    • Merging rows and columns of tables
    • Table wrapped by texts
    • Table with colored background
  • 8 Table Preparation II
    • Nested tables
    • Column alignment about decimal point
    • Side-by-side tables
    • Sideways (rotated) table
    • Long table on multiple pages
    • Tabels in multi-column documents
    • Foot notes in tables
    • Changing printing format of tables
    • Tables at the end of a document
  • 9 Figure Insertion
    • Commands and environment for inserting figures
    • Inserting a simple figure
    • Side-by-side figures
    • Figure wrapped by texts
    • Rotated figure
    • Mathematical notations in figures
    • Figures in tables
    • Figures in multi-column documents
    • Changing printing format of figures
    • Figures at the end of a document
    • Editing LaTeX input file involving many figures
  • 10 Figure Drawing
    • Circles and circular arcs
    • Straight lines and vectores
    • Curves
    • Oval boxes
    • Texts in figures
    • Compound figures
  • 11 Equation Writing I
    • Basic mathematical notations and delimiters
    • Mathematical operators
    • Mathematical expressions in text-mode
    • Simple equations
    • Array of equations
    • Left aligning an equation
    • Sub-numbering a set of equations
  • 12 Equation Writing II
    • Texts and blank space in math-mode
    • Conditional expression
    • Evaluation of functional values
    • Splitting an equation into multiple lines
    • Vector and matrix
    • Overlining and underlining
    • Stacking terms
    • Side-by-side equations
  • 13 User-defined Macros
    • Defining new commands
    • Redefining existing commands
    • Defining new environments
    • Redefining existing environments
  • 14 Bibliography with LaTeX
    • Preparation of bibliographic reference database
    • Citing bibliographic references
    • Compiling thebibliography based LaTeX input file
  • 15 Bibliography with BibTeX
    • Preparation of BibTeX compatible reference database
    • Standard bibliographic styles of LaTeX
    • Use of natbib package
    • Compiling BibTeX based LaTeX input files
    • Editing the .bbl file
    • Multiple bibliographies
  • 16 Lists of Contents and Index
    • Lists of contents
    • Making index
  • 17 Miscellaneous I
    • Boxed items
    • Rotated items
  • 18 Miscellaneous II
    • Horizontal rules and dots
    • Hyperlinking referred and cited items
    • Current date and format
    • Highlighted texts
    • Verbatim texts
    • Fragile commands
    • Watermarking on pages
    • Logo in header and footer
    • Paragraphs in different forms
  • 19 Letter and Articles
    • Letter writing
    • Article preparation
  • 20 Book and Report
    • Template of a book
    • Book preparation using a root file
    • Dividing a book into parts
    • Compilation of a book
  • 21 Slide Preparation I
    • Frames in presentation
    • Sectional units in presentation
    • Presentation structure
    • Appearance of a presentation (BEAMER themes)
    • Frame customization
  • 22 Slide Preparation II
    • Piece-wise presentation (BEAMER overlays)
    • Environments in BEAMER class
    • Table and figure in presentation
    • Dividing a frame column-wise
    • Jumping (hyperlink) to other slides
  • 23 Error and Warning Messages
  • Error message
  • Warning message
  • Error without any message
  • Tips for debugging

Latex Wikibook

  • Source
  • Getting started
    1. Introduction
    2. Installation
    3. Installing Extra Packages
    4. Basics
  • Common Elements
    1. Document Structure
    2. Text Formatting
    3. Paragraph Formatting
    4. Colors
    5. Fonts
    6. List Structures
    7. Special Characters
    8. Internationalization
    9. Rotations
    10. Tables
    11. Title Creation
    12. Page Layout
    13. Importing Graphics
    14. Floats, Figures and Captions
    15. Hyperlinks
    16. Labels and Cross-referencing
  • Mechanics
    1. Errors and Warnings
    2. Lengths
    3. Counters
    4. Boxes
    5. Rules and Struts
  • Technical Texts
    1. Mathematics
    2. Advanved Mathematics
    3. Theorems
    4. Chemical Graphics
    5. Algorithms
    6. Source Code Listings
    7. Linguistics
    8. References
    9. External Links
  • Special Pages
    1. Indexing
    2. More Bibliographies
  • Special Documents
    1. Letters
    2. Presentations
    3. Teacher's Corner
    4. Curriculum Vitae
  • Creating Graphics
    1. Introducing Procedural Graphics
    2. MetaPost
    3. Picture
    4. PGF/TikZ
    5. PSTricks
    6. Xy-Pic
    7. Creating 3D Graphics
  • Programming
    1. Macros
    2. Plain Tex
    3. Creating Packages
    4. Themes
  • Miscellaneous
    1. Modular Documents
    2. Collaborative Writing of Latex Documents
    3. Export to Other Formats
  • Help and Recommendations
    1. FAQ
    2. Tips and Tricks
  • Appendices
  1. Authors
  2. Links
  3. Sample Latex Documents
  4. Index

Math into Latex

  • I A short course
    • 1 Typing your first article
  • II Text and math
    • 2 Typing text
    • 3 Text environments
    • 4 Typing math
    • 5 Multiline math displays
  • III Document structure
    • 6 Latex documents
    • 7 Standard Latex document classes
    • 8 AMS-Latex documents
  • IV Customizing
    • 9 Customizing Latex
  • V Long bibligraphies and indexes
    • 10 BibTex
    • 11 MakeIndex
  • VI Appendices
  • A Math symbol tables
  • B Text symbol tables
  • C The AMS-Latex sample article
  • D Sample article with user-defined commands
  • E Background
  • F Postscript fonts
  • G Getting it
  • H Conversions
  • I Final word

More Math into LaTeX

  • I Mission Impossible
    • 1 Short course
    • 2 And a few more things...
  • II Text and Math
    • 3 Typing text
    • 4 Text environments
    • 5 Typing math
    • 6 More math
    • 7 Multiline math displays
  • III Document structure
    • 8 Documents
    • 9 The AMS article document class
    • 10 Legacy documents
  • IV PDF documents
    • 11 The PDF file format
    • 12 Presentations
    • 13 Illustrations
  • V Customization
    • 14 Commands and environments
  • VI Long documents
    • 15 BibTeX
    • 16 MakeIndex
    • 17 Books in LaTeX
  • A Math symbol tables
  • B Text symbol tables
  • C Some background
  • D LaTeX and the internet
  • E PostScript fonts
  • F LaTeX localized
  • G LaTeX on the iPad
  • H Final thoughts

Practical LaTeX

  • 1 Mission impossible
  • 2 Text
  • 3 Text environments
  • 4 Inline formulas
  • 5 Displayed formulas
  • 6 Documents
  • 7 Customizing LaTeX
  • 8 Presentations
  • 9 Illustrations

The Latex graphics companion

  • 1 Graphics with Latex
  • 2 The Latex 2e graphics bundle
  • 3 Working with METAFONT and METAPOST
  • 4 Harnessing PostScript inside Latex: the pstricks package
  • 5 The Xy-pic package
  • 6 Applications in chemistry, physics and engineering
  • 7 Preparing music schores
  • 8 Playing games
  • 9 The world of color
  • 10 Using postscript fonts
  • 11 PostScript drivers and tools

Using imported graphics in Latex

  • I Background information
    • 1 Introduction
    • 2 Latex terminology
    • 3 Encapsulated postscript
    • 4 How eps files are used by latex
    • 5 PDF graphics
    • 6 Graphics software
  • II The Latex graphics bundle
    • 7 Graphics inclusion
    • 8 Rotating and scaling objects
    • 9 Advanced graphics-inclusion commands
  • III Using graphics-inclusion commands
    • 10 Horizontal spacing and centering
    • 11 Rotation, scaling, alignment
    • 12 Overlaying two imported graphics
    • 13 Using subdirectories
    • 14 Compressed and non-EPS graphics files in dvips
    • 15 The psfrag package
    • 16 Including an eps file multiple times
  • IV The figure environment
    • 17 The figure environment
    • 18 Customizing float placement
    • 19 Customizing the figure environment
    • 20 Customizing captions with the caption package
    • 21 Non-floating figures
    • 22 Marginal figures
    • 23 Wide figures
    • 24 Landscape figures
    • 25 Captions beside figures
    • 26 Figures on odd or even pages
    • 27 Boxed figures
  • V Complex figures
  • 28 Side-by-side graphics
  • 29 Separate minipages for captions
  • 30 Placing a table beside a figure
  • 31 Stacked figures and subfigures
  • 32 The subfic package
  • 33 Continued figures and subfigures

TeX in a Nutshell

  • 1 Terminology
  • 2 Formats, engines
  • 3 Searching data
  • 4 Processing the input
  • 5 Vertical and horizontal modes
  • 6 Groups in TeX
  • 7 Box, kern, penalty glue
  • 8 Syntactic rules
  • 9 Principles of macros
  • 10 Math modes
  • 11 Registers
  • 12 Expandable primitive commands
  • 13 Primitive commands at the main processor level
  • 14 Summary of plain TeX macros

The TexBook

  • 1 The name of the game
  • 2 Book printing versus ordinary typing
  • 3 Controlling Tex
  • 4 Fonts of type
  • 5 Grouping
  • 6 Running Tex
  • 7 How Tex reads what you type
  • 8 The characters you type
  • 9 Tex's roman fonts
  • 10 Dimensions
  • 11 Boxes
  • 12 Glue
  • 13 Modes
  • 14 How Tex breaks paragraphs into lines
  • 15 How Tex makes lines into pages
  • 16 Typing math formulae
  • 17 More about math
  • 18 Fine points of mathematics typing
  • 19 Displayed equations
  • 20 Definitions (also called macros)
  • 21 Making boxes
  • 22 Alignment
  • 23 Output routines
  • 24 Summary of vertical mode
  • 25 Summary of horizontal mode
  • 26 Summary of math mode
  • 27 Recovery from errors

Introduction to METAPOST

  • 1 Introduction
  • 2 Overview of the Language
  • 3 Interesting Figures
  • 4 Macro Packages
  • 5 Conclusion

MetaPost for Beginners

Learning METAPOST by doing

  • 1 Introduction
  • 2 A simple example
  • 3 Basic graphical primitives
  • 4 Style directives
  • 5 Transformations
  • 6 Advanced graphics
  • 7 Control structures
  • 8 Macros
  • 9 More examples
  • 10 Solutions to exercises
  • 11 Appendix

The METAFONT book

  • 1 The Name of the Game
  • 2 Coordinates
  • 3 Curves
  • 4 Pens
  • 5 Running METAFONT
  • 6 How METAFONT Reads What You Type
  • 7 Variables
  • 8 Algebraic Expressions
  • 9 Equations
  • 10 Assignments
  • 11 Magnification and Resolution
  • 12 Boxes
  • 13 Drawing, Filling, Erasing
  • 14 Paths
  • 15 Transformations
  • 16 Calligraphic Effects
  • 17 Grouping
  • 18 Definitions (also called Macros)
  • 19 Conditions and Loops
  • 20 More about Macros
  • 21 Random Numbers
  • 22 Strings
  • 23 Online Displays
  • 24 Discreteness and Discretion
  • 25 Summary of Expressions
  • 26 Summary of the Language
  • 27 Recovering from Errors

A Few Notes on Book Design

  • 1 Historical background
    • Galloping through the millenia
    • Making type
    • Book types
    • Setting type
    • Today
    • Setting maths
  • 2 The parts of a book
    • Front matter
    • Main matter
    • Back matter
    • Signatures and casting off
    • Paper
  • 3 The page
    • The shape of a book
    • The spread
    • The typeblock
    • Folios
    • Headers and footers
    • Electronic books
  • 4 Styling the elements
    • Front matter
    • Main matter
    • Back matter
    • Type size
    • Poems and plays
    • Selecting a typeface
  • 5 Picky points
  • Word and line spacing
  • Letterspacing
  • Abbreviations and acronyms
  • Dashes and ellipses
  • Punctuation
  • Narrow measures
  • Emphasis
  • Captions and legends
  • Tables
  • Number formatting

The Memoir Class for Configurable Typesetting

  • 1 Starting off
  • 2 Laying out the page
  • 3 Texts and fonts
  • 4 Titles
  • 5 Abstracts
  • 6 Document divisions
  • 7 Pagination and headers
  • 8 Paragraphs and lists
  • 9 Contents lists
  • 10 Floats and captions
  • 11 Rows and columns
  • 12 Page notes
  • 13 Decorative text
  • 14 Poetry
  • 15 Boxes, verbatims and files
  • 16 Cross referencing
  • 17 Back matter
  • 18 Miscellaneous
  • 19 For package users
  • 20 An example book design
  • 21 An example thesis design
  • A Packages and macros
  • B Showcases
  • C Sniplets
  • D Pictures
  • E LaTeX and TeX
  • F The terrors of errors
  • G Comments

Courses

Curso Introductorio de Latex Falappa

  • TeX viene de tau-epsilon-chi
  • LaTeX fue desarrollado por Leslie Lamport a comienzos de los 80 y agrega ciertas herramientas de abstraccion a TeX.
  • LaTeX sirve para escribir reportes tecnicos, monografias, articulos y libros.

Primeros Pasos

  • Comandos y caracteres especiales
    • palabras reservadas: documentclass, begin, end, emph, etc
    • caracteres especiales: \ { } # \$ % ^ & _ ~
  • Estructura de un libro
  • Oraciones y parrafos
  • Puntos
  • Negrita, italica, subrayado
  • Encabezado de un documento
  • Tipos de documento
  • Paquetes
  • Division de un documento
  • Numeraciones
  • Texto indentado

Ejemplo 1

\documentclass{article}
\begin{document}
\title{Title}
\date{14 de Noviembre de 2020}
\author{Iñaki Garay}
\maketitle
Hello, world!
\end{document}

Ejemplo 2

\documentclass[12pt]{article}
\begin{document}
\title{Title}
\date{14 de Noviembre de 2020}
\author{Iñaki Garay}
\maketitle
\begin{abstract}
This is the abstract.
\end{abstract}
\section{Introduction}
This is the introduction.
\end{document}

Ejemplo 3

\documentclass{book}
\begin{document}
\title{Title}
\date{14 de Noviembre de 2020}
\author{Iñaki Garay}
\maketitle
\tableofcontents
\listoffigures
\listoftables

\chapter{The First Chapter}
\section{The first section of the first chapter}

\begin{thebibliography}
\end{thebibliography}

\end{document}

Homer J. Simpson vs Homer J.\ Simpson
Negrita, Italica, Subrayado
  • \textbf{}
  • \textit{}
  • \emph{}
  • \underline{}

To highlight text don’t use \textit but \emph, because the latter also provides the right logical markup (“emphasize”) and takes care of nesting.

\emph{This is some example text which is emphasized by using an italic font. A \emph{nested highlighted} will be recto (upright)}.
Encabezado
  • El texto que precede a \begin{document} se denomina preambulo.
  • Tiene dos partes;
    • La definicion del tipo de documento a utilizar, y las opciones correspondientes.
    • Los paquetes adicionales que utilizara el mismo.
  • Condiciona los comandos disponibles en el documento (e.g. book admite chapter y article no).
  • Tipos de documentos: report,
  • Paquetes
  • Division de un documento
  • Numeraciones
  • Texto indentado

Formulas, Listas, Enumeraciones, Alineaciones

  • Formulas
  • Listas
  • Enumeraciones
  • Descripciones
  • Anidamientos de items
  • Uso de subitems
  • Anidamiento de enumeraciones
  • Alinenacion de texto
  • Prevenir saltos de lineas

Estilos, Tamaños, Simbolos Especiales, Opciones de Pagina

  • Estilos de letras
  • Tamaños de letras
  • Combinaciones de tipos y tamaños
  • Simbolos especiales
  • Margenes de un documento
  • Ejemplo de ajust de margenes
  • Tipos de documentos
  • Tamaños de pagina
  • Opciones de pagina
  • Paquetes adicionales
  • Estilos de pagina
  • Titulo
  • Verbatim

Formulas Matematicas Avanzadas

  • Formulas
  • Letras griegas
  • Simbolos relacionales
  • Flechas
  • Simbolos miscelaneos
  • Simbolos de tamaño variable
  • Funciones
  • Diferencia entre \[ \] y \( \)
  • Entorno array
  • Alineacion vertical
  • Delimitadores
  • Formulas en varias lineas
  • Superposicion de formulas
  • Espacios en entornos matematicos
  • Estilos en entornos matematicos

Tablas, Separaciones Silabicas

  • Entorno tabular
  • Entorno tabbing
  • Espacios
  • Hyphenation
  • Referencias cruzadas

Bibliografias, Graficos, Entornos, Indices Tematicos

  • Bibliografia
  • Bibliografia sin BibTeX
  • Bibliografia con BibTeX
  • Tipos de fuentes
  • Formatos de bibliografia
  • Archivo bib
  • Formato plain
  • Formato alpha
  • Formato unsrt
  • Formato abbrv
  • Estilo elsevier
  • Estilo kluwer
  • Estilo ASL/JSL
  • Figuras y graficos
  • Figuras en LaTeX
  • Uso de comandos definidos
  • Separar un documento LaTeX
  • Indice tematico
  • Otros temas

Learn X in Y Minutes: Latex

% All comment lines start with %
% There are no multi-line comments

% LaTeX is NOT a "What You See Is What You Get" word processing software like
% MS Word, or OpenOffice Writer

% Every LaTeX command starts with a backslash (\)

% LaTeX documents start with a defining the type of document it's compiling
% Other document types include book, report, presentations, etc.
% The options for the document appear in the [] brackets. In this case
% it specifies we want to use 12pt font.
\documentclass[12pt]{article}

% Next we define the packages the document uses.
% If you want to include graphics, colored text, or
% source code from another language file into your document,
% you need to enhance the capabilities of LaTeX. This is done by adding packages.
% I'm going to include the float and caption packages for figures
% and hyperref package for hyperlinks
\usepackage{caption}
\usepackage{float}
\usepackage{hyperref}

% We can define some other document properties too!
\author{Chaitanya Krishna Ande, Colton Kohnke, Sricharan Chiruvolu \& \\
Svetlana Golubeva}
\date{\today}
\title{Learn \LaTeX{} in Y Minutes!}

% Now we're ready to begin the document
% Everything before this line is called "The Preamble"
\begin{document}
% if we set the author, date, title fields, we can have LaTeX
% create a title page for us.
\maketitle

% If we have sections, we can create table of contents. We have to compile our
% document twice to make it appear in right order.
% It is a good practice to separate the table of contents form the body of the
% document. To do so we use \newpage command
\newpage
\tableofcontents

\newpage

% Most research papers have abstract, you can use the predefined commands for this.
% This should appear in its logical order, therefore, after the top matter,
% but before the main sections of the body.
% This command is available in the document classes article and report.
\begin{abstract}
 \LaTeX{} documentation written as \LaTeX! How novel and totally not
 my idea!
\end{abstract}

% Section commands are intuitive.
% All the titles of the sections are added automatically to the table of contents.
\section{Introduction}
Hello, my name is Colton and together we're going to explore \LaTeX!

\section{Another section}
This is the text for another section. I think it needs a subsection.

\subsection{This is a subsection} % Subsections are also intuitive.
I think we need another one.

\subsubsection{Pythagoras}
Much better now.
\label{subsec:pythagoras}

% By using the asterisk we can suppress LaTeX's inbuilt numbering.
% This works for other LaTeX commands as well.
\section*{This is an unnumbered section}
However not all sections have to be numbered!

\section{Some Text notes}
%\section{Spacing} % Need to add more information about space intervals
\LaTeX{} is generally pretty good about placing text where it should
go. If
a line \\ needs \\ to \\ break \\ you add \textbackslash\textbackslash{}
to the source code.

Separate paragraphs by empty lines.

You need to add a backslash after abbreviations (if not followed by a comma), because otherwise the spacing after the dot is too large:
E.g., i.e., etc.\ are are such abbreviations.

\section{Lists}
Lists are one of the easiest things to create in \LaTeX! I need to go shopping
tomorrow, so let's make a grocery list.
\begin{enumerate} % This creates an "enumerate" environment.
  % \item tells the enumerate to increment
  \item Salad.
  \item 27 watermelon.
  \item A single jackrabbit.
  % we can even override the item number by using []
  \item[how many?] Medium sized squirt guns.

  Not a list item, but still part of the enumerate.

\end{enumerate} % All environments must have an end.

\section{Math}

One of the primary uses for \LaTeX{} is to produce academic articles
or technical papers. Usually in the realm of math and science. As such,
we need to be able to add special symbols to our paper!

Math has many symbols, far beyond what you can find on a keyboard;
Set and relation symbols, arrows, operators, and Greek letters to name a few.

Sets and relations play a vital role in many mathematical research papers.
Here's how you state all x that belong to X, $\forall$ x $\in$ X.
% Notice how I needed to add $ signs before and after the symbols. This is
% because when writing, we are in text-mode.
% However, the math symbols only exist in math-mode.
% We can enter math-mode from text mode with the $ signs.
% The opposite also holds true. Variable can also be rendered in math-mode.
% We can also enter math mode with \[\]

\[a^2 + b^2 = c^2 \]

My favorite Greek letter is $\xi$. I also like $\beta$, $\gamma$ and $\sigma$.
I haven't found a Greek letter yet that \LaTeX{} doesn't know
about!

Operators are essential parts of a mathematical document:
trigonometric functions ($\sin$, $\cos$, $\tan$),
logarithms and exponentials ($\log$, $\exp$),
limits ($\lim$), etc.\
have pre-defined LaTeX commands.
Let's write an equation to see how it's done:
$\cos(2\theta) = \cos^{2}(\theta) - \sin^{2}(\theta)$

Fractions (Numerator-denominators) can be written in these forms:

% 10 / 7
$$ ^{10}/_{7} $$

% Relatively complex fractions can be written as
% \frac{numerator}{denominator}
$$ \frac{n!}{k!(n - k)!} $$

We can also insert equations in an ``equation environment''.

% Display math with the equation 'environment'
\begin{equation} % enters math-mode
    c^2 = a^2 + b^2.
    \label{eq:pythagoras} % for referencing
\end{equation} % all \begin statements must have an end statement

We can then reference our new equation!
Eqn.~\ref{eq:pythagoras} is also known as the Pythagoras Theorem which is also
the subject of Sec.~\ref{subsec:pythagoras}. A lot of things can be labeled:
figures, equations, sections, etc.

Summations and Integrals are written with sum and int commands:

% Some LaTeX compilers will complain if there are blank lines
% In an equation environment.
\begin{equation}
  \sum_{i=0}^{5} f_{i}
\end{equation}
\begin{equation}
  \int_{0}^{\infty} \mathrm{e}^{-x} \mathrm{d}x
\end{equation}

\section{Figures}

Let's insert a figure. Figure placement can get a little tricky.
I definitely have to lookup the placement options each time.

\begin{figure}[H] % H here denoted the placement option.
    \centering % centers the figure on the page
    % Inserts a figure scaled to 0.8 the width of the page.
    %\includegraphics[width=0.8\linewidth]{right-triangle.png}
    % Commented out for compilation purposes. Please use your imagination.
    \caption{Right triangle with sides $a$, $b$, $c$}
    \label{fig:right-triangle}
\end{figure}

\subsection{Table}
We can also insert Tables in the same way as figures.

\begin{table}[H]
  \caption{Caption for the Table.}
  % the {} arguments below describe how each row of the table is drawn.
  % Again, I have to look these up. Each. And. Every. Time.
  \begin{tabular}{c|cc}
    Number &  Last Name & First Name \\ % Column rows are separated by &
    \hline % a horizontal line
    1 & Biggus & Dickus \\
    2 & Monty & Python
  \end{tabular}
\end{table}

\section{Getting \LaTeX{} to not compile something (i.e.\ Source Code)}
Let's say we want to include some code into our \LaTeX{} document,
we would then need \LaTeX{} to not try and interpret that text and
instead just print it to the document. We do this with a verbatim
environment.

% There are other packages that exist (i.e. minty, lstlisting, etc.)
% but verbatim is the bare-bones basic one.
\begin{verbatim}
  print("Hello World!")
  a%b; % look! We can use % signs in verbatim.
  random = 4; #decided by fair random dice roll
\end{verbatim}

\section{Compiling}

By now you're probably wondering how to compile this fabulous document
and look at the glorious glory that is a \LaTeX{} pdf.
(yes, this document actually does compile).

Getting to the final document using \LaTeX{} consists of the following
steps:
  \begin{enumerate}
    \item Write the document in plain text (the ``source code'').
    \item Compile source code to produce a pdf.
     The compilation step looks like this (in Linux): \\
     \begin{verbatim}
        > pdflatex learn-latex.tex
     \end{verbatim}
  \end{enumerate}

A number of \LaTeX{} editors combine both Step 1 and Step 2 in the
same piece of software. So, you get to see Step 1, but not Step 2 completely.
Step 2 is still happening behind the scenes\footnote{In cases, where you use
references (like Eqn.~\ref{eq:pythagoras}), you may need to run Step 2
multiple times, to generate an intermediary *.aux file.}.
% Also, this is how you add footnotes to your document!

You write all your formatting information in plain text in Step 1.
The compilation part in Step 2 takes care of producing the document in the
format you defined in Step 1.

\section{Hyperlinks}
We can also insert hyperlinks in our document. To do so we need to include the
package hyperref into preamble with the command:
\begin{verbatim}
    \usepackage{hyperref}
\end{verbatim}

There exists two main types of links: visible URL \\
\url{https://learnxinyminutes.com/docs/latex/}, or
\href{https://learnxinyminutes.com/docs/latex/}{shadowed by text}
% You can not add extra-spaces or special symbols into shadowing text since it
% will cause mistakes during the compilation

This package also produces list of thumbnails in the output pdf document and
active links in the table of contents.

\section{End}

That's all for now!

% Most often, you would want to have a references section in your document.
% The easiest way to set this up would be by using the bibliography section
\begin{thebibliography}{1}
  % similar to other lists, the \bibitem command can be used to list items
  % each entry can then be cited directly in the body of the text
  \bibitem{latexwiki} The amazing \LaTeX{} wikibook: \emph{https://en.wikibooks.org/wiki/LaTeX}
  \bibitem{latextutorial} An actual tutorial: \emph{http://www.latex-tutorial.com}
\end{thebibliography}

% end the document
\end{document}

Example

\documentclass[a4,12pt]{book}
% Document class can be: article, report, book, letter
% Two-column article: \documentclass[twocolumn]{article}
% A4 book with 11pt font: \documentclass[a4,11pt]{book}
% Presentation: \documentclass[25pt,landscape,headrule]{foils}

% Page options:
% final/draft
% oneside/twoside Default is oneside, except for book where it is twoside.
% openright/openany Openright; chapters begin on right side pages only. Only for report (default openany) and book (default openright).
% onecolumn/twocolumn
% notitlepage/titlepage Titlepage makes \maketitle use a separate page for title and abstract.
% leqno Makes equation numbers appear to the left.
% fleqn Makes mathematical forumalas align left.

\usepackage[spanish]{babel}
\selectlanguage{spanish}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{eucal}
\usepackage{latexsym} % for more symbols

% Margins

\pageheight
\pagewidth

\voffset % The distance between the top of the page and the beginning of the top margin.
\hoffset % The distance between the left side of the page and the odd side margin.

\topmargin % The height of the top margin.
\headheight % The height of the header.
\headsep % The separation between the header and the text.

\textheight
\textwidth

\oddsidemargin
\evensidemargin

\footskip

\marginparwidth
\marginparsep

\begin{document}

% Title

\title{}
\author{ Number One \and Number Two }
\date{}
\maketitle
\tableofcontents
\listoffigures
\listoftables

% Abstract is for article document class
\begin{abstract}
\end{abstract}

% Page styles

Pagestyles define the dimensions and page styles of the header and footer of a page.

\pagestyle{option}

option may be:
plain header is empty and footer has page number
empty both header and footer are empty
headings the header is determined by the document class, normally the section name and the page number
myheadings like headings, except the header content is specified by the user.

When myheadings is used, the header may be specified with:
\markright{}
\markboth{lefthead}{righthead}
\thispagestyle{options} % is like \pagestyle but applies only to the current page

The page numbering is determined with the command:
\pagenumbering{numstyle}

where numstyle may be:
arabic
roman
Roman
alph
Alph

% Document structure

% Chapter is for book or report document class
% Section and subsection are allowed in article document class
\part{}
\chapter{}
\section{}
\subsection{}
\subsubsection{}
\paragraph{}
\subparagraph{}

% Special characters

``Quote marks''

\'{A} Accent.

\~{N} N tilde.

% Text styles

\emph{Emphasized.}
{\em Also emphasized.}
\emph{Emphasized text \emph{within} Emphasized text.}
\begin{emph}
Also emphasized.
\end{emph}

\textbf{Bold face.}
{\bf Also bold face.}
\begin{bf}
Also boldface.
\end{bf}

\underline{Underlined.}

\textsf{Sans Serif}
\textsc{Uppercase}
\textsl{Inclinated}
\texttt{Terminal}
\verb"Verbatim"

\begin{verbatim}
\end{verbatim}

\begin{verbatim*}
In this mode whitespace characters are printed with a special character.
\end{verbatim*}

\begin{quote}
\end{quote}

\begin{itemize}
 \item[] Text
 \item[] Text
\end{itemize}

\begin{enumerate}
 \item[] Text
 \item[] Text
\end{enumerate}

{\tiny         Tiny}
{\scriptsize   Scriptsize}
{\footnotesize Footnotesize}
{\small        Small}
{\normalsize   Normalsize}
{\large        large}
{\Large        Large}
{\huge         huge}
{\Huge         Huge}

\noindent
Some text.

\textmd{Is the same as normalsize.}

\# \$ \% \& \_ \{ \}
\verb"^"
\verb"~"
\verb"\"

% Symbols
\LaTeX

% Greek letters
\alpha
\beta
\gamma
\delta
\epsilon
\varepsilon
\zeta
\eta
\theta
\vertheta
\iota
\kappa
\lambda
\mu
\nu
\xi
\pi
\varpi
\rho
\varrho
\sigma
\varsigma
\tau
\upsilon
\phi
\varphi
\chi
\psi
\omega

\Gamma
\Delta
\Theta
\Lambda
\Xi
\Pi
\Sigma
\Upsilon
\Phi
\Psi
\Omega

% Binary operators
\pm
\mp
\times
\div
\ast
\star
\circ
\bullet
\cdot
\cap
\cup
\uplus
\sqcap
\sqcup
\vee
\wedge
\setminus
\wr
\diamond
\bigtriangleup
\bigtriangledown
\triangleleft
\triangleright
\oplus
\odot
\bigcirc
\dagger
\ddagger
\amalg

% Relational symbols

% Arrows

% Miscellaneous

% Variable size symbols

% Functions

% Formulas

% Difference between \[ \] and \( \)

% Array environment

El entorno tiene un unico argumento que especifica el numero de columnas y la alineacion de los elementos dentro de la columna.
c el elemento va centrado
l el elemento va justificado a la izquierda
r el elemento va justificado a la derehca
Las filas adyacentes van separadas por \\
Los elementos dentro de una fila van separados por &
No debe haber un & despues del ultimo item en una fila ni un \\ despues de la ultima fila

\[
\begin{array}{clcr}
a+b+c & uv    & 27       \\
a+b   & u+v   & 134      \\
a     & 3u+vw & xyz & 65 \\
a^2   & u^{-1} & z  & 0
\end{array}
\]

% Alineacion vertical
Latex dibuja una linea horizontal central imaginaria a lo largo de cada formula, a la altura de un hipotetico signo menos.
Los items un una fila de un arreglo se posicionan verticalmente de modo que sus lineas centrales estan a la misma altura.
Se puede cambiar la posicion de un entorno array (por default esta centrado)
El argumento opcional t indica que se alineara con la primera linea del arreglo.
El argumento opcional b indica que se alineara con la ultima linea del arreglo.

\[
x -
\begin{array}{c}
a_1    \\
\vdots \\
a_n    \\
\end{array}
+
\begin{array}[t]{cl}
u - v & 13                                       \\
u + v & \begin{array}[b]{r}12 \\ 144 \end{array}
\end{array}
\]

% Delimitadores

% Tables
El entorno tabular es similar al entorno array pero se puede usar en modo texto y en modo matematico.
Dentro de los argumentos, pueden ir c, l, r igual que en array, y ademas | coloca una barra vertical simple.
Un comando \hline despues de un \\ o al comienzo del entorno dibuja una lina horizontal a lo largo de la tabla.
El comando \cline{i-j} dibuja una linea horizontal entre las columnas i y j inclusive.
Un item de una tabla puede expandirse en varias en varias columnas utilizando el comando:
\multicolumn{cols}{pos}{text}
cols es el numero de columnas a expandir
pos es su posicion
text es el texto del item

\begin{tabular}{||l|lr||} \hline \hline
\textbf{Equipo} & \textbf{Zone} & \textbf{Puntos} \\ \hline \hline
Licenciados     & Norte         & 25              \\ \hline
Ingenieros      & Sur           & 24              \\ \hline
Profesores      & Centro        & 22              \\ \hline \hline
\end{tabular}

\begin{tabular}{||l|lr||} \hline
Papas   & Horno   & \$ 2.00 \\ \cline{2-3}
        & Fritas  & \$ 3.00 \\ \hline
Asado   & Parilla & \$ 8.00 \\ \hline
Chorizo &         & \$ 4.00 \\ \hline
Vino    & Rosado  & \$ 9.00 \\ \hline
\end{tabular}


\begin{tabular}{||l|l|r||} \hline \hline
\textbf{Equipo} & \multicolumn{2}{c||}{\emph{Camiseta}} \\ \hline
River           & Blanco & Rojo     \\
Boca            & Azul   & Amarillo \\
San Lorenzo     & Rojo   & Azul     \\
Olimpo          & Negro  & Amarillo \\ \hline \hline
\end{tabular}

Tabbing environment

El entorno tabbing permite alinear texto como el entorno tabular, pero sin separar mediante lineas verticales y horizontales.
Permite alinear texto definiendo tabulaciones.
Las tabulaciones se definen con \= y el \> permite desplazarse hasta la proxima tabulacion.
Las lineas se separan con \\
Un comando \= puede resetear una antigua tabulacion
Un comando \kill en lugar de \\ al final de una linea define las tabulaciones pero no muestra a esa linea en la salida

\begin{tabbing}
Los hermanos sean \= unidos                          \\
                  \> porque esa es la ley primera    \\
                  \> tengan union verdader           \\
                  \> en cualquier tiempo que sea     \\
                  \> porque si entre ellos se pelean \\
                  \> los devoran los de afuera
\end{tabbing}

\begin{tabbing}
Si no se     \= definen bien las tabulaciones    \\
las          \> palabras pueden verse bien o ... \\
superponerse \> //// entre ellas
\end{tabbing}

\begin{tabbing}
Columna 1 \= Columna 2 \= Columna 3 \\
A \> B  \> C \\
a \> b  \> c \\
Nueva Columna 1 \= Nueva Columna 2 \= Nueva Columna 3 \\
A               \> B               \> C \\
a               \> b               \> c
\end{tabbing}

\begin{tabbing}
Carpinteria \= Anaranjado \= Hipertension \kill
Uno         \> Dos        \> Tres         \\
Cuatro      \> Cinco      \> Seis         \\
Siete       \> Ocho       \> Nueve
\end{tabbing}

% The bibliography is for book document class
\begin{thebbibliography}
\end{thebibliography}

\end{document}

Logica para Ciencias de la Computacion: TP3

%
% Lógica para Ciencias de la Computación,
% Primer Cuatrimestre de 2006
%
% Trabajo Práctico Número 3bis
%

\documentclass[12pt]{article}

\usepackage[spanish]{babel}
\selectlanguage{spanish}

\input{Defs}

\newcommand{\then}{\rightarrow}
\newcommand{\tab}{\hspace{1.5cm}}
\setlength{\tabcolsep}{0.5mm}

\begin{document}

\begin{center}

\begin{tabular}{lcl}
\hspace{.85in} & \large Departamento de Cs.\ e Ingenier\'{\i}a de
la Computaci\'on & \hspace{.85in} \\
& \large Universidad Nacional del Sur \\
\smash{\includegraphics[width=.70in,height=.75in]{DeptoLogo.eps}} &
& \hspace{.10in} \smash{\includegraphics[width=.78in,height=.78in]{UniLogo.eps}} \\
\end{tabular}

\vspace{4mm}

{\Large\sc L\'ogica para Ciencias de la Computaci\'on} \\[4pt]
{\bf Pruebas Sint\'acticas} \\[4pt]
Primer Cuatrimestre de 2006

\end{center}

\section*{Convenciones adoptadas}

\setlength{\tabcolsep}{1.5mm}
\begin{tabular}{cl}
\sf Def. Deducci\'on & Definici\'on de deducci\'on.\\
\sf MP ($x$,$y$) & Consecuencia directa, por Modus Ponens, de $x$ e $y$.\\
\sf TD ($x$) & Teorema de la Deducci\'on aplicado a $x$.\\
\sf Transitividad & Tercer propiedad de la deducci\'on (no confundir con Lema 2.3.4). \\
\sf Lema $n$ & Lema $n$, seg\'un ``Truth, Deduction and Computation'' (R. Davis). \\
\sf L1 & Instancia del axioma $A\then(B\then A)$.\\
\sf L2 & Instancia del axioma $(A\then(B\then C))\then((A\then B)\then(A\then C))$.\\
\sf L3 & Instancia del axioma $(\neg B\then\neg A)\then((\neg B\then A)\then B)$.\\
\end{tabular}

\section*{Soluciones}

\subsubsection*{(a) $\neg\neg A\then A$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & $\neg\neg A$ & $\vdash$ & $\neg\neg A$ & Def. Deducci\'on \\
2. & \ & $\vdash$ & $(\neg A\then\neg\neg A)\then((\neg A\then\neg A)\then A)$ & {\bf L3} \\
3. & \ & $\vdash$ & $\neg\neg A\then(\neg A\then\neg\neg A)$ & {\bf L1} \\
4. & $\neg\neg A$ & $\vdash$ & $\neg A\then\neg\neg A$ & MP (1,3) \\
5. & $\neg\neg A$ & $\vdash$ & $(\neg A\then\neg A)\then A$ & MP (4,2) \\
6. & \ & $\vdash$ & $\neg A\then\neg A$ & Lema 2.3.2 \\
7. & $\neg\neg A$ & $\vdash$ & $A$ & MP (5,2) \\
8. & \ & $\vdash$ & $\neg\neg A\then A$ & TD (7) \\
\end{tabular*}

\subsubsection*{(b) $A,B \vdash \neg(A\then\neg B)$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & $A$ & $\vdash$ & $A$ & Def. Deducci\'on \\
2. & $B$ & $\vdash$ & $B$ & Def. Deducci\'on \\
3. & \  & $\vdash$ & $(\neg\neg(A\then\neg B)\then\neg B)\then$ \\
\ & \ & \ & \tab $((\neg\neg(A\then\neg B)\then B)\then\neg(A\then\neg B))$ & {\bf L3} \\
4. & \ & $\vdash$ & $\neg\neg(A\then\neg B)\then(A\then\neg B)$ & Ejercicio (a) \\
5. & $A\then\neg B$ & $\vdash$ & $A\then\neg B$ & Def. Deducci\'on \\
6. & $A,A\then\neg B$ & $\vdash$ & $\neg B$ & MP (1,5) \\
7. & $A$ & $\vdash$ & $(A\then\neg B)\then\neg B$ & TD (6) \\
8. & $A$ & $\vdash$ & $\neg\neg(A\then\neg B)\then\neg B$ & Lema 2.3.4 (4,7) \\
9. & $A$ & $\vdash$ & $(\neg\neg(A\then\neg B)\then B)\then\neg(A\then\neg B)$ & MP (3,8) \\
10. & \ & $\vdash$ & $B\then(\neg\neg(A\then\neg B)\then B)$ & {\bf L1} \\
11. & $B$ & $\vdash$ & $\neg\neg(A\then\neg B)\then B$ & MP (2,10) \\
12. & $A,B$ & $\vdash$ & $\neg(A\then\neg B)$ & MP (11,9) \\
\end{tabular*}

\ \\
\noindent {\sc Observaci\'on:} Los pasos {\bf 5}, {\bf 6} y {\bf 7} prueban un
resultado intermedio necesario para la demostraci\'on. No constituye una
prueba auxiliar independiente pues se considera que el resultado es muy
espec\'{\i}fico.

\subsubsection*{(c) $\vdash A\equiv\neg\neg A$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & \ & $\vdash$ & $(\neg\neg\neg A\then\neg A)\then((\neg\neg\neg A\then A)\then\neg\neg A)$ & {\bf L3} \\
2. & \ & $\vdash$ & $\neg\neg\neg A\then\neg A$ & Ejercicio (a) \\
3. & \ & $\vdash$ & $(\neg\neg\neg A\then A)\then\neg\neg A$ & MP (2,1) \\
4. & \ & $\vdash$ & $A\then(\neg\neg\neg A\then A)$ & {\bf L1} \\
5. & \ & $\vdash$ & $A\then\neg\neg A$ & Lema 2.3.4 (4,3) \\
6. & \ & $\vdash$ & $\neg\neg A\then A$ & Ejercicio (a) \\
7. & \ & $\vdash$ & $A\equiv\neg\neg A$ & Ejercicio (b) (5,6) \\
\end{tabular*}

\subsubsection*{(d) $\vdash (A\then B)\then(\neg B\then\neg A)$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & \ & $\vdash$ & $(\neg\neg A\then\neg\neg B)\then((\neg\neg A\then \neg B)\then\neg A)$ & {\bf L3} \\
2. & \ & $\vdash$ & $\neg\neg A\then A$ & Ejercicio (a) \\
3. & $A\then B$ & $\vdash$ & $A\then B$ & Def. Deducci\'on \\
4. & $A\then B$ & $\vdash$ & $\neg\neg A\then B$ & Lema 2.3.4 (2,3) \\
5. & \ & $\vdash$ & $B\then\neg\neg B$ & Prueba Aux. (1) \\
6. & $A\then B$ & $\vdash$ & $\neg\neg A\then\neg\neg B$ & Lema 2.3.4 (4,5) \\
7. & $A\then B$ & $\vdash$ & $(\neg\neg A\then\neg B)\then\neg A$ & MP (6,1) \\
8. & \ & $\vdash$ & $\neg B\then(\neg\neg A\then\neg B)$ & {\bf L1} \\
9. & $A\then B$ & $\vdash$ & $\neg B\then\neg A$ & Lema 2.3.4 (8,7) \\
10. & \ & $\vdash$ & $(A\then B)\then(\neg B\then\neg A)$ & TD (9) \\
\end{tabular*}

\subsubsection*{(e) $\neg(A\then\neg B) \vdash B$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & \ & $\vdash$ & $(\neg B\then\neg(A\then\neg B))\then ((\neg B\then(A\then\neg B))\then B)$ & {\bf L3} \\
2. & \ & $\vdash$ & $\neg(A\then\neg B)\then(\neg B\then\neg(A\then\neg B))$ & {\bf L1} \\
3. & $\neg(A\then\neg B)$ & $\vdash$ & $\neg(A\then\neg B)$ & Def. Deducci\'on \\
4. & $\neg(A\then\neg B)$ & $\vdash$ & $\neg B\then\neg(A\then\neg B)$ & MP (3,2) \\
5. & $\neg(A\then\neg B)$ & $\vdash$ & $(\neg B\then(A\then\neg B))\then B$ & MP (1,4) \\
6. & \ & $\vdash$ & $\neg B\then(A\then\neg B)$ & {\bf L1} \\
7. & $\neg(A\then\neg B)$ & $\vdash$ & $B$ & MP (6,5) \\
\end{tabular*}

\subsubsection*{(f) Si $A,B \vdash C$, entonces $\neg(A\then\neg B) \vdash C$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & $\neg(A\then\neg B)$ & $\vdash$ & $A$ & Prueba Aux. (2) \\
2. & $\neg(A\then\neg B)$ & $\vdash$ & $B$ & Ejercicio (e) \\
3. & $A,B$ & $\vdash$ & $C$ & Hip\'otesis \\
4. & $\neg(A\then\neg B)$ & $\vdash$ & $C$ & Transitividad (1,2,3) \\
\end{tabular*}

\subsubsection*{(g) $A \vdash \neg\neg(A\then\neg B)\then\neg B$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & $\neg\neg(A\then\neg B)$ & $\vdash$ & $\neg\neg(A\then\neg B)$ & Def. Deducci\'on \\
2. & \ & $\vdash$ & $\neg\neg(A\then\neg B)\then(A\then\neg B)$ & Ejercicio (a) \\
3. & $\neg\neg(A\then\neg B)$ & $\vdash$ & $A\then\neg B$ & MP (1,2) \\
4. & $A$ & $\vdash$ & $A$ & Def. Deducci\'on \\
5. & $A,\neg\neg(A\then\neg B)$ & $\vdash$ & $\neg B$ & MP (4,3) \\
6. & $A$ & $\vdash$ & $\neg\neg(A\then\neg B)\then\neg B$ & TD (5) \\
\end{tabular*}

\subsubsection*{(h) Si $\vdash P\then A$ y $\vdash P\then B$, entonces $\vdash P\then \neg(A\then\neg B)$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & $P$ & $\vdash$ & $P$ & Def. Deducci\'on \\
2. & \ & $\vdash$ & $P\then A$ & Hip\'otesis \\
3. & $P$ & $\vdash$ & $A$ & MP (1,2) \\
4. & \ & $\vdash$ & $P\then B$ & Hip\'otesis \\
5. & $P$ & $\vdash$ & $B$ & MP (1,4) \\
6. & $A,B$ & $\vdash$ & $\neg(A\then\neg B)$ & Ejercicio (b) \\
7. & $P$ & $\vdash$ & $\neg(A\then\neg B)$ & Transitividad (3,5,6) \\
8. & \ & $\vdash$ & $P\then\neg(A\then\neg B)$ & TD (7) \\
\end{tabular*}

\subsubsection*{(i) $\vdash \neg A\then(A\then B)$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & \ & $\vdash$ & $\neg A\then(\neg B\then\neg A)$ & {\bf L1} \\
2. & \ & $\vdash$ & $(\neg B\then\neg A)\then(A\then B)$ & Lema 2.3.5 \\
3. & \ & $\vdash$ & $\neg A\then(A\then B)$ & Lema 2.3.4 (1,2) \\
\end{tabular*}

\section*{Pruebas Auxiliares}

\subsubsection*{(1) $\vdash A\equiv\neg\neg A$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & \ & $\vdash$ & $(\neg\neg\neg A\then\neg A)\then((\neg\neg\neg A\then A)\then\neg\neg A)$ & {\bf L3} \\
2. & \ & $\vdash$ & $\neg\neg\neg A\then\neg A$ & Ejercicio (a) \\
3. & \ & $\vdash$ & $(\neg\neg\neg A\then A)\then\neg\neg A$ & MP (2,1) \\
4. & \ & $\vdash$ & $A\then(\neg\neg\neg A\then A)$ & {\bf L1} \\
5. & \ & $\vdash$ & $A\then\neg\neg A$ & Lema 2.3.4 (4,3) \\
\end{tabular*}

\subsubsection*{(2) $\neg(A\then\neg B) \vdash A$}
\begin{tabular*}{170mm}{lrcl@{\extracolsep{\fill}}r}
1. & $\neg(A\then\neg B)$ & $\vdash$ & $\neg(A\then\neg B)$ & Def. Deducci\'on \\
2. & \ & $\vdash$ & $(\neg A\then\neg(A\then\neg B))\then((\neg A\then(A\then\neg B))\then A)$ & {\bf L3} \\
3. & \ & $\vdash$ & $\neg(A\then\neg B)\then(\neg A\then\neg(A\then\neg B))$ & {\bf L1} \\
4. & $\neg(A\then\neg B)$ & $\vdash$ & $\neg A\then\neg(A\then\neg B)$ & MP (1,3) \\
5. & $\neg(A\then\neg B)$ & $\vdash$ & $(\neg A\then(A\then\neg B))\then A$ & MP (4,2) \\
6. & \ & $\vdash$ & $\neg A\then(\neg\neg B\then\neg A)$ & {\bf L1} \\
7. & \ & $\vdash$ & $(\neg\neg B\then\neg A)\then(A\then\neg B)$ & Lema 2.3.5 \\
8. & \ & $\vdash$ & $\neg A\then(A\then\neg B)$ & Lema 2.3.4 (6,7) \\
9. & $\neg(A\then\neg B)$ & $\vdash$ & $A$ & MP (8,5) \\
\end{tabular*}

\end{document}