Wikipedia:School and university projects/Discrete and numerical mathematics/Learning plan

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 * style="padding:2px;" | Introduction
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Glossary of abbreviations

 * CC BY, CC BY-­SA, CC BY­-ND, CC BY­-NC, CC BY­-NC­-SA, CC BY­-NC­-ND: Creative Commons public licenses (https://creativecommons.org/licenses/).
 * GratisOA: Gratis Open Access (http://legacy.earlham.edu/~peters/fos/overview.htm) (https://cyber.harvard.edu/hoap/Open_Access_(the_book)).
 * ARR: All Rights Reserved.

Ex ante I: Mathematics and Computing
(Just as an appetiser)

— Harangues

 * Q&amp;A With Nine Great Programmers; 2006, by Jaroslaw &quot;sztywny&quot; Rzeszótko (AKA &quot;Stiff&quot;) (in English) (just one Spanish translation: 10 preguntas a los más grandes programadores.)
 * You Don't Need Math Skills To Be A Good Developer But You Do Need Them To Be A Great One; March 24, 2010, by Alan Skorkin.

— Discrete mathematics

 * What is discrete mathematics and why is it so important for computer science?; Quora.

— Algorithms

 * What algorithms should I know to become a good programmer?; Quora.

— Wikipedia (Starting points)

 * Discrete mathematics (© CC BY-SA)
 * Portal:Mathematics (© CC BY-SA)
 * Portal:Technology (© CC BY-SA)

— Others:

 * Progopedia (© GFDL &amp; BSD-style).

Ex ante III: Graduates in computing (Spain)

 * Colegio Profesional de Ingenieros en Informática de Extremadura (CPIIEX) (Gratis Access) (In Spanish).
 * Código deontológico (Gratis Access) (In Spanish).
 * Consejo General de Colegios Profesionales de Ingeniería en Informática (CCII) (Gratis Access) (In Spanish).

Ex ante IIII: Pre-university mathematical literacy

 * Calvo Jurado, Carmen, Cristina Gutiérrez, Concepción Marín, Pedro Martín, Rodrigo Martínez, Pablo Monfort, José Navarro e Ignacio Ojeda; (2012) HEDIMA: Temas básicos de Análisis Matemático, Álgebra Lineal y Geometría. Grupo HEDIMA, Departamento de Matemáticas, UEX. (© ARR). (In Spanish).
 * Calle David, Unicoos. (© gratis OA). (In Spanish).
 * Casaravilla Gil, A. et al. (2010) Apoyo para la preparación de los estudios de Ingeniería y Arquitectura: Matemáticas (Preparación para la Universidad). OCW UPM. (© CC BY-NC-SA). (In Spanish).
 * Díaz Hernández, A. M. et al. (2010) Curso 0 de Matemáticas. OCW UNED. (© CC BY-NC-ND). (In Spanish).
 * EDU365CAT (2013) (Inici &gt; Batxillera &gt; Ciència i tecnologia &gt; Matemàtiques). Generalitat de Catalunya. (© gratis OA). (In Catalan).
 * Gobernación de Antioquia (2015) 100 Problemas que todo bachiller debe entender y resolver. (© gratis OA). (In Spanish).
 * González Ortíz, F. J. (2006) Proyecto MATEX. Gobierno de Cantabria. (© gratis OA). (In Spanish).
 * Grupo Lemat (2009) Proyecto Lemat. Universidad de Cantabria, Gobierno de Cantabria y Gobierno de España. (© gratis OA). (In Spanish).
 * MECD. Proyecto Descartes. (© CC BY-NC-ND). (In Spanish).
 * WikiDidácTICa (2013) Matemáticas en Bachillerato. Gobierno de España (Intef). (© gratis OA). (In Spanish).

Ex ante V: General motivation
'Listening to my father during those early years, I began to realise how important it was to be an enthusiast in life. He taught me that if you are interested in something, no matter what it is, go at it full speed ahead. Embrace it with both arms, hug it, love it and above all become passionate about it. Lukewarm is no good. Hot is no good, either. White hot and passionate is the only thing to be.'

Roald Dahl : My Uncle Oswald. London, England (GB-ENG), UK: Penguin Books Limited, 1980, p. 37.


 * Andoni Alonso Puelles: Apuntes para una historia social de la computación. 2008. (© CC BY-SA). (In Spanish; the debate starts at 57:43).
 * Arthur Benjamin: Teach statistics before calculus! 2009. (© CC BY-NC-SA). (In English, with Spanish subtitles).
 * Pedro Cavadas:Address by Dr. Pedro Cavadas on the occasion of his investiture as Doctor 'honoris causa' of the Valencian International University (VIU), 00:26:00-01:01:33. (© ARR, gratisOA). (In Spanish).
 * Comunes Collective: Hack for your rights. © CC BY-SA.
 * © CC BY-SA.
 * Ana María Cruz Martín: Los ingenieros y Cervantes . The Weekend Archaeologist. 2017. (© ARR, gratisOA). (In Spanish).
 * Maria Popova: 10 Rules for Students, Teachers, and Life by John Cage and Sister Corita Kent. 2012. Brain Picking. (© ARR, gratisOA). (In Spanish: Jesús Fernández: '10 Reglas para estudiantes y profesores | Corita Kent | John Cage'. 2017. Deviolines. (© ARR, gratisOA).)
 * David Maddox (Director): Alternative Math. 2017. (Short film). Produced by Ideaman Studios. (© ARR, gratisOA). (In English, with subtitles in several languages, including English and Spanish).
 * Tim Minchin: Tim Minchin UWA Address 2013 (© ARR, gratisOA). (In English, with English subtitles). (Another video on YouTube includes an independent translation into Spanish). (In addition, there exists a transcript [in English] of Tim Minchin's address is available).
 * (© ARR, gratisOA). (To watch a video about the sports life of Katelyn Ohashi, click here (© ARR, gratisOA)).


 * We keep picking berries... — (© ARR, gratisOA).—.

Ex ante VI: Specific motivation
— Films
 * Hanna Fry: The mathematics of love. 2014. (© CC BY-NC-SA). (In English, with Spanish subtitles).
 * Medialab-Prado, Madrid: Jugando con números. 2010-2011. (Mainly in Spanish).
 * Eduardo Sáenz de Cabezón: Math is forever. 2014. (© CC BY-NC-SA). (In Spanish, with English subtitles).
 * Cristóbal Vila: Nature by numbers, 2010, (© ARR, gratisOA) — a short movie inspired on numbers, geometry and nature —. The theory behind the movie: http://www.etereaestudios.com/docs_html/nbyn_htm/about_index.htm (© ARR, gratisOA). (More work by Cristóbal Vila, at his gallery online: Etérea).
 * Dylan Selterman: Why I give my students a 'tragedy of the commons' extra credit challenge. (The Washington Post, 2015). (© gratisOA). (En inglés / In English). Relativo a esta noticia, en español, por ejemplo, en / Related to this news, in Spanish, for example, at: Europa Press: La pregunta extra para ganar puntos en un examen de este profesor te hará pensar.
 * (Alternate URL)
 * Morgan Matthews (dir.), X+Y (distributed in the United States as A Brilliant Young Mind), 2014 —X+Y (Nathan solves math problem; scene in English); X+Y (the same scene, subtitled in Spanish).
 * Robert Luketic (dir.), 21, 2008 —scene (in English) in which Ben Campbell resolves the Monty Hall Problem (the same scene in Spanish).
 * Detlev Buck (dir.), Die Vermessung der Welt, 2012, based on homonymous novel by Daniel Kehlmann (translated into English by Carol Brown Janeway, with the title 'Measuring the World'), on the life of Carl Friedrich Gauss, Alexander von Humboldt and Aimé Bonpland, scene of interest, in Spanish from minute 4:00 to 7:35 (the same scene, in German).
 * Álex de la Iglesia (dir.), The Oxford Murders, 2008, scene of interest, in Spanish (14:15-18:33) (the same scene in English).
 * Sam Buntrock (dir.), ReCURSION, 2014
 * Mathematics in Movies (web page by Oliver Knill, Department of Mathematics, Harvard University)

— Prove why it is so
 * «La tabla del 9», fragment of «Un día en el colegio», Las Aventuras de los Payasos, 1973 (from 4:25 to 7:25) (in Spanish).
 * Multiplying by 6, 7, 8 and 9 with your hands. Guiainfantil (click here).
 * (11-15)Multiplication Using hands (Explanation in the comments).


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— International

 * European University Association (EUA)
 * International Association of Universities (IAU)
 * Association des États Généraux des Étudiants de l'Europe (AEGEE)
 * Board of European Students of Technology (BEST) (es)
 * International Association of Student Affairs and Services (IASAS) (Also: IASAS-Global(@IASASGlobal)|Twitter)
 * Erasmus Student Network (ESN)
 * International Union of Students (IUS)
 * More: Student organizations  ·   University associations and consortia

— Spain: Institutions, organizations, associations

 * Asociación para la Transparencia Universitaria (ATU). © gratisOA. (In Spanish).
 * Confederación Estatal de Asociaciones de Estudiantes (CANAE). © gratisOA. (In Spanish).
 * Crue Universidades Españolas. © gratis OA. (In Spanish).
 * Consejo de la Juventud de España (CJE). © gratisOA. (In Spanish).
 * Coordinadora de Representantes de Universidades Públicas (CREUP). © gratisOA. (In Spanish).
 * Federación de Asociaciones de Estudiantes Progresistas del Estado Español, Ginés de los Ríos (FAEST). © gratisOA. (In Spanish).
 * Ministerio de Educación y Formación Profesional - Ministerio de Ciencia, Innovación y Universidades. © gratisOA. (In Spanish).
 * Red Universitaria de Asuntos Estudiantiles (RUNAE). © gratisOA. (In Spanish).
 * Sindicato de Estudiantes (SE). © gratisOA. (In Spanish).

— Spain: Legislation

 * University Student Statute. © gratisOA. (In Spanish).
 * Universities Organic Law. © gratisOA. (In Spanish).
 * University Professors Regime. © gratisOA.(University Professors Statute (Draft). © gratisOA.) (In Spanish).

<span style="color: rgb(0, 0, 0); font-style: italic;">University of Extremadura (Spain)

 * General regulations of the University of Extremadura. © gratis OA. (In Spanish).
 * University Defence Office of the University of Extremadura. © gratisOA. (In Spanish).
 * Student Council of the University of Extremadura. © gratisOA. (In Spanish).
 * Volunteerism UEX (In Spanish).

<span style="color: rgb(0, 0, 0); font-style: italic;">School of Technology (EPCC)

 * Official website. © gratisOA. (In Spanish).
 * Official blog. © gratisOA. (In Spanish).
 * Official Twitter. © gratisOA. (In Spanish).
 * Student Council of the EPCC. © gratisOA. (In Spanish).


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<span style="color: rgb(0, 0, 0); font-style: italic;">— Professor
<blockquote style="margin-top: 0em; margin-bottom: 0em;"> Juan Miguel León Rojas

Office: 1904/1/9 (according to the planimetry of Cáceres campus facilities and services: building [Civil Engineering premises]/floor/office) (you may consult the course programme (ficha12a) to find out where it is).

E-mail: jmleon&#x40; unex .es.

Office hours.

<span style="color: rgb(0, 0, 0); font-style: italic;">— Course description
<blockquote style="margin-top: 0em; margin-bottom: 0em;"> This course is a primer on discrete mathematics and its applications including a very short introduction to a few numerical methods.

UEX code: 501272.

<span style="color: rgb(0, 0, 0); font-style: italic;">— Rationale
<blockquote style="margin-top: 0em; margin-bottom: 0em;"> The recommendations included in the Computer Engineering Curricula 2016* and in the Computer Science Curricula 2013†, among others, have been considered.

Regarding Discrete Mathematics, the latter report identifies the following topics as the knowledge base for discrete structures (pp.76-81): to which we would add: On the other hand, we have to keep in mind that some of these topics are studied in other courses taught at the School of Technology: DS6, in Statistics (UEX 501270); DM2, in Linear Algebra (UEX 502382); DM3, in Introduction to Programming (UEX 502304) and in Analysis and Design of Algorithms (UEX 501273); DS5, in Analysis and Design of Algorithms (UEX 501273) and in Data Structures and Information (UEX 501271), although from an algorithmic point of view.
 * (DS1) Functions, relations and sets,
 * (DS2) Basic logic,
 * (DS3) Proof techniques,
 * (DS4) Basics of counting,
 * (DS5) Graphs and trees, and
 * (DS6) Discrete probability,
 * (DM1) Algebraic structures,
 * (DM2) Matrices,
 * (DM3) Algorithms and complexity, and
 * (DM4) Basic number theory.

With respect to Numerical Calculus and in order to provide students with a sufficient introduction to the algorithms and methods for computing discrete approximations used to solving continuous problems, in terms of linear and non linear approaches to a problem, we identify as essential contents: On the other hand, again, we have to keep in mind that some of these topics are studied in other courses taught at the School of Technology: NC2, in Linear Algebra (UEX 502382); NC3, in Statistics (UEX 501270) (with regard to regression).
 * (NC1) Roots of Equations,
 * (NC2) Linear Algebraic Equations, and
 * (NC3) Curve Fitting (regression and interpolation).

With all this in mind and meeting all the essential requirements of the academic program (ficha12a), 60 hours are programmed as can be seen in a synthetic way in the course outline and scheduled in the tentative course outline (chronogram for the 2019-2020 academic year). <hr style="width:25%;">
 * https://www.computer.org/cms/Computer.org/professional-education/curricula/ComputerEngineeringCurricula2016.pdf

† https://www.acm.org/education/CS2013-final-report.pdf

<span style="color: rgb(0, 0, 0); font-style: italic;">— Course objectives
<blockquote style="margin-top: 0em; margin-bottom: 0em;"> After taking this course students should have reached the following objectives:


 * Targets: Representation, formulation, abstraction, modelling, verification and generalization.


 * General: Acquire scientific culture and mathematical culture in particular. Enhance reflective and creative attitudes. Enhance skills and abilities of analysis, search, discovery, verification and generalization. Promote the development and enhancement of problem-solving skills and of positive attitudes towards mathematical, analytical and concrete critical thinking. Be prepared for independent, critical study and assessment of elementary academic and informative publications about the topics covered in the course. Develop the capacity for lifelong learning.


 * Common: Enhance the ability to develop strategies for problem solving and decision making. Increase the ability to interpret the results obtained. Increase the rigor in the arguments and develop the reading and writing skills, the ability to use information and the capacity to make written or oral presentation of ideas and reasoning.


 * Specific for themes 1 (Fundamentals) and 2 (Number Theory): Enhance the ability to understand and use the logical-mathematical language. Develop the capacity for abstraction through the construction of logical-mathematical arguments. Enhance the capacity of logical-mathematical reasoning in its deductive, inductive, abductive and algorithmic types.


 * Specific for themes 3 (Combinatorics) and 4 (Difference Equations): Enhance the capacity of logical-mathematical reasoning in its inductive, algorithmic and recursive types. Enhance the ability to count.

<span style="color: rgb(0, 0, 0); font-style: italic;">— Prerequisites
<blockquote style="margin-top: 0em; margin-bottom: 0em;"> Although in respect of scientific knowledge, it has no particular prerequisites, some prior background in maths (mainly in algebra, calculus and probability) and computing (mainly in programming) is welcomed but in no way presupposed. Regarding English language, it may be desirable that you are at a intermediate conversational level, e.g. at least as skilled as an independent (self-reliant) user (level B) according to the Common European Framework of Reference for Languages*. You might find out your English level taking this free online English test and then you might improve your knowledge of the English language, for instance, practising your English skills at your level, and many more things available on these pages by the British Council (Prince of Asturias Award for Communication and Humanities 2005).

<hr style="width:25%;">
 * Please keep in mind that it is enough to know the English language at a CEFR B1 level to apply for British citizenship or to settle in the UK and at a CEFR B2 level to study in the UK at a degree level or above.

<span style="color: rgb(0, 0, 0); font-style: italic;">— Course program
<blockquote style="margin-top: 0em; margin-bottom: 0em;">

Academic year 2019-2020

 * In Spanish.
 * In English.

Academic year 2018-2019

 * In Spanish.
 * In English.

<span style="color: rgb(0, 0, 0); font-style: italic;">— School hours
<blockquote style="margin-top: 0em; margin-bottom: 0em;">
 * Class meetings and planners.

<span style="color: rgb(0, 0, 0); font-style: italic;">Textbook
For the discrete mathematics part of the course, students are encouraged to use the following book as a textbook :


 * Rosen, Kenneth H. (2012). Discrete Mathematics and its Applications (7th edition) (USA edition). New York: McGraw­Hill. ISBN 978­-0­-07­-338309­-5 © ARR. http://www.mheducation.com/highered/product.M0073383090.html?searchContext=discrete+mathematics

(However, its eighth edition is already available — 2019, http://highered.mheducation.com/sites/125967651x/information_center_view0/index.html).

As this book cover the vast majority of the material of the course — which, incidentally, corresponds to what is currently taught in hundreds of universities in the field of discrete mathematics —, students are encouraged to adopt and study it. Rosen's book is both a textbook and a workbook with lots of exercises and practical cases (computer projects, computations and explorations). It is even a guidebook including suggested readings, Despite its encyclopaedic spirit, it is also a handbook including lists of key terms and results and review questions.

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Companion website
In addition, it has a companion website : http://www.mhhe.com/rosen.

For instance, you can download a complete set of lecture slides: http://highered.mheducation.com/sites/0073383090/student_view0/lecture_powerpoint_slides.html

Please be aware that:
 * There is a further international edition, the Global edition (2013, adapted by Kamala Krithivasan, ISBN 978­-0­-07­-131501­2, © ARR, companion website: http://www.mhhe.com/rosenGE), that although it is also a 7th edition, it includes new topics and the exercises are in different order.
 * As you know, the content of newer editions are usually updated and improved versions of the content of older editions, occasionally including new content, so it is highly recommendable that, as far as possible, you read and study the new versions of the sections and exercises. However, within that aim of making continuous improvements, some contents may have been removed so it is also important to take a look to the older and newer editions, therefore keep mainly in mind these editions (even though only the seventh is highlighted):
 * fifth (2003, ISBN 978-0-07-242434-8, © ARR, companion website: http://www.mhhe.com/math/advmath/rosen/r5/) — the fifth has been the last edition translated into Spanish: (2010, Matemática discreta y sus aplicaciones (5th ed.), Madrid: McGraw-Hill/Interamericana de España, S. A. U. ISBN 84-481-4073-7 —,
 * sixth (2006, ISBN 978-0-07-288008-3, © ARR, companion website: http://highered.mheducation.com/sites/0072880082/information_center_view0/index.html),
 * seventh (2012, ISBN 978­-0­-07­-338309­-5, © ARR, companion website: http://www.mhhe.com/rosen),
 * seventh global (2013, ISBN 978­-0­-07­-131501­2, © ARR, companion website: http://www.mhhe.com/rosenGE), and
 * eighth (2019, ISBN 978-1-259-67651-2, © ARR, companion website: http://highered.mheducation.com/sites/125967651x/information_center_view0/index.html).

All these companion websites include, among other material and resources, interactive demos, self assessments and extra examples.

<span style="color: rgb(0, 0, 0); font-style: italic;">Companion books describing solutions for each of the proposed exercises
On the other side, this book is accompanied by books describing solutions for each of the proposed exercises, for instance, for the 5th and 7th US editions:
 * Kenneth H. Rosen, Jerrold Grossman, Student's Solutions Guide (odd exercises) (5th ed., 2003, ISBN 0-07-247477-7) (7th ed., 2012, ISBN 978-0-07-735350-6, © ARR.
 * Kenneth H. Rosen, Jerrold Grossman, Instructor's Resource Guide (even exercises) (5th ed., 2003, ISBN 0-07-247480-7) (7th ed., 2012, ISBN 978-0-07-735349-0), © ARR.

<span style="color: rgb(0, 0, 0); font-style: italic;">Companion books exploring and discussing contents and solutions to the proposed 'computer projects' and 'computations and explorations'
And also by the supplementary books exploring and discussing contents and solutions to the 'computer projects' and 'computations and explorations' sections, from the 7th US edition:
 * Daniel R. Jordan, Exploring Discrete Mathematics using Maple, 2nd edition, http://highered.mheducation.com/sites/0073383090/student_view0/exploring_discrete_mathematics_using_maple.html, © ARR.
 * Daniel R. Jordan, Exploring Discrete Mathematics using Mathematica, 1st edition, http://highered.mheducation.com/sites/0073383090/student_view0/exploring_discrete_mathematics_using_mathematica.html, © ARR.

<span style="color: rgb(0, 0, 0); font-style: italic;">Companion book about applications of discrete mathematics
Finally, you can download another supplement, one book about applications of discrete mathematics, last edition, paired with Rosen's book 6th edition, in any case for you to study it once you finish the course, except for the chapters that are of interest to it:
 * John G. Michaels, Kenneth H. Rosen Applications of Discrete Mathematics, 2007, http://highered.mheducation.com/sites/0072880082/student_view0/applications_of_discrete_mathematics.html, http://highered.mheducation.com/sites/0073383090/student_view0/applications_of_discrete_mathematics.html, http://highered.mheducation.com/sites/0071315012/student_view0/applications_of_discrete_mathematics.html, ISBN 978-0-07-041823-3.

<span style="color: rgb(0, 0, 0); font-style: italic;">— Numerical calculus
<blockquote style="margin-top: 0em; margin-bottom: 0em;"> For the short numerical calculus part of the course, students are encouraged to use the following book as a textbook :


 * Chapra, Steven C., & Canale, Raymond P. (2006) Numerical Methods for Engineers (5th international edition). New York: The McGraw­Hill Companies, Inc. ISBN 0-07­-124429­-8. © ARR.

Companion website: http://www.mhhe.com/engcs/general/chapra/

Please be aware that:
 * Although we use the fifth international edition, note that the seventh edition of this book has been published (http://www.mheducation.com/highered/product.M007339792X.html).

At the UEX library, you have electronic access to the 6th edition, in Spanish: http://0-www.ingebook.com.lope.unex.es/ib/NPcd/IB_BooksVis?cod_primaria=1000187&codigo_libro=4250

<span style="color: rgb(0, 0, 0); font-style: italic;">— To find out more, while course is running (or once it is finished)
<blockquote style="margin-top: 0em; margin-bottom: 0em;"> In addition to the references that appear in the course outline and in the academic program (ficha12a), and to those that can be mentioned in the classroom (large group and seminar/lab meetings) or posted on the talk page of the learning plan or at the UEX online campus in the course private forum, and to those that are referenced in the 13 question selections that are used throughout the course, you should consider:
 * as the course develops:
 * WP+: Paths on Wikipedia, bibliography (theory and proposed and solved exercises), multimedia and even more;
 * © ARR;
 * once the course is finished:
 * John G. Michaels, Kenneth H. Rosen Applications of Discrete Mathematics, 2007, http://highered.mheducation.com/sites/0072880082/student_view0/applications_of_discrete_mathematics.html, http://highered.mheducation.com/sites/0073383090/student_view0/applications_of_discrete_mathematics.html, http://highered.mheducation.com/sites/0071315012/student_view0/applications_of_discrete_mathematics.html, ISBN 978-0-07-041823-3.
 * © gratisOA.

<span style="color: rgb(0, 0, 0); font-style: italic;">— Communicating

 * Course private forum at the UEX online campus
 * Talk page of the learning plan

<span style="color: rgb(0, 0, 0); font-style: normal;">Participating in MATDIN is an optional continuous evaluation out-of-class practical activity which is worth a try for contributing to your personal developmentand because it might help you boost your course grade; furthermore, if you are thinking of grading with distinction ['matrícula de honor', in Spanish], your participation in this project is strongly recommended. Find out more on its descriptive web page and in the welcome message to the course.
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<span style="color: rgb(0, 0, 0); font-style: italic;">— On the English-language Wikipedia

 * School and university projects/Discrete and numerical mathematics, and
 * Welcome to the course and to its learning plan strengthened by the English Wikipedia (academic year 2019-2020) (above all regarding the assessment of your work in the course).

<span style="color: rgb(0, 0, 0); font-style: italic;">— Communicating
<blockquote style="margin-top: 0em; margin-bottom: 0em;"> To keep track of the project you have joined to, please follow the recommendations on its descriptive page, particularly on 'The basics' subsection.

<span style="color: rgb(0, 0, 0); font-style: italic;">— Equivalent project on the Spanish-language Wikipedia

 * <span style="color: rgb(0, 0, 0); font-style: italic;">(Only if you take the course in Spanish).
 * Wikipedia:Proyecto educativo/Matemática discreta y numérica, and
 * Bienvenida a la asignatura y a su plan de aprendizaje apoyado en la Wikipedia en español (año académico 2019-2020) (above all regarding the assessment of your work in the course).




 * style="padding:2px;" | <h2 id="pamdan-ede-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:110%; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Contents and learning paths on Wikipedia<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" />

<div style="margin-left: 48%; text-align: right; color: rgb(0, 102, 0); background-color: rgb(255, 255, 223); background-image: none; padding: 0px; border-top: 4px solid rgb(0, 102, 0); border-bottom: 4px solid rgb(0, 102, 0); vertical-align: middle;"> 'Do I contradict myself? Very well then I contradict myself, (I am large, I contain multitudes.)'

Walt Whitmann (1819-1892): Song of Myself (in Leaves of Grass, 1855)
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<span style="color: rgb(0, 0, 0); font-style: italic;">Considerations

 * Using Wikipedia for mathematics self-study
 * Proofs

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<span style="color: rgb(0, 0, 0); font-style: italic;">Course outline

 * Theme 1: FUNDAMENTALS (16 h LG and 5 h SL) Must-study: Rosen's book 7th ed. USA (§ 1.1-1.8, 2.1-2.3, 2.5, 9.1, 9.3, 9.5-9.6); Rosen's book 7th Global ed. (§ 1.7, 12.1-12.4); and León-Rojas, J. M., Notas incompletas de clase (Incomplete class notes) (in Spanish). Must-do: Every recommended exercise in every section covered for this theme in the calendar of activities; Question selections no. 1 to 5; Sample exam questions for the theme 1; Preparatory exam questions for the theme 1; and Real exam questions for the theme 1.
 * Contents: ► Logic: propositions, propositional equivalences, predicates and quantifiers, nested quantifiers, translating English statements into the language of logic and vice versa, valid arguments and rules of inference; direct and indirect proofs, verification and refutation strategies (truth tables, proof by contraposition, proof by contradiction, normal forms, natural deduction, semantic tableaux). ► Sets: concepts and definitions, cardinality and power set; relations (membership, inclusion and equality), operations (union, intersection, complement, difference, symmetric difference) and properties, partition, cardinality of the union, cartesian product. ► Maps and functions: types (injective, surjective and bijective), monotony, representation (cartesian, arrow-set, matrix-based and graph-based), composition, inverse; multiset. ► Relations: properties, representing relations using matrices and graphs; equivalence relations, equivalence classes and partitions; tolerance relations; orderings, Hasse diagrams; preference relations. ► Cardinality: infinite sets, countability, Cantor's diagonal argument, Cantor's theorem and the continuum hypothesis. ►  Induction: weak, strong and structural; well ordering. ► Algebraic structures: magma, semigroup, monoid, group, ring, integral domain, field; homomorphism.
 * Seminars/Labs: ► [1]: Proofs and refutations, I; ► [2]: Proofs and refutations, II; ► [3]: Proofs and refutations, III; ► [4]: Induction and recursion; ► [5]: Cardinality and algebraic structures.
 * Connections: ...


 * Theme 2: NUMBER THEORY (9h LG and 3 h SL) Must-do: Every recommended exercise in every section covered for this theme in the calendar of activities; Question selections no. 6 to 8; Sample exam questions for the theme 2; Preparatory exam questions for the theme 2; and Real exam questions for the theme 2.
 * Contents: ► Divisibility and modular arithmetic: divisibility, division algorithm, modular arithmetic. ► Primes and greatest common divisor: integer representations, prime numbers and their properties, the fundamental theorem of arithmetic, conjectures and open problems about primes, greatest common divisor and least common multiple, the Euclidean algorithm, Bézout's theorem and the extended Euclidean algorithm. ► Solving congruences: linear congruences, Euler's φ function, the Chinese remainder theorem, Euler-Fermat's theorem, Fermat's little theorem, Wilson's theorem and Wolstenholme's theorem. ► Applications of congruences: cryptography. ► Divisibility rules: power residues, divisibility rules. ► Diophantine equations: linear equations, systems.
 * Seminars/Labs: ► [6]: Divisibility, modular arithmetic, primes, gcd and congruences; ► [7]: Diophantine and congruence equations, I; ► [8]: Diophantine and congruence equations, II.
 * Connections: ...


 * Theme 3: COMBINATORICS (8h LG and 3 h SL) Must-study: Rosen's book 7th ed. USA (§ 6.1-6.5) and Franco Brañas, J. R., Espinel Febles, M. C., Almeida Benítez, P. R. Manual de Combinatoria (in Spanish). Must-do: Every recommended exercise in every section covered for this theme in the calendar of activities; Question selections no. 9 to 11; Sample exam questions for the theme 3; Preparatory exam questions for the theme 3; and Real exam questions for the theme 3.
 * Contents: ► The basics of counting: the sum rule, the product rule, the subtraction rule (inclusion-exclusion principle) and the division rule; the pigeonhole principle and its generalization; binomial coefficients and identities; variations, permutations and combinations. ► Combinatorial proofs: bijective proofs and double counting proofs. ► Combinatorial modeling: 1st, sample selection and unit labelling with and without repetition; 2nd, grouping units (distribution, storage or placement of objects into recipients); 3rd, partitions of sets, and 4th, partitions of numbers.
 * Seminars/Labs: ► [9]: Combinatorics, I; ► [10]: Combinatorics, II; ► [11]: Combinatorics, III.
 * Connections: Game theory; Network science; Number theory; Topology.


 * Theme 4: DIFFERENCE EQUATIONS (8 h LG and 2 h SL) Must-study: Rosen's book 7th ed. USA (§ 2.4, 8.1-8.3) and Navas Ureña, J., Esteban Ruiz, F. J., Quesada Teruel, J. M. 'Tema 9: Ecuaciones y sistemas en diferencias' (in: Modelos matemáticos en biología. Teoría) (in Spanish). Must-do: Every recommended exercise in every section covered for this theme in the calendar of activities — also some exercises from Navas Ureña, J., Esteban Ruiz, F. J., Quesada Teruel, J. M. 'Tema 2: Modelos discretos II' [in: Modelos matemáticos en biología. Ejercicios resueltos y propuestos] [in Spanish] —; Question selections no. 12 to 13; Sample exam questions for the theme 4; Preparatory exam questions for the theme 4; and Real exam questions for the theme 4.
 * Contents: ► Linear difference equations: homogeneous and non-homogeneous; with constant coefficients; direct; simple or multiple; indirect: systems of linear difference equations. ► Linear discrete dynamical systems: population dynamics, linear discrete dynamical models, BIDE models, Markov chains. ► Solving equations numerically: method of successive approximations (fixed point iteration); secant method.
 * Seminars/Labs: ► [12]: Difference equations, I; ► [13]: Difference equations, II.
 * Connections: ...


 * High-contrast-input-keyboard.svg Optional seminars/labs, two editathons (whenever we have time, the first in the middle of the semester and the second at the end; anyway, the exact content of this personal-enhancement tasks will not be mandatory covered on any exam): ► [14]: Start of the optional reading and writing of a mathematical and computational, reflective, critical and analytical short essay of the text A. K. Dewdney (1993) The Tinkertoy Computer and other machinations. Chapter 17: Automated Math. New York: W. H. Freeman. ('Juegos de ordenador. De cómo un par de programas obtusos pasan por genios en los tests de inteligencia.' Investigación y Ciencia, No. 116, May 1986, pp. 94-98, Prensa Científica, S. A. [In Spanish]). ► [15]: Start of the optional reading and writing of a mathematical and computational, reflective, critical and analytical short essay about the Collatz conjecture and its 'visualization' (for example: Collatz Graph: All Numbers Lead to One, de Jason Davies). (See also the sandbox/workshop of this course).
 * Appendices (optional) (some complimentary or curious facts apart from the present programme, although related to it)
 * Contents: ► Graphs; Numerical calculus; Complimentary knowledge pills; Editathons.
 * Connections: ...


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<span style="color: rgb(0, 0, 0);">Theme 1.- Fundamentals
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Logic

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<span id="Logic_Key_concepts"> Key concepts
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— Propositional logic

See also: Conditioned disjunction, Indicative conditional and Logical constant. — Verification and rebuttal strategies, I
 * Proposition
 * Logical connective
 * Well-formed formula
 * Propositional formula
 * Truth value
 * Logical truth
 * False (logic)
 * Consistency
 * Truth function
 * Table of binary truth functions
 * Functional completeness (Functionally complete set of connectives; adequate set of connectives)
 * Logical equivalence

<ul> <li> Normal forms, I: </li></ul> <ul> <li> Normal forms, II: </li></ul> — Predicate logic
 * Truth table
 * Contraposition
 * Reductio ad absurdum (see also: Proof by contradiction)
 * Conjunctive normal form
 * Disjunctive normal form
 * Negation normal form
 * Algebraic normal form


 * Quantifier (linguistics)
 * Quantifier (logic)
 * Free variables and bound variables
 * Universal quantification
 * Existential quantification
 * Uniqueness quantification
 * Quantifier elimination

— Translating English statements into the language of logic and vice versa
 * Syllogism
 * (es; pt; ca)


 * Square of opposition
 * Donkey sentence

— Valid arguments and inference rules See also: — Direct and indirect proofs
 * Law of identity
 * Law of noncontradiction
 * Law of excluded middle
 * Argument
 * Premise
 * Logical consequence
 * Validity (logic)
 * Counterargument
 * Soundness
 * Logical reasoning
 * Deductive reasoning
 * Problem of deduction (es; ca)
 * Rule of inference
 * Inductive reasoning
 * Problem of induction
 * Abductive reasoning
 * Reasoning methods (es) and Cogency (es; pt; ca)
 * Stylistic device, Modes of persuasion, Argumentation theory, Fallacy, Enthymeme, (es; pt; fr; it; ca),  (pt; fr; it) and Antinomy;
 * Paradox;
 * Category:Reasoning.
 * Proof theory
 * Methods of proof (see also: Constructive proof, Proof by exhaustion, Counterexample and Proof by infinite descent)
 * Mathematical fallacy (i.e., invalid proof)

— Verification and rebuttal strategies, II — Verification and rebuttal strategies, III <ul> <li> Normal forms, III: </li></ul> — Some unusual situations in Logic
 * Natural deduction
 * Logic puzzle
 * MU puzzle
 * Knights and knaves (Questions about truthful and deceitful people)
 * Portia's caskets (external link: Bellos, Alex (2017). Can you solve it? The mystery of Portia's caskets. The Guardian)
 * Prenex normal form
 * Skolem normal form
 * Method of analytic tableaux (Semantic tableaux)
 * Infinite regress
 * Münchhausen trilemma
 * Liar paradox
 * Epimenides paradox
 * Definite description
 * Curry's paradox
 * See also: Paradox, List of paradoxes: Logic and More complimentary knowledge pills

<span id="Logic_Connections"> Connections
<hr style="margin-left: 0px; width:50%;"> — Automated reasoning

— Boolean algebra

— Diagrammatic reasoning
 * Canonical normal form
 * Simplifying Boolean functions
 * Karnaugh map
 * Quine–McCluskey algorithm

— Logic gates

<span id="Logic_Bibliography"> Bibliography: theory and proposed and solved exercises
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<span id="Logic_Software"> Software
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<span id="Logic_Multimedia"> Multimedia-icon.svg Multimedia
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<span id="Logic_See_also"> See also
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<span id="Logic_To_find_out_more"> To find out more
<hr style="margin-left: 0px; width:50%;"> <ol style="list-style-type:circle;"> <li>Portal:Mathematics</li> <li>Portal:Philosophy</li> <li>And more:</li> <ol style="list-style-type:circle;"> <li>Outline of logic</li> <li>Category:Concepts in logic</li> <li>WikiProject Logic</li> <li>Logic alphabet</li> <li>Metamath. © Public domain (with some exceptions)</li> <li>Equational logic; for instance, chapter 5 (Equational Logic: Part 1) from Backhouse, Roland, ''Program Construction. The Correct Way'', 2002.</li> </ol> <li>And even more:</li> <ol style="list-style-type:circle;"> <li>Index of logic articles</li> <li>List of logicians</li> </ol> </ol>


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Sets, relations and functions

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<span id="SetsRelationsFunctions_Key_concepts"> Key concepts
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 — Sets 

- Concepts and definitions
 * Naive set theory
 * Element (mathematics)
 * Extensional and intensional definitions
 * Universal set
 * Empty set
 * Cardinal number
 * Power set

- Relations
 * Set membership
 * Indicator function
 * Subset (inclusion relation)
 * Equality in set theory

- Set operations and representation
 * Algebra of sets
 * Carroll diagram
 * Euler diagram
 * Venn diagram
 * Union (set theory)
 * Intersection (set theory)
 * Disjoint sets
 * Disjoint union
 * Complement (set theory)
 * Set difference
 * Symmetric difference

- Properties
 * Of set union
 * Of set intersection
 * Cardinality of set union (see also: Inclusion–exclusion principle)
 * Of absolute complement and Of relative complement
 * Of set difference
 * Of symmetric difference
 * De Morgan's laws

- Partition and cover
 * Partition of a set
 * Cover (topology)

- Cartesian product
 * Cartesian product
 * n-ary Cartesian power
 * Ordered pair
 * Tuple

 — Relations 

- Concepts and definitions
 * Arity
 * n-ary relation
 * Logical matrix
 * Converse relation
 * Complementary relation

- Representation
 * Cartesian coordinate system
 * Mathematical correspondence (es; ref) (graphical representation using sets and arrows)
 * Adjacency matrix
 * Incidence matrix
 * Directed graph

- Outstanding properties
 * Reflexive relation
 * Irreflexive relation
 * Coreflexive relation
 * Quasi-reflexive relation
 * Symmetric relation
 * Asymmetric relation
 * Antisymmetric relation
 * Transitive relation
 * (es)
 * Quasitransitive relation
 * Euclidean relation
 * Serial relation
 * Trichotomous relation
 * Semiconnex relation (Complete relation)
 * Connex relation (Total, linear, or strong complete relation)
 * Circular relation
 * Circular relation

- Outstanding closures
 * Closure (mathematics)
 * Binary relation closures
 * Reflexive closure
 * Symmetric closure
 * Transitive closure

- Operations
 * Union of relations
 * Intersection of relations
 * Composition of relations

- Outstanding types

 - Equivalence relations 
 * Partial equivalence relation
 * Equivalence relation

 - Tolerance relations 
 * Dependency relation
 * Tolerance relation

 - Orderings 
 * Preorder
 * Total preorders
 * Partially ordered set
 * Chain (poset)
 * Ascending and descending chain conditions
 * Antichain
 * Lexicographical order
 * Product order
 * Covering relation
 * Transitive reduction
 * Hasse diagram
 * Total order
 * Dense order

 - Outstanding elements 
 * Greatest and least elements
 * Maximal and minimal elements
 * Upper and lower bounds
 * Infimum and supremum
 * Minimum and maximum
 * See also: Anexo:Extremos de conjunto acotado (some examples of extrema of bounded sets, on the Spanish Wikipedia)
 * Contour set
 * Boundedness in order theory
 * Lattice (order)

 - Indiference and preference relations 
 * Preference

 - Well order 
 * Well-order
 * Well-founded relation
 * Axiom of choice
 * Zermelo's well-ordering theorem (Note: The fact that every set may be well ordered, is what most people apparently also call 'Well-ordering principle' (WOP); ref., e.g.: Bagaria, Joan, 'Set Theory', The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2019/entries/set-theory/>, and so many others; what Wikipedia and Wolfram MathWorld and others call WOP is the particular case of being $$\mathbb{N}$$ a well-ordered set by <; in short, the current situation in the literature is that WOP has two meanings, either Zermelo's Theorem or $$\mathbb{N}$$ is well-orderd by <).
 * Zorn's lemma

 — Functions 

- Concepts and definitions
 * Mathematical correspondence (es; ref) (graphical representation using sets and arrows)
 * Domain of a function
 * Partial function
 * Total function (mapping or map [aplicación, in Spanish]; see: Map (mathematics))
 * Codomain
 * Image (mathematics)
 * Range (mathematics)
 * Restriction (mathematics)
 * Identity function
 * Discrete function (es)
 * Floor and ceiling functions
 * Graph of a function

- Outstanding types
 * Bijection, injection and surjection
 * Injective function
 * Surjective function
 * Bijection
 * Gallery of mathematical correspondences (es)

- Operations
 * Function composition
 * Inverse function
 * Inverse image (es; pt; fr; it) (see also: Image (mathematics) and Inverse function)

- Multisets
 * Multiset
 * Multiplicity (mathematics)

 — Paradoxes 
 * Russell's paradox (see also: Class (set theory))
 * Interesting number paradox

<span id="SetsRelationsFunctions_Connections"> Connections
<hr style="margin-left: 0px; width:50%;"> — Extensive systems

— Entity-relationship model

<span id="SetsRelationsFunctions_Bibliography"> Bibliography: theory and proposed and solved exercises
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<span id="SetsRelationsFunctions_Software"> Software
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<span id="SetsRelationsFunctions_Multimedia"> Multimedia-icon.svg Multimedia
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<span id="SetsRelationsFunctions_To_find_out_more"> To find out more
<hr style="margin-left: 0px; width:50%;"> <ol style="list-style-type:circle;"> <li>Portal:Mathematics</li> </ol>

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Cardinality, induction and recursion

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<span id="Cardinality_Key_concepts"> Key concepts
<hr style="margin-left: 0px; width:50%;"> — Cardinality

- Infinite sets
 * Potential infinity and actual infinity
 * Infinity
 * Paradoxes about infinity and infinitesimals
 * Galileo's paradox
 * Hilbert's paradox of the Grand Hotel
 * Ross–Littlewood paradox (see also: Supertask and Connections: Hypercomputability)
 * Equinumerosity
 * Finite set
 * Infinite set

- $$\mathbb{N}$$, $$\mathbb{Z}$$ and $$\mathbb{Q}$$ are countable sets
 * Peano axioms
 * Countably infinite set (Denumerable set)
 * Enumeration
 * Aleph-sub-zero

- $$\mathbb{R}$$ is an uncountable set
 * Cantor's diagonal argument
 * Uncountable set (Non-denumerable set)

- Cantor's Theorem and the Continuum Hypothesis
 * Cantor's theorem
 * Aleph number
 * (see also: Número cardinal [Cardinal number (set theory)] on the Spanish Wikipedia)
 * Transfinite number
 * Continuum hypothesis

 — Induction 
 * Weak induction
 * Strong induction
 * Structural induction

 — Recursion 
 * Recursion
 * Recursion (computer science)
 * Structural induction and recursive definitions (es)

<span id="Cardinality_Connections"> Connections
<hr style="margin-left: 0px; width:50%;"> — Hypercomputability


 * Supertask
 * Thomson's lamp
 * Zeno machine

<span id="Cardinality_Bibliography"> Bibliography: theory and proposed and solved exercises
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<span id="Cardinality_Software"> Software
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<span id="Cardinality_Multimedia"> Multimedia-icon.svg Multimedia
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<span id="Cardinality_To_find_out_more"> To find out more
<hr style="margin-left: 0px; width:50%;"> <ol style="list-style-type:circle;"> <li> Manuel José González Ortiz (2000). La hipótesis del continuo. Números 43-44, artículo n. 63 (pp. 315-318). Sociedad Canaria "Isaac Newton" de Profesores de Matemáticas y Nivola Libros y Ediciones S.L. Disponible en: http://www.sinewton.org/numeros/index.php?option=com_content&view=article&id=72:volumen-43-septiembre-2000&catid=35:sumarios-webs&Itemid=66</li>

<li>Continuum hypothesis. Encyclopedia of Mathematics. Disponible en: http://www.encyclopediaofmath.org/index.php?title=Continuum_hypothesis</li> <li>Koellner, Peter, "The Continuum Hypothesis", The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.). Disponible en: https://plato.stanford.edu/archives/win2016/entries/continuum-hypothesis/.</li> <li>The Continuum Hypothesis (la página web «oficial» de la hipótesis del continuo, en Infinity Ink [Nancy McGough, 1992]). Disponible en: http://www.ii.com/math/ch/</li> <li>Portal:Mathematics</li> <li>And more:</li> <ol style="list-style-type:circle;"> * Category:Set theory </ol> </ol>

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Algebraic structures

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<span id="AlgebraicStructures_Key_concepts"> Key concepts
<hr style="margin-left: 0px; width:50%;"> — Algebraic structures


 * Law of composition (es)
 * Binary operation
 * External binary operations (see also: External (mathematics))
 * Mathematical structure
 * Identity element (neutral element)
 * Absorbing element (annihilating element)
 * Inverse element
 * Additive inverse (opposite number, sign change, negation)
 * Multiplicative inverse (reciprocal)

— Magma, semigroup and monoid
 * Magma
 * Semigroup and Monoid
 * Quasigroup and Loop

— Group

See also: Cyclic permutation (Cycle), Symmetric group, Permutation group, Isometry group, Stirling numbers of the first kind and List of small groups
 * Group
 * Abelian group (commutative group)

— Ring, integral domain and field
 * Ring
 * Unit ring (es)
 * Commutative ring
 * Ring of sets
 * Prime ring
 * Boolean ring
 * Ordered ring
 * Domain (ring theory)
 * Integral domain
 * Unique factorization domain
 * Field
 * Ordered field
 * Vector space, Module and Algebra over a field

— Homomorphisms
 * Homomorphism
 * Group homomorphism
 * Isomorphism
 * Group isomorphism
 * Ring homomorphism

<span id="AlgebraicStructures_Connections"> Connections
<hr style="margin-left: 0px; width:50%;"> — Cryptography

— Category theory
 * Pairing-based cryptography

Live article: Baez, John (November 8, 2019). [https://elpais.com/elpais/2019/11/06/ciencia/1573042148_224789.html 'Qué es la teoría de categorías y cómo se ha convertido en tendencia'. El País (In: Café y Teoremas)] (In Spanish). — Coding theory
 * Olog

See also: Block code, Group code and Hamming code
 * Linear code

<span id="AlgebraicStructures_Bibliography"> Bibliography: theory and proposed and solved exercises
<hr style="margin-left: 0px; width:50%;">

<span id="AlgebraicStructures_Software"> Software
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<span id="AlgebraicStructures_Multimedia"> Multimedia-icon.svg Multimedia
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<span id="AlgebraicStructures_To_find_out_more"> To find out more
<hr style="margin-left: 0px; width:50%;"> <ol style="list-style-type:circle;"> <li>Portal:Mathematics</li> <li>And more:</li> <ol style="list-style-type:circle;"> <li>Multiplicative group of integers modulo n</li> </ol> </ol>
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<span style="color: rgb(0, 0, 0);">Theme 2.- Number theory
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Number theory

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<span id="NumberTheory_Key_concepts"> Key concepts
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— Divisibility and modular arithmetic


 * Arithmetic
 * Divisibility
 * Euclidean division
 * Modular arithmetic

— Primes and greatest common divisor
 * Prime number
 * Euclid's theorem
 * Primorial
 * Euclid's lemma
 * Fundamental theorem of arithmetic
 * Integer factorization
 * Category:Conjectures about prime numbers
 * Greatest common divisor
 * Coprime integers
 * Least common multiple
 * Euclidean algorithm
 * Bézout's identity
 * Extended Euclidean algorithm

— Solving congruences
 * Chinese remainder theorem
 * Solving congruences (es)
 * Modular multiplicative inverse (Inverse modulo m of an integer)
 * Solving linear congruences (es)
 * Method of successive substitution
 * Euler's totient function (Euler's φ)
 * Fermat-Euler theorem
 * Fermat's little theorem
 * Wilson's theorem
 * Wolstenholme's theorem
 * Table of congruences
 * Josephus problem (first contact)

— Applications of congruences - Cryptology
 * Hash function
 * Pseudorandom number
 * Cryptology
 * Cryptography
 * RSA public-key cryptosystem
 * RSA Factoring Challenge

— Divisibility rules
 * Numeral system
 * Power residue
 * Pascal's tape (es; fr)
 * Divisibility rule

— Diophantine equations <ul> <li> Some useful background: </li> <li> Diophantine equation</li> <li> System of linear Diophantine equations</li> </ul>
 * Algebraic equation
 * System of equations
 * Vieta's formulas
 * Ruffini's rule

— Paradoxes
 * Will Rogers phenomenon

<span id="NumberTheory_Bibliography"> Bibliography: theory and proposed and solved exercises
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<span id="NumberTheory_Software"> Software
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<span id="NumberTheory_Multimedia"> Multimedia-icon.svg Multimedia
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<span id="NumberTheory_See_also"> See also
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 * 1729 (number)
 * 5040 (number)

<span id="NumberTheory_To_find_out_more"> To find out more
<hr style="margin-left: 0px; width:50%;"> <ol style="list-style-type:circle;"> <li>Portal:Mathematics</li> <li>And more:</li> <ol style="list-style-type:circle;"> <li>Divisibility</li> <ol style="list-style-type:circle;"> <li>Division algorithm (Algorithms for division)</li> </ol> <li>Primality</li> <ol style="list-style-type:circle;"> <li>Quadratic residue</li> <li>Quadratic reciprocity</li> <li>Primality test</li> </ol> <li>Pseudo-random number generation</li> <ol style="list-style-type:circle;"> <li>List of random number generators</li> </ol> <li>Cryptography</li> <ol style="list-style-type:circle;"> <li>Highly totient number</li> <li>Highly composite number</li> <li>Smooth number</li> <li>Rough number</li> <li>Semiprime</li> <li>Elliptic curve cryptography</li> </ol> <li>List of prime numbers</li> <li>List of numbers</li> </ol> </ol>


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<span style="color: rgb(0, 0, 0);">Theme 3.- Combinatorics
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Combinatorics

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<span id="Combinatorics_Key_concepts"> Key concepts
<hr style="margin-left: 0px; width:50%;">

 — The basics of counting 

- Rules of sum, product, substraction and division
 * Rule of sum (addition principle)
 * Rule of product (multiplication principle)
 * Inclusion-exclusion principle (sieve principle or substraction rule)
 * Complement rule
 * Division rule

- Drawer principle and its generalisation
 * Pigeonhole principle (Dirichlet's box/drawer/shelf principle or Dirichlet's distribution principle)

- Binomial coefficients and identities
 * Factorial
 * Binomial coefficient
 * Binomial theorem
 * Pascal's triangle
 * Trinomial expansion
 * Multinomial theorem (and multinomial coefficient)
 * Pascal's rule

- (Ordinary) (i.e., without repetition) variations, permutations and combinations, and with repetition, and circular permutations
 * (Ordinary) variations (k-permutations of n, also called restricted partial permutations)
 * Variations with repetition
 * (Ordinary) permutations
 * Circular permutations
 * Permutations with repetition
 * (Ordinary) combinations
 * Combinations with repetition

- Counting with restrictions
 * Rook polynomial

 — Combinatorial proofs: 1st, bijective proofs; 2nd, double counting proofs; 3rd, using distinguished element, and 4th, using the inclusion-exclusion principle 


 * Bijective proof
 * Double counting (proof technique)
 * Method of distinguished element
 * Derangement (an example of proof, here)

 — Combinatorial modeling 
 * [Mathematical model
 * Combinatorial modelling
 * Twelvefold way

- I: Sample selection and unit labelling with and without repetition
 * Sample selection and unit labelling (combinatorics)

- II: Grouping units (distribution, storage or placement of objects into recipients) (Occupancy problems)
 * Distributions of objects into bins (es)
 * Stars and bars (combinatorics)

- III: Partition of sets '''· Catalan and Narayana numbers. Noncrossing partitions'''
 * Partitions of a set
 * Stirling numbers of the second kind
 * Bell numbers
 * Lah numbers
 * Noncrossing partition
 * Catalan number
 * Narayana number

- IV: Additive decompositions of numbers
 * Partitions of integers (additive decompositions)

 — Paradoxes 
 * Ellsberg paradox
 * Birthday problem
 * False positive paradox
 * Monty Hall problem
 * Simpson's paradox

<span id="Combinatorics_Bibliography"> Bibliography: theory and proposed and solved exercises
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<span id="Combinatorics_Software"> Software
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<span id="Combinatorics_Multimedia"> Multimedia-icon.svg Multimedia
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<span id="Combinatorics_See_also"> See also
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 * Kirkman's schoolgirl problem
 * Magic square
 * Latin square
 * Trinomial triangle
 * Permutation generation methods (ref)
 * Heap's algorithm
 * Steinhaus–Johnson–Trotter algorithm (and its relation to change ringing).

<span id="Combinatorics_To_find_out_more"> To find out more
<hr style="margin-left: 0px; width:50%;"> <ol style="list-style-type:circle;"> <li>Portal:Mathematics</li> <li>And more:</li> <ol style="list-style-type:circle;"> <li>Generating functions</li> <li>Examples of generating functions</li> </ol> </ol>
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<span style="color: rgb(0, 0, 0);">Theme 4.- Difference equations
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Finite difference equations (recurrence relations)

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<span id="RecurrenceRelations_Key_concepts"> Key concepts
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Some useful previous concepts: Recursive definition, Recursion and Recursion (computer science)  — Linear difference equations 

 — Linear discrete dynamical systems  - Population dynamics - Linear discrete dynamical models - BIDE models - Markov chains  — Solving equations numerically 
 * Recurrence relation (difference equation)
 * Solving homogeneous linear recurrence relations with constant coefficients
 * Solving non-homogeneous linear recurrence relations with constant coefficients
 * Population dynamics
 * Dynamical system
 * Matrix population models
 * Markov chain


 * Fixed-point iteration
 * Secant method

<span id="RecurrenceRelations_Connections"> Connections
<hr style="margin-left: 0px; width:50%;"> — Computational complexity

- Analysis of algorithms

<span id="RecurrenceRelations_Bibliography"> Bibliography: theory and proposed and solved exercises
<hr style="margin-left: 0px; width:50%;">

<span id="RecurrenceRelations_Software"> Software
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<span id="RecurrenceRelations_Multimedia"> Multimedia-icon.svg Multimedia
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<span id="RecurrenceRelations_To_find_out_more"> To find out more
<hr style="margin-left: 0px; width:50%;"> <ol style="list-style-type:circle;"> <li>Integer sequences</li> <li>List of integer sequences in the OEIS that have their own English Wikipedia entries</li> <li>Index to OEIS: Section Recurrent Sequencies</li> <li>Recursion (computer science)</li> <li>Exponential factorial</li> <li>Ackermann function</li> <li>McCarthy 91 function</li> <li>Tower of Hanoi</li> <li>Josephus problem</li> </ol>
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<span style="color: rgb(0, 0, 0);">Appendices
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Graphs

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<span id="Graphs_Key_concepts"> Key concepts
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 * Graph theory
 * Graph (discrete mathematics)
 * Seven Bridges of Königsberg
 * Five room puzzle
 * Eulerian path
 * Hamiltonian path
 * Hamiltonian path problem
 * Graph isomorphism
 * Graph isomorphism problem
 * Graph connectivity
 * Matching (graph theory)
 * Assignment problem
 * Graph algorithms

<span id="Graphs_Bibliography"> Bibliography: theory and proposed and solved exercises
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<span id="Graphs_Software"> Software
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<span id="Graphs_Multimedia"> Multimedia-icon.svg Multimedia
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<span id="Graphs_See_also"> See also
<hr style="margin-left: 0px; width:50%;">
 * Handshaking lemma
 * Route inspection problem

<span id="Graphs_To_find_out_more"> To find out more
<hr style="margin-left: 0px; width:50%;"> <ol style="list-style-type:circle;"> <li>Gallery of named graphs</li> <li>Portal:Mathematics</li> <li>Mesh networking</li> </ol>
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Numerical calculus

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<span id="NumericalCalculus_Key_concepts"> Key concepts
<hr style="margin-left: 0px; width:50%;"> Main articles: Numerical analysis and Numerical linear algebra Main categories: Numerical analysis and Numerical linear algebra — Interpolation 
 * Polynomial interpolation
 * Newton's divided differences interpolation polynomial
 * Lagrange polynomial

<span id="NumericalCalculus_Bibliography"> Bibliography: theory and proposed and solved exercises
<hr style="margin-left: 0px; width:50%;">

<span id="NumericalCalculus_Software"> Software
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<span id="NumericalCalculus_Multimedia"> Multimedia-icon.svg Multimedia
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<span id="NumericalCalculus_To_find_out_more"> To find out more
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More complimentary knowledge pills

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 * Red pill and blue pill
 * Red pill and blue pill

Conjectures
<hr style="margin-left: 0px; width:50%;">


 * List of conjectures (list of mathematical conjectures)
 * Undecidable conjectures

Open problems
<hr style="margin-left: 0px; width:50%;">


 * List of unsolved problems in mathematics
 * Category:Unsolved problems in mathematics
 * List of unsolved problems in computer science
 * Category:Unsolved problems in computer science
 * Lists of unsolved problems
 * Category:Lists of unsolved problems

Paradoxes
<hr style="margin-left: 0px; width:50%;">
 * List of paradoxes

Some more problems, either not solved or solved
<hr style="margin-left: 0px; width:50%;">
 * Category:Hypotheses
 * Category:Philosophical problems
 * Category:Mathematical problems
 * Category:Computational problems
 * Category:Recreational mathematics
 * Category:Mathematics-related lists, for instance:
 * List of proved mathematical statements (es)
 * List of theorems
 * List of lemmas
 * List of mathematical proofs
 * List of inequalities
 * List of disproved mathematical ideas
 * List of incomplete proofs
 * List of undecidable problems
 * Category:Lists of unsolved problems
 * Category:Statistics-related lists
 * Category:Computing-related lists
 * Category:Philosophy-related lists

Philosophy
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 * Philosophy of logic
 * Philosophy of language
 * Philosophy of Arithmetic
 * Philosophy of statistics
 * Philosophy of mathematics
 * Philosophy of computer science
 * Philosophy of science
 * Philosophy of technology
 * Philosophy of technique (es; de)
 * Philosophy of mind
 * Philosophy of artificial intelligence
 * Philosophy of information
 * Philosophy of social science
 * Science and technology studies
 * Technoscience

History
<hr style="margin-left: 0px; width:50%;">
 * History of logic
 * Historical linguistics
 * History of algebra
 * History of arithmetic
 * History of numbers
 * History of number theory
 * History of the Theory of Numbers (three-volume work by Leonard Eugene Dickson summarizing work in number theory up to about 1920)
 * History of combinatorics
 * History of statistics
 * History of probability
 * History of mathematics
 * History of computer science
 * History of science
 * History of technology
 * History of science and technology
 * History of computing
 * History of artificial intelligence
 * History of philosophy
 * History of sociology
 * History of ideas

Imagination
<hr style="margin-left: 0px; width:50%;">
 * Category:Imagination
 * Analog computer
 * Future of mathematics
 * Category:Future

Languages
<hr style="margin-left: 0px; width:50%;">
 * Language of mathematics
 * Mathematical notation
 * Category:Glossaries, for instance:
 * Category:Glossaries of mathematics
 * Of particular relevance: Closed-form expression • ...
 * Category:Terminology, for instance:
 * Category:Mathematical terminology
 * Category:Computing terminology
 * Category:Philosophical terminology
 * Category:Social sciences terminology

Multimedia-icon.svg Multimedia
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To know more
<hr style="margin-left: 0px; width:50%;">
 * Richard Elwes (2016) The top 10 mathematical achievements of the last 5ish years, maybe. En: www.mathematics-in-europe.eu, European Mathematical Society (EMS)
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Editathons (intensive collaborative learning meetings)

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 * Here.
 * Here.


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Multimedia-icon.svg More multimedia by the mentioned authors and by others

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<span style="color: rgb(0, 0, 0); font-style: italic;">(Illustrative examples, cases, exercises, problems). <div style="margin-left: 48%; text-align: right; color: rgb(0, 102, 0); background-color: rgb(255, 255, 223); background-image: none; padding: 0px; border-top: 4px solid rgb(0, 102, 0); border-bottom: 4px solid rgb(0, 102, 0); vertical-align: middle;"> 'It is axiomatic that the greater the student's individual effort, the more thorough will be his (sic) learning.'
 * style="padding:2px;" | <h2 id="pamdan-ede-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:110%; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Sample exam questions, instrumental and relational, and some answers<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" />

Timothy J. Fitikides: Common mistakes in English, Longmans, 6ª edición / 6th edition, 2000, p. vii.
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Propositional logic
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question L1. (2.5 points). On the island of truthfuls and deceitfuls — another nomenclature in the literature for the couple have been (knights, knaves) and (truth-tellers/'truthers', liars) — there are two types of inhabitants, 'truthfuls' who always tell the truth and 'deceitfuls' who always lie. It is assumed that every inhabitant is either a truthful or a deceitful person. There were two inhabitants, $$A$$ and $$B$$, standing together in the front yard of a house. You passed by and asked them, 'Are you truthful or deceitful persons?' <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;"> <li>a) $$A$$ answered, 'If $$B$$ is a truthful person then I am a deceitful person.' Can it be determined whether $$A$$ and $$B$$ were truthfuls or deceitfuls? (1.25 p.)</li> <li>b) Afterward, $$B$$ said, 'Don't believe $$A$$; he's lying.' With this new information, can it be determined whether $$A$$ and $$B$$ were truthfuls or deceitfuls? (1.25 p.)</li> </ul>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question L2. (2.5 points) With the help of propositional logic, prove that the following argument is valid or not. 'This program will compile whenever we have declared the variables. However, in truth, we will declare the variables precisely if we do not forget to do so. It turns out that the program has not compiled. Then it follows that we have forgotten to declare the variables.' Important: Do not solve it using truth tables.

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question L3. (2.5 points). <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;"> <li>a) Define adequate set of connectives (asc), also called completely expressive or functionally complete set of connectives.</li> <li>b) Provide two examples of two-element asc, explaining why they are so and assuming that we know the asc which elements are the most usual connectives $$\left\lbrace \neg, \wedge, \vee, \rightarrow, \leftrightarrow \right\rbrace$$.</li> </ul>

Predicate logic
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(0, 153, 153); padding: 1em; text-align:left;"> Question L4. (2.5 points). There are $77$ animals on a hill, they are two-legged or four-legged. A villager says: 'At least one of the animals has two legs and given any pair of animals, at least one of them has four legs.' <ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;"> <li>(a) Formalise in predicate logic what the local said.</li> <li>(b) How many of them are two-legged and how many are four-legged?</li> </ul>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(0, 153, 153); padding: 1em; text-align:left;"> Question L5. (2.5 points). Source: Lewis Carroll, Symbolic Logic: Part I. Elementary (Macmillan, 1896), pg. 118. Public Domain.

40.

(1) No kitten, that loves fish, is unteachable;

(2) No kitten without a tail will play with a gorilla;

(3) Kittens with whiskers always love fish;

(4) No teachable kitten has green eyes;

(5) No kittens have tails unless they have whiskers.

Universe = 'kittens'; A = loving fish; B = green-eyed; C = tailed; D = teachable; E = whiskered; H = will play with a gorilla.

You are required to: <ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;"> <li>(a) Formalise all these statements into Predicate Logic.</li> <li>(b) In the universe of kittens and using Predicate Logic, deduce the one conclusion that follows from these statements and makes the argument valid.</li> <li>(c) Translate your symbolic answer into English.</li> </ul>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(0, 153, 153); padding: 1em; text-align:left;"> Question L6. (2.5 points). Formalise into Predicate Logic: <ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;"> <li>(a) 'All $$A$$ that is $$B$$, it is also $$C$$.'</li> <li>(b) 'If all $$A$$ is $$B$$, then it is also $$C$$.'</li> <li>(c) 'There is none which is $$A$$ or $$B$$ and is not $$C$$.'</li> </ul>

Sets, relations and functions
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question SRF1. (2.5 points) <ul style = "list-style: none; margin-left: 5; padding-left: 1.75em; text-indent: -1.75em;"> <li>(a) Propose three sets $$A$$, $$B$$ and $$C$$, such that $$A \in B$$, $$B \in C$$ and $$A \notin C$$. (0,5 points).</li> <li>(b) According to a survey of a certain group of students, they said that, if they had to decide between two courses, equally interesting because of their contents, they prefer that one for which the time they dedicate to study it is the lowest and for which they foresee the best results in exams. In case of equality of study times and of exam results forecasts, they are indifferent to them. Study the properties of this binary relation. (2 points).</li> </ul>

Algebraic structures
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question AS1. (2.5 points) Let $$ x \ast y = u $$ be a binary relation defined on the set $$A = \left\lbrace 1,3,5,7,9 \right\rbrace$$, for every two elements $$x$$ and $$y$$ in $$A$$, where $$u$$ is the figure of the units of the usual product $$x \cdot y$$ between two natural numbers (for example, $$3 \ast 9 = 7$$). <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;"> <li>a) Find out theCayley table for the operation $$\ast$$ on $$A$$.</li> <li>b) Is $$(A;\ast)$$ an abelian group? (You can reason using the Cayley table).</li> </ul>

Cardinality, induction and recursion
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question C1. (2.5 points). Proof by definition that $$\mathbb{N}$$ is an infinite set.

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question C2. (2.5 points). Knowing that $$\mathbb{Z}$$ (integers) is a denumerable set and that the denumerable union of denumerable sets is a denumerable set, prove that $$\mathbb{Q}$$ (rationals) is a denumerable set.

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(0, 153, 153); padding: 1em; text-align:left;"> Cuestión C3. (2.5 points). Let $$A$$ be a denumerable set and let $$x \notin A$$. Prove that $$A \cup \{x\}$$ is a denumerable set.

Congruences
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question NT1. (2.5 points). Use congruence relation theory to respond. <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;"> <li>a) Prove that, for any $$n \in \mathbb{N}$$, $$21 \cdot 2^{5n+1} - 4 \cdot 3^{3n+1}$$ is divisible by $$5$$. (1.25 p.)</li> <li>b) Calculate the remainder of $$3^{6n+1} + 3^{2n+1} \cdot 19^{2n} - 3$$ (for any $$n \in \mathbb{N}$$), when it is divided by $$28$$. (1.25 p.)</li> </ul>

Power residues and divisibility rules
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question NT2. (2.5 points) In base-ten (decimal numeral system), find the digits $$x,y$$ such that the number $$12xy567$$ be divisible by $$33$$.

Diophantine equations
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question NT3. (2.5 points). One company spent $$100000$$ euros in buying $$100$$ electronic devices, some of them ground breaking and providing maximum performance. Smartphones were $$50$$ euros each, tablets were $$1000$$ euros each and laptops were $$5000$$ euros each. How many of each device did they buy? Solve this question using the theory of: <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;"> <li>a) diophantine equations;</li> <li>b) congruence equations.</li> </ul>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question NT4. (2.5 points). How could we distribute $$100$$ litres of water in a total of $$40$$ different containers of $$1$$, $$4$$ and $$12$$ litres? Solve this question using the theory of Diophantine equations.

Cryptography
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question NT5. (2.5 points). Abigail wants to send Balbina the most simple call message:. They can only send numbers. Abigail and Balbina use the letters' position in the alphabet to code them (thus, Abigail codes  as   and   as  ). They use RSA to encrypt their messages. If Abigail choose $$p = 3$$ and $$q = 7$$ as the ground primes for RSA: <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;"> <li>a) imagine you are Abigail and obtain the encrypted message that you have to send to Balbina;</li> <li>b) imagine you are Balbina and decrypt the encrypted message that Abigail has sent to you.</li> </ul>

Combinatorics
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question CT1. (2.5 points) Let $$D$$ be the set of decimal digits, that is, $$D = \left\lbrace 0,1,2,3,4,5,6,7,8,9 \right\rbrace$$. Using combinatorial reasoning, calculate: <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;"> <li>a) The number of subsets of $$D$$ which elements are all primes.</li> <li>b) The number of subsets of $$D$$ having a prime number of elements.</li> </ul>

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question CT2. (2.5 points) A group of twelve people visit a museum. Everybody is wearing a woolen overcoat. Upon entering, they leave their coats in the attended cloakroom. On leaving, the cloakroom attendant puts the twelve coats on the counter. Each person in the group picks out one at random, completely absent-minded because of a very interesting discussion. Using combinatorial reasoning, calculate in how many ways can the coats be chosen by them so that none of them get their own coat back.

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question CT3. (2.5 points) An urn contains seven balls numbered one through seven. They are randomly chosen, one by one and without reposition until the urn is empty. As they are removed from the urn, we write their figures down from left to right on a first out, first writen basis. Using combinatorial reasoning calculate how many numbers thus formed start and end with an even digit.

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question CT4. (2.5 points) A secret ballot is made in a meeting of seventeen people. Two people have cast invalid ballots, three have cast blank ballots, five have cast dissenting votes and seven have cast assenting votes. Using combinatorial reasoning calculate in how many ways this could have occured.

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question CT5. (2.5 points). Use a combinatorial reasoning to respond. <ul style = "list-style: none; margin-left: 5; padding-left: 1.35em; text-indent: -1.35em;"> <li>a) A number is palindrome if it reads the same from left to right and from right to left. In base ten, how many seven-digit numbers are palindromes? (1.25 p.)</li> <li>b) Let us assume a $$n$$ sided polygon network ($$n$$-gon network). Calculate $$n$$, the number of nodes (vertices) of the network, knowing that the number of line segments (sides + diagonals) is $$253$$. (1.25 p.)</li> </ul>

Theme 4.- Finite difference equations (recurrence relations)
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question RR1. (2.5 points) Let the following be the definition of the sum of two natural numbers $$n$$ and $$m$$: $$ \begin{align} S(n,0) &= n \\ S(n,m) &= S(n,m-1) + 1 \end{align} $$ Prove that the solution of this recurrence is $$S(n) = n + m$$.

<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question RR2. (2.5 points) Let $$x(t)$$ and $$y(t)$$ be the numbers of malicious software belonging to two malware types, in the day $$t$$, that coexist in a certain insecure wide area network (WAN) under malware evolution daily control. Let us assume that the original memberships were of $$x(0)=3$$ and $$y(0)=7$$ and that the coexistence evolution is as follows: Find out and solve the system of recurrence equations of the evolution of the malware.
 * every day, the growth in malware type $$x$$ is the sum of the triple of the growth in type $$x$$ on the previous day and the growth in type $$y$$ also on the previous day plus seven new malware (that were classified as type $$x$$),
 * and also every day, the growth in malware type $$y$$ is the result of subtracting the growth in type $$x$$ on the previous day from the growth in type $$y$$ on the previous day, plus three new malware (that were classified as type $$y$$).

Graphs
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;">

Numerical algebra and calculus
<div style="margin-left: 5%; margin-right: 5%; background: #FAFBFC; border: 1px solid rgb(153, 153, 153); padding: 1em; text-align:left;"> Question NAC1. (2.5 points) Find a possible general formula for computing the nth term, that is, $$a_n$$, of the sequence $$a_1 = 1, a_2 = 2, a_3 = 4, a_4 = 7$$ using the Newton's divided differences interpolation polynomial.

Academic year 2016-2017
(The questions and their identifiers are placed from the next section onwards).


 * Part 1: Themes 1 and 2.
 * Sample preparatory exam, 1: L2, SRF1, ACN1, NT1.
 * Sample preparatory exam, 2: L3, C1, NT5, NT3.
 * Part 2: Themes 3 and 4.
 * Sample preparatory exam, 1: CT1, CT2, RR1, AE1.
 * Sample preparatory exam, 2: CT3, CT4, RR2, G1.

Academic year 2017-2018

 * The midterm qualifying activity was the first sample preparatory exam.

Academic year 2018-2019

 * Mid course preparatory exam
 * End course preparatory exam

Past qualifying activities and real exams with some solutions
Most of them are bilingual documents with side by side texts in a double column format, the left column being the Spanish text and the right column being the corresponding English text. (Why? Please read for instance what Maria Martinello says).

Academic year 2016-2017

 * Final exam, 1 June, 2017 (A resolution approach to the first question has been published here: L1).
 * Resit final exam, 7 July, 2017 (Resolution approaches to the second, third and fifth questions have been published here: C2, NT4 y CT5, respectively).

Academic year 2017-2018

 * Midterm qualifying activity, White, 10 April, 2018
 * Midterm qualifying activity, Green, 10 April, 2018
 * Final exam, 24 May, 2018
 * Resit final exam, 29 June, 2018

Academic year 2018-2019

 * Midterm Qualifying Activity (10 April, 2019) Questions Resolution approach
 * Final Exam (3 June, 2019)
 * Resit Final Exam (25 June, 2019)

Academic year 2019-2020

 * Two-Week Take-Home Final Exam (due to the COVID-19 pandemic) (before 12 June, 2020)
 * Two-Week Take-Home Resit Final Exam (due to the COVID-19 pandemic) (before 14 July, 2020)

Tentative course outline (chronogram for the 2019-2020 academic year)
Important: Let us remember that exercises from Rosen's books and many others are available under the all rights reserved regime. However, their study and work on are an endless source of ideas for making contributions to Wikipedia.

Ex post I: Arts & Humanities Classroom 'Juanelo Turriano'
(For the time being, please see Ex post I on the Spanish Wikipedia).

Ex post II: Humour, entertainment and curiosities

 * Piled Higher and Deeper. © ARR.
 * Tim Hunkin and Shane Frazer's The Rudiments of Wisdom Cartoon Encyclopaedia. © ARR.
 * Wizards. Abstruse Goose. © CC BY-NC.

To keep track, know more or write a comment
Feel free to correct any typographical error you have detected in any of the project or plan pages (this is Wikipedia!).

Also all the feedback for doing better the next time, that is, all the comments, impressions, opinions, sensations and advices, wishes, suggestions or proposals concerning how we might improve this initiative will be most welcomed and greatly appreciated. The talk page of the learning plan is an ideal place to write them on. Please do not hesitate to do so. It means a lot to us.


 * Talk page of the learning plan 'Discrete and numerical mathematics'

Declaration of conformity
<div lang="en" dir="ltr" style="position:relative;margin:20px auto;width:66%;min-width:18em;border:1px solid #AAA;background:white;padding:2px;"> Juan Miguel León Rojas declares under his own responsibility that the learning plan specified here meets all the essential requirements of the academic program (ficha12a) corresponding to the course Further Mathematics taught at the School of Technology, University of Extremadura.

About this page on the English Wikipedia

 * ([ more information])


 * }
 * style="border:1px solid transparent;" |

{| id="pamdan-right" style="width:100%; vertical-align:top; background:#f7fffa;"
 * class="MainPageBG" style="width:20%; border:1px solid #deecc8; background:#f7fffa; vertical-align:top;"|

<div style="margin-left: 0%; text-align: right; color: rgb(0, 102, 0); font-size:small;">
 * style="padding:2px;" | <h2 id="pamdan-it1-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:small; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Transverse information 1.- Some points about Free/Libre & Open Knowledge (FLOK)<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" />
 * style="color:#000; padding:2px 5px 5px;" |
 * style="color:#000; padding:2px 5px 5px;" |

<span style="color: rgb(0, 0, 0); font-style: italic;">Some referencies about free/libre & open licenses and free knowledge and culture

 * Creative Commons España (CC)
 * Definición de Conocimiento Abierto (OKD) (Open Definition)
 * Definición de Obras Culturales Libres (freedomdefined.org)
 * Free Software Foundation Europe (FSFE)
 * Open CourseWare (OCW)
 * Open Education
 * Open Educational Resources (OER)
 * Open Knowledge Foundation (OKF)
 * Open Source Initiative (OSI)
 * Open Textbooks
 * es:Usuario:Jmleonrojas/Taller/Software Libre y Conocimiento Libre
 * Xnet

<span style="color: rgb(0, 0, 0); font-style: italic;">Practising lawyers who are specialised in intellectual property and computer law (IP & IT lawyers)

 * David Bravo
 * Javier de la Cueva
 * David Maeztu

<span style="color: rgb(0, 0, 0); font-style: italic;">Some open repositories

 * Internet Archive: Non-profit digital library offering free universal access to books, movies &amp; music, as well as 386 billion archived web pages (Wayback Machine).
 * Algunos sobre proyectos de arquitectura (además de software libre para arquitectura) (CC BY-NC-SA)
 * Infotecarios: Hackathon, Repositorios de Datos Abiertos – Open Data (segunda parte) (CC BY-NC-ND)

<span style="color: rgb(0, 0, 0); font-style: italic;">Take into account the right of quotation

 * Lema, Carlos (2006) Guía de buenas prácticas acordes con la Ley de Propiedad Intelectual; III Jornada Campus Virtual UCM (gratis OA).
 * Sánchez Almeida, Carlos (2009) El derecho de cita con fines docentes (CC BY-NC-SA).
 * Universidad de Granada (2009) Derechos de autor en plataformas e-learning (CC BY-NC-SA)

<span style="color: rgb(0, 0, 0); font-style: italic;">And the possible plagiarism

 * Beall, J. Plagiarism. Scholarly Open Access. © gratis OA.
 * Copy, Shake, and Paste. A blog about plagiarism and scientific misconduct (CC BY-NC-SA)
 * Usuario:Jmleonrojas/Taller/Anti-plagio
 * VroniPlag Wiki (CC BY-SA) (more information on Wikipedia)


 * Plagiarism
 * Plagiarism detection
 * Copying within Wikipedia

<span style="color: rgb(0, 0, 0); font-style: italic;">Libraries (texts, courses)

 * Alqua. (© FLOK).
 * American Institute of Mathematics (AIM): Approved Textbooks (open textbooks). (© FLOK).
 * Biblioteca Digital Mundial / World Digital Library (WDL) (UNESCO)
 * Biblioteca Virtual Miguel de Cervantes (España)
 * Biblioteca Virtual Universal (Argentina)
 * Bookboon (© gratis OA)
 * Boundless (© CC BY-SA)
 * CK-12
 * College Open Textbooks: Math
 * Community College Consortium for Open Educational Resources: Open Textbooks
 * CopyFight
 * Free High School Science Texts Project
 * FreeTechBooks
 * Proyecto / Project LATIn, libros publicados, comunidades y grupos de escritura.
 * Merlot
 * MagMath
 * MIT OpenCourseWare: Online Textbooks
 * OER Commons
 * Open Book Publishers
 * Open Course Library
 * Open Culture: Free Textbooks
 * Open Learning Initiative (Carnegie Mellon University)
 * Open SUNY Textbooks
 * Open Textbook Store
 * OpenLibra
 * OpenStax
 * OpenStax CNX
 * Orange Grove Texts
 * O'Reilly Open Books Project
 * Project Gutenberg
 * Red Matematica Antioquia (© gratisOA)
 * Saylor Academy (© CC BY-SA)
 * Scholar Commons (University of South Florida)
 * Student PIRGS: affordable textbooks initiative, conjuntamente con la Universidad de Minesota creando la Open Textbook Library
 * Textbook Equity LLC
 * Textbook Revolution
 * Textos Marea Verde
 * The Assayer (© OPL, without A or B)
 * The Online Books Page. (Véase, también: Copyrights and licenses y Archives and Indexes).
 * Universidad Nacional de Educación a Distancia (UNED) Abierta
 * Más...


 * style="padding:2px;" | <h2 id="pamdan-it1-h2" style="margin:3px; background:#deecc8; font-family:inherit; font-size:small; font-weight:bold; border:1px solid #deecc8; text-align:left; color:rgb(0, 0, 0); padding:0.2em 0.4em;">Transverse information 2.- About more topics of interest<hr style="width: 100%; height: 4px; background-color: rgb(0, 102, 0);" />
 * style="color:#000; padding:2px 5px 5px;" |
 * style="color:#000; padding:2px 5px 5px;" |

<span style="color: rgb(0, 0, 0); font-style: italic;">Accessibility and usability

 * A List Apart
 * Nielsen, Jacob (2012) Usability 101: Introduction to Usability.Nielsen Norman Group Weekly Newsletter, January 4. © ARR, Acceso Gratis).
 * World Wide Web Consortium (W3C)
 * W3Schools Online Web Tutorials


 * Accessibility. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).
 * Usability. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).
 * Web usability. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).

<span style="color: rgb(0, 0, 0); font-style: italic;">Library

 * Caplow, Theodore (1968) Two against one: Coallitions in Triads. Prentice Hall, Inc., Englewood Cliffs, New Jersey, U.S.A. (Versión española de Natividad Sánchez Sáinz-Trápaga: Dos contra uno: Teoría de coaliciones en las tríadas. Alianza Editorial, S. A., Madrid, España, 1974).

<span style="color: rgb(0, 0, 0); font-style: italic;">— Technology

 * Fairphone

<span style="color: rgb(0, 0, 0); font-style: italic;">Hacker ethic

 * Alan Lazalde (2014) Libros imprescindibles para entender la cultura hacker, Diario Turing. Eldiario.es. © CC BY-SA.
 * Pablo Romero (2014) Richard Stallman: 'La RAE debe corregir la definición de hacker', El mundo. © ARR.


 * Hacker culture. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).
 * Hacker ethic. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).

<span style="color: rgb(0, 0, 0); font-style: italic;">Philosophy

 * DMOZ (en varias lenguas)
 * Proyecto Filosofía en español (Fundación Gustavo Bueno)
 * Internet Encyclopedia of Philosophy (en inglés)
 * Philosophica
 * Philosophy Stack Exchange (en inglés)
 * PhilPapers (en inglés)
 * Stanford Encyclopedia of Philosophy (en inglés)
 * The Information Philosopher (en inglés)


 * Philosophy. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).

<span style="color: rgb(0, 0, 0); font-style: italic;">Periodical library

 * Teknokultura. (© CC BY-NC-ND)

<span style="color: rgb(0, 0, 0); font-style: italic;">Tools

 * ATOM (text editor hackable to the core) (© FLOSS).
 * Etherpad Lite (real-time collaborative online editor).
 * Código fuente.
 * Sitios que lo proporcionan, por ejemplo, el colectivo Systemli.org o el Swedish Pirate Party / Partido Pirata sueco con PiratePad.

<span style="color: rgb(0, 0, 0); font-style: italic;">LaTeX

 * CervanTeX
 * Detexify
 * Editor en línea de ecuaciones en LaTeX (CodeCogs)
 * Editor en línea de ecuaciones en LaTeX (Rincón Matemático)
 * Ejemplos (en Overleaf)
 * LaTeX para Gmail
 * LaTeX Stack Exchange
 * Modelos (en Papeeria)
 * Moodle LaTeX
 * Pakin, Scott (2015). The Comprehensive LaTeX Symbol List.
 * ShareLaTeX
 * TeXample.net
 * TeXblog
 * TeXnique
 * The LaTeX Project
 * Walsh, N. (2002) Making TeX Work. O'Reilly.
 * Asymptote
 * Asymptote (en asy.marris.fr)
 * Asymptote (en piprime.fr)
 * PGFPlots.net


 * LaTeX. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).
 * Wikipedia:LaTeX_symbols. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).

<span style="color: rgb(0, 0, 0); font-style: italic;">Spanish language

 * (Spanish language)


 * Real Academia Española (RAE) [Royal Spanish Academy]
 * Asociación de Academias de la Lengua Española (ASALE). (© gratisOA). (Para saber de otras asociaciones de la lengua española en el mundo, consúltese, por ejemplo: Wikipedia [© CC BY-SA]).
 * Centro Internacional de Investigación de la Lengua Española (Cilengua).
 * Centro Virtual Cervantes (CVC).
 * Diccionario de la Lengua Española.
 * Diccionario panhispánico de dudas (DPD).
 * Diccionario español de ingeniería (DEI).
 * Español (con virgulilla).
 * Fundación Camino de la Lengua Castellana.
 * Fundación del Español Urgente.
 * Wikilengua.
 * Instituto Cervantes.
 * Fernando Lázaro Carreter (2002): El neologismo en el Diccionario.
 * Lexipedia.
 * Practica Español
 * SIELE (Servicio Internacional de Evaluación de la Lengua Española) (Práctica de examen).
 * RAE (2010): Principales novedades de la ortografía de la lengua española.

<span style="color: rgb(0, 0, 0); font-style: italic;">— U.S. Spanish

 * Academia Norteamericana de la Lengua Española (ANLE) [North American Academy of the Spanish Language]

<span style="color: rgb(0, 0, 0); font-style: italic;">— Collocations dictionaries

 * (Collocation)


 * DICE

<span style="color: rgb(0, 0, 0); font-style: italic;">— Spelling and grammar checkers

 * SpanishChecker

<span style="color: rgb(0, 0, 0); font-style: italic;">— Plain Spanish

 * (Plain language)


 * Asociación Lectura Fácil
 * Cómo escribir con claridad, por la Comisión Europea
 * Extremadura vive la fácil lectura
 * ¿Qué es el lenguaje claro?, por Plain Language Association International (PLAIN)

<span style="color: rgb(0, 0, 0); font-style: italic;">English language

 * (English language)


 * Test and practise your English (by the British Council)

<span style="color: rgb(0, 0, 0); font-style: italic;">— Academic English

 * Michael McCarthy & Felicity O'Dell: Academic Vocabulary in Use, Cambridge University Press, 2nd ed., 2016. (© ARR).
 * Using English for Academic Purposes: A Guide for Students in Higher Education (UEfAP) y el / and the English Language Level Test.
 * Writing Guide.
 * Purdue Online Writing Lab

<span style="color: rgb(0, 0, 0); font-style: italic;">— American English

 * American English (U.S. Department of State)
 * How to Fake a Convincing American Accent. WikiHow. (© gratisOA)
 * Merriam-Webster
 * Merriam-Webster: The Open Dictionary - New Words & Slang
 * The Punctuation Guide
 * William Strunk (Jr.) (1918/1920), Elwin Brooks White (1959) The Elements of Style. Harcourt, Brace &amp; Howe (1920), Macmillan (1959).

<span style="color: rgb(0, 0, 0); font-style: italic;">— British English

 * How to Speak in a British Accent. WikiHow. (© gratisOA)
 * The Queen's English Society. (© gratisOA)

<span style="color: rgb(0, 0, 0); font-style: italic;">— Collocations dictionaries

 * Ozdic

<span style="color: rgb(0, 0, 0); font-style: italic;">— Dictionaries and thesauri

 * Cambridge dictionaries, grammar and thesaurus, (British & American)
 * Collins (multiple languages)
 * Linguee (multiple languages)
 * OneLook
 * Oxford dictionaries, grammar and thesaurus, (British & American)
 * Urban dictionary
 * WordReference.com (multiple languages)

<span style="color: rgb(0, 0, 0); font-style: italic;">— English and mathematics

 * Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics, Oxford University Press, 5th ed., 2014. (© ARR).
 * IXL | Maths and English Practice.

<span style="color: rgb(0, 0, 0); font-style: italic;">— Plain English

 * Judy Steiner-Williams: Writing in plain English (This course qualifies for professional development units (PDUs))
 * University of Bradford (GB): Write in Plain English.
 * University of Bristol (GB): Improve your writing: grammar exercises
 * University of Sidney (AU): Writing in Plain English.
 * Martin Cutts. Oxford Guide to Plain English. Oxford University Press, Oxford, United Kingdom,, 4th edition, 2013. ISBN 978-0-19-966917-1
 * Charles K. Ogden's Basic English, e. g., the Simple English Wikipedia.
 * Clarity: General Resources
 * Plain Language Association International (PLAIN)
 * Writing for GOV.UK
 * Plain Language.gov (USA): Plain-Language Training Resources
 * Center for Plain Language
 * Plain language supports science communication
 * The importance of plain language for content creation
 * Plain English Campaign (Fighting for crystal-clear communication since 1979).
 * Free Guides
 * Drivel Defence for Text
 * Hemingway Editor
 * Plain English Foundation: Free Writing Tools
 * Plain Language Commission
 * Publications
 * Globish
 * Globish

<span style="color: rgb(0, 0, 0); font-style: italic;">— Web sites

 * Category:Unsolved problems in mathematics
 * ::ZTFNews.org
 * A kind of library
 * AlgoRythmics
 * American Mathematical Society (AMS)
 * AoPS: Art of Problem Solving
 * bit-player
 * blocdemat
 * Blog para anti-matemáticos
 * Café matemático
 * Cifras y teclas
 * divulgaMAT
 * El Paraíso de las Matemáticas
 * Encyclopedia of Mathematics
 * Epsilones
 * Gacetilla matemática
 * Gausianos
 * GeoGebra
 * Grafitti matemático
 * Instituto de Estadística de Extremadura
 * Instituto Nacional de Estadística (INE)
 * Juegos topológicos
 * lasmatematicas.es en pdf
 * MacTutor History of Mathematics archive
 * Matematicalia (© gratis OA)
 * Matemáticas cercanas. (© CC BY-NC-ND).
 * Matemáticas digitales
 * Matemáticas en tu mundo
 * Matemáticas interactivas y manipulativas
 * Matemáticas y sus fronteras
 * ¡Mates, Mates!
 * MaTeTaM
 * Mathematik
 * Mathsfact
 * Mathsmovies
 * MathWorld (© ARR)
 * Mati, una profesora muy particular
 * Mati y sus mateaventuras
 * Memorandum Matemático
 * Museo interactivo de Matemáticas
 * Naukas
 * OpenMind (© ARR)
 * Adrián Paenza: Libros de divulgación
 * Planeta Matemático
 * PlanetMath (© CC BY-SA)
 * Portal Matemático (fmat)
 * Proyecto Descartes
 * Quanta Magazine: Computer Science (© ARR)
 * Quanta Magazine: Mathematics (© ARR)
 * Real Sociedad Matemática Española
 * Revista Digital Matemática, Educación e Internet
 * Rincón Matemático
 * The MacTutor History of Mathematics archive
 * The On-line Encyclopedia of Integer Sequences
 * Tio Petros
 * Tito Eliatron Dixit
 * Turismo Matemático
 * Wikillerato (© CC BY)
 * Wolfram|Alpha
 * WolframMathWorld
 * Yair.es

<span style="color: rgb(0, 0, 0); font-style: italic;">— Android apps

 * MalMath


 * Mathematics. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).

<span style="color: rgb(0, 0, 0); font-style: italic;">— Languages

 * Free Pascal
 * Wikibook 'Pascal Programming'
 * Python (programming language)
 * Python Shell
 * Shelf of wikibooks about 'Python programming language'
 * w3school Python Tutorial
 * SageMath
 * SageMathCell
 * CoCalc

<span style="color: rgb(0, 0, 0); font-style: italic;">— Online interpreters

 * Ideone
 * Rextester
 * Try It Online

<span style="color: rgb(0, 0, 0); font-style: italic;">Digital preservation

 * Archivematica
 * DAITSS
 * Hoppla
 * JHOVE
 * RODA


 * Digital preservation. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).

<span style="color: rgb(0, 0, 0); font-style: italic;">Knowledge representation

 * Simple Knowledge Organization System (SKOS)
 * Sowa, John F. (2000) Knowledge Representation. Logical, Philosophical, and Computational Foundations. Brooks/Cole, Thomson Learning, Pacific Grove, CA, USA.


 * Knowledge representation and reasoning. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).

<span style="color: rgb(0, 0, 0); font-style: italic;">Computer security

 * Kriptópolis | Criptografía y Seguridad
 * Nmap, incluyendo el repositorio NSE.
 * Un informático en el lado del mal.


 * Computer security. In Wikipedia, The Free Encyclopedia. (© CC BY-SA).

<span style="color: rgb(0, 0, 0); font-style: italic;">Free and open-source software

 * AlternativeTO
 * Best Open Source
 * Libre Projects
 * Linux Adictos
 * Linux App Finder
 * Open Hub
 * Open Source Alternative
 * Open Source Living
 * Osalt
 * OStatic
 * The Free Software Directory
 * The Linux Alternative Project
 * Zwodnik
 * AlternativePedia


 * Free and open-source software. En Wikipedia, The Free Encyclopedia. (© CC BY-SA).
 * Alternative terms for free software. En Wikipedia, The Free Encyclopedia. (© CC BY-SA).

<span style="color: rgb(0, 0, 0);">— &amp; —

<span style="color: rgb(0, 0, 0); font-style: italic;">Snoozing

 * Ars Technica (© ARR)
 * Barrapunto (© ARR)
 * Elástico 3.0 (© CC BY-NC)
 * Geek Gazette (© ARR)
 * Genbeta (© ARR)
 * Genbeta Dev (© ARR)
 * Hackernoon (© ARR)
 * Medium (© ARR)
 * Microsiervos (© ARR)
 * OpenMind (© ARR)
 * Redes Zone (© ARR)
 * Slashdot (© ARR)
 * Wired (© ARR)

<span style="color: rgb(0, 0, 0); font-style: italic;">Bubbles

 * Donald D. Hoffman (en inglés)
 * The Best Jobs of 2015 (CareerCast) (en inglés)

<span style="color: rgb(0, 0, 0); font-style: italic;">Miscellanea

 * 101claves
 * 99 % invisible (© TDR)
 * Cultura Colectiva + (© TDR)
 * Domestika
 * Mario Kogan y José Ochoa: ¿Dónde está mi equipo? / Where is my team?
 * Estética y Teoría del Arte
 * Joshua Davis Studios
 * Kunsthalle Wien
 * OpenMind books (© ARR)
 * TreceBits (© ARR)
 * Yugop

<span style="color: rgb(0, 0, 0); font-style: italic;">— Wikimedia

 * Wikimedia outreach: Education: Case_Studies
 * Wikimedia outreach: Coursework feedback

<span style="color: rgb(0, 0, 0); font-style: italic;">— Wikipedia

 * Welcome to Wikipedia
 * Wikipedia: Training
 * Wikipedia: Education program
 * Wikipedia: School and university projects
 * Wikipedia: Requested articles
 * Wikipedia: WikiProject Mathematics
 * Help: Cheatsheet

<span style="color: rgb(0, 0, 0); font-style: italic;">— Wikibooks

 * Wikibooks: Welcome
 * Wikibooks: Requested books

<span style="color: rgb(0, 0, 0); font-style: italic;">— Wikipedia

 * Wikipedia: Bienvenidos
 * Wikipedia: Bienvenidos alumnos
 * Wikipedia: Bienvenidos profesores
 * Wikipedia: Programa educativo
 * Wikipedia: Proyectos educativos
 * Wikipedia: Artículos solicitados
 * Wikiproyecto: Matemáticas
 * Wikipedia en CD (2009): Matemáticas
 * Ayuda: Referencia rápida

<span style="color: rgb(0, 0, 0); font-style: italic;">— Wikilibros

 * Wikilibros: Bienvenida
 * Wikilibros: Wikilibros solicitados
 * }
 * }