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

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This university learning plan consists of a primer on discrete mathematics and its applications including a brief introduction to a few numerical analysis. It has a special focus on dialogic learning (learning through argumentation) and computational thinking, promoting the development and enhancement of: problem solving and decision making skills; positive attitudes towards mathematical, analytic, reflective, creative, lateral, divergent and concrete critical thinking. My personal experience is the foundation of it. Wikipedia and other information sources, in English and Spanish, support it. Course educational materials — including video — by others, help to improve this plan. Using Wikipedia, bibliography, multimedia and others, stimulate and enhance learning through crossover learning, incidental learning, learning by doing, learning by teaching and microlearning, leaving renewed flavours of blended learning such as flipped learning. Please, be free for suggesting improvements (see this section).

Ex ante

Background information

Specific information

Project on Wikipedia

Course outline

Paths on Wikipedia

Sample exams

Real exams

Study programme

Ex post

Introduction 'I was speaking one day to a chemical expert about Avogrado's hypothesis concerning the number of molecules of the gases in equal volume, and its relation with the so-called Mariotte's law and its consequences in modern chemistry, and he came to answer: "Theories, theories! All of that does not matter to me ... That is for those who do science, I just apply it." I kept silent, torturing my mind to finding out how science can be applied without doing it, and finally, when after some time I knew why our expert had been close to dying, I understood it finally.' Miguel de Unamuno (1864-1936): De la enseñanza superior en España [On higher education in Spain], Madrid, Revista Nueva, 1899, http://www.liburuklik.euskadi.eus/handle/10771/24524, p. 45. Also, in: Obras Completas de Miguel de Unamuno, Vol. VIII (Ensayos), pp. 1-58 (the quotation, on page 32). Glossary of abbreviations Ex ante I: Mathematics and Computing (Just as an appetiser) — Harangues Q&A With Nine Great Programmers; 2006, by Jaroslaw "sztywny" Rzeszótko (AKA "Stiff") (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. Ex ante II: Theme wikis — Wikipedia (Starting points) Discrete mathematics (© CC BY-SA) Portal:Mathematics (© CC BY-SA) Portal:Technology (© CC BY-SA) — Others: Progopedia (© GFDL & 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 > Batxillera > Ciència i tecnologia > 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. Cueva, Javier de la. "Free Legal Ontology, Licenses to Live or Move Commons". © 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 [1] includes an independent translation into Spanish). (In addition, there exists a transcript [in English] of Tim Minchin's address is available). UCLA Athletics. "KatelynOhashi - 10.0 Floor (1-12-19). [The meaning of a 10: effort, sacrifice, discipline, work, perseverance, technique, security, confidence, innovation, improvisation, happiness, emotion, passion, perfection, excellence]". (© ARR, gratisOA). (To watch a video about the sports life of Katelyn Ohashi, click here (© ARR, gratisOA)). --- We keep picking berries... —Bates, Marcia J.(en) (2017). "The design of browsing and berrypicking techniques for the online search interface". (© ARR, gratisOA).—. Ex ante VI: Specific motivation 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. Simonson, Shai. "What kinds of problems are solved in discrete math?". (Alternate URL) — Films 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). Background information Universities — 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). 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). 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). Specific information — Professor 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@unex.es. Office hours. — Course description This course is a primer on discrete mathematics and its applications including a very short introduction to a few numerical methods. UEX code: 501272. — Rationale 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): (DS1) Functions, relations and sets, (DS2) Basic logic, (DS3) Proof techniques, (DS4) Basics of counting, (DS5) Graphs and trees, and (DS6) Discrete probability, to which we would add: (DM1) Algebraic structures, (DM2) Matrices, (DM3) Algorithms and complexity, and (DM4) Basic number theory. 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. 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: (NC1) Roots of Equations, (NC2) Linear Algebraic Equations, and (NC3) Curve Fitting (regression and interpolation). 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). 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). * https://www.computer.org/cms/Computer.org/professional-education/curricula/ComputerEngineeringCurricula2016.pdf † https://www.acm.org/education/CS2013-final-report.pdf — Course objectives 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. — Prerequisites 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). * 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. — Course program Academic year 2019-2020 In Spanish. In English. Academic year 2018-2019 In Spanish. In English. — School hours Class meetings and planners. — Textbooks — Discrete mathematics 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, 0­-07­-338309­-0 © 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. 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, 0-07-242434-6, © 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, 0-07-242434-6, © 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, 0-07-288008-2, © ARR, companion website: http://highered.mheducation.com/sites/0072880082/information_center_view0/index.html), seventh (2012, ISBN 978­-0­-07­-338309­-5, 0­-07­-338309­-0, © 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. 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, 0-07-735350-1, © 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, 0-07-735349-8), © ARR. 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. 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. — Numerical calculus 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 — To find out more, while course is running (or once it is finished) 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; Epp, Susanna S. (2019). Discrete Mathematics with Applications (5th ed.). Andover, Hampshire (GB-HAM), England (GB-ENG), UK: Cengage Learning. ISBN 978-1337694193. © 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. Wolfram, Stephen (2002). A New Kind of Science (HTML/PDF). Champaign, Illinois (US-IL), USA: Wolfram Media, Inc. ISBN 1-57955-008-8. © gratisOA. — Communicating Course private forum at the UEX online campus Talk page of the learning plan University project on the English Wikipedia 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. It is important that you become aware that joining the university project 'Discrete numerical mathematics' is optional. Therefore, it is entirely up to you to do it. But if you do it, remember, you are required to: (a) use your true identity on free, open and public access web pages (Wikipedia) — although you can use an alias as your username, you must report your real identity (first, middle and last name) on your user page on the English Wikipedia —; (b) be polite and respect diversity (please remember, diversity is a wealth, neither a problem nor a threat); (c) comply with the rules and obligations laid down by the project coordination for this project (click and read them here), in particular the dynamic commitments (click and read them here); (d) help the individuals involved in the project as much as possible; (e) above all, commit yourself to you. — On the English-language Wikipedia 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). — Communicating To keep track of the project you have joined to, please follow the recommendations on its descriptive page, particularly on 'The basics' subsection. — Equivalent project on the Spanish-language Wikipedia (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). Contents and learning paths on Wikipedia '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) Considerations Using Wikipedia for mathematics self-study Proofs 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. See also: León-Rojas, J. M., KDEM DA/HD – 703-01; WP+ (Logic); WP+ (Sets, relations and functions); WP+ (Cardinality); and WP+ (Algebraic structures). More detail can be found in the Calendar of activities. 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-study: Rosen's book 7th ed. USA (§ 4.1, 4.3-4.6); (lessons 10-13); and UPM (theme 3). 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. See also: WP+ (Number theory). More detail can be found in the Calendar of activities. 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ふぁい 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. See also: WP+ (Combinatorics). More detail can be found in the Calendar of activities. 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. See also: WP+ (Finite difference equations [recurrence relations]). More detail can be found in the Calendar of activities. 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: ... 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: ... WP+: Paths on Wikipedia, bibliography (theory and proposed and solved exercises), multimedia and even more Using Wikipedia, bibliography, multimedia and others, stimulate and enhance learning through crossover learning, incidental learning, learning by doing, learning by teaching and microlearning, leaving renewed flavours of blended learning such as flipped learning. Very important warning Logic Sets, relations and functions Cardinality, induction and recursion Algebraic structures Number theory Combinatorics Difference equations Appendix: Graphs Appendix: Numerical calculus Appendix: More knowledge pills Theme 1.- Fundamentals Logic Key concepts Main article: Mathematical logic Main category: Mathematical logic See also: Classical logic, Logic, Formalism (philosophy of mathematics), List of logic symbols, and Category:Logic — Propositional logic Main article: Propositional logic Main category: Propositional calculus 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 See also: Conditioned disjunction, Indicative conditional and Logical constant. — Verification and rebuttal strategies, I See also: Category:Theorems in propositional logic Truth table Contraposition Reductio ad absurdum (see also: Proof by contradiction) Normal forms, I: Conjunctive normal form Disjunctive normal form Normal forms, II: Negation normal form Algebraic normal form — Predicate logic Main article: First-order logic Main category: Predicate logic Quantifier (linguistics) Quantifier (logic) Free variables and bound variables Universal quantification Existential quantification Uniqueness quantification Quantifier elimination See also: Completeness (logic) — Translating English statements into the language of logic and vice versa Syllogism Logical calculus (es; pt; ca) Square of opposition Donkey sentence — Valid arguments and inference rules 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 See also: Reasoning methods (es) and Cogency (es; pt; ca) Stylistic device, Modes of persuasion, Argumentation theory, Fallacy, Enthymeme, Paralogism (es; pt; fr; it; ca), Sophism (pt; fr; it) and Antinomy; Paradox; Category:Reasoning. — Direct and indirect proofs Proof theory Methods of proof (see also: Constructive proof, Proof by exhaustion, Counterexample and Proof by infinite descent) Mathematical fallacy (i.e., invalid proof) See also: Category:Theorem proving software systems, Problem solving, and Theory of justification — Verification and rebuttal strategies, II 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) — Verification and rebuttal strategies, III Normal forms, III: Prenex normal form Skolem normal form Method of analytic tableaux (Semantic tableaux) — Some unusual situations in Logic 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 Connections — Automated reasoning Main article: Automated reasoning — Boolean algebra Main article: Boolean algebra Canonical normal form Simplifying Boolean functions Karnaugh map Quine–McCluskey algorithm — Diagrammatic reasoning Main article: Diagrammatic reasoning — Logic gates Main article: Logic gate Bibliography: theory and proposed and solved exercises In English: In Spanish: —¤— Kenneth H. Rosen. Discrete mathematics and its applications. New York, New York State (US-NY), United States: McGraw-Hill, 7th edition, 2012. ISBN 978-0-07-338309-5. (Chapter 1 and related exercises). —¤— Amador Antón y Pascual Casañ, Lógica Matemática. Ejercicios. I. Lógica de enunciados. Valencia, Valencian Community (ES-VC), Spain: NAU llibres, 3rd edition, 1987. ISBN 84-85630-42-4 —¤— María Manzano y Antonia Huertas, Lógica para principiantes. Humanes de Madrid, Madrid, Community of Madrid (ES-MD), Spain: Alianza, 2006. ISBN 84-206-4570-2. —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. Aravaca, Madrid, Community of Madrid (ES-MD), Spain: McGraw-Hill/Interamericana de España, S.A.U., 5th edition, 2004. ISBN 84-481-4073-7. (Sections 1.1, 1.2, 1.3, 1.4, 1.5, 3.1 and related exercises). Software In English: In Spanish: —¤— Truth Tables (truth table calculator). © gratisOA. Michael Rieppel's Truth Table Generator. © gratisOA. Logicmind (command line model calculator). © MIT License. Tyler Sorensen's PBL, a Boolean algebra/propositional logic library written in Python. © BSD License. The Incredible Proof Machine, by Joachim Breitner —¤— Wolfgang Schwarz's Tree Proof Generator (© gratisOA): Tree proof generator for closed semantic tableaux (it provides with a countermodel in the case of an open tableau); Tree proof generator for open and closed semantic tableaux. —¤— Christian Gottschall's Gateway to Logic (© gratisOA) (Gateway to Logic): Logic calculator (for classic bivalent propositional logic) (see also: about what the calculator can do and about the calculator syntax); Automated Theorem Prover for Classical Predicate Logic (see also: examples of prover syntax); Proof Checker (see also: about the checker syntax). WolframAlpha: Examples for Logic & Set theory (Boole algebra, set theory and transfinite numbers). © gratisOA. —¤— Logisim (a graphical tool for designing and simulating logic circuits)] (in Spanish, English and more languages). © GNU GPL. Multimedia In English: In Spanish: Baugher, Greg A. "Section 1.1 Intro to Propositional Logic" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Baugher, Greg A. "Section 1.2 Applications of Propositional Logic" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Baugher, Greg A. "Sections 1.3-1.4 Prop Equivalences and Quantifiers" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Baugher, Greg A. "Section 1.6 Rules of Inference" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Baugher, Greg A. "Section 1.7 Introduction to Proofs" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Baugher, Greg A. "Section 1.8 Proof Methods and Strategy" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Krithivasan, Kamala. "Lecture 1 - Propositional Logic" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 2 - Propositional Logic (Contd.)" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 3 - Predicates & Quantifiers" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 4 - Predicates & Quantifiers (Contd.)" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 5 - Logical Inference" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 6 - Resolution Principles & Application to PROLOG" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 7 - Methods of Proof" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 8 - Normal Forms" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 9 - Proving programs correct" (Video). Indian Institute of Technology Madras. Simonson, Shai. "Lecture 2 - Boolean Algebra and formal logic" (Video). ArsDigita University. (Alternate URL) Jordán Lluch, Cristina. "¿Usamos la Lógica en la vida cotidiana?" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Lógica, ¿para qué?" (Video). Universidad Politécnica de Valencia (UPV). Hervás Jorge, Antonio. "Lógica proposicional" (Video). Universidad Politécnica de Valencia (UPV). Bustince Sola, Nicanor Humberto. "Lógica proposicional (Máster universitario en investigación en inteligencia artificial: Técnicas avanzadas de representación del conocimiento y razonamiento, Módulo 2, Introducción, Tema 1)" (Video). Universidad Pública de Navarra (UPNA). Jordán Lluch, Cristina. "Si y sólo si" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Condición necesaria y suficiente" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "¿Cuándo un razonamiento es correcto?" (Video). Universidad Politécnica de Valencia (UPV). Hervás Jorge, Antonio. "Demostración por contradicción aplicando tablas de verdad" (Video). Universidad Politécnica de Valencia (UPV). Hervás Jorge, Antonio. "Proposiciones: métodos de demostración" (Video). Universidad Politécnica de Valencia (UPV). Bustince Sola, Nicanor Humberto. "Lógica de primer orden (Máster universitario en investigación en inteligencia artificial: Técnicas avanzadas de representación del conocimiento y razonamiento, Módulo 2, Introducción, Tema 2)" (Video). Universidad Pública de Navarra (UPNA). See also In English: In Spanish: León-Rojas, Juan Miguel (2017-01-31). Matching tables for corresponding exercises from the 5th, 6th, 7th and 7th global editions of Rosen's book Discrete Mathematics and its Applications, Chapter 1 on The Foundations: Logic and Proofs (Bilingual edition, Spanish/English) (Technical report). KDEM (Knowledge Discovery Engineering and Management). DA/HD – 703-01. NSA's Puzzle Periodical León-Rojas, Juan Miguel (2017-01-31). Cuadros de emparejamiento para los ejercicios de las ediciones 5ª, 6ª, 7ª y 7ª global del libro Matemáticas discretas y sus aplicaciones, de Rosen, Capítulo 1 sobre Los Fundamentos: Lógica y Demostraciones (Edición bilingüe, español/inglés) (Technical report). KDEM (Knowledge Discovery Engineering and Management). DA/HD – 703-01. León-Rojas, J. M. (2009): Lógica proposicional (Notas incompletas de clase), v. 1.6.3-alpha2, rev. 17-02-2014. © CGL, CC0, CC BY. León-Rojas, J. M. (2009): Lógica de primer orden (Notas incompletas de clase), v. 0.3.1-alpha0, rev. 17-02-2014. © CGL, CC0, CC BY. Summa Logicae (María Manzano, Gustavo Santos, Belén Pérez Lancho, José Meseguer, Enrique Alonso, Huberto Marraud, Antonia Huertas, Dick de Jongh, Giovanna D'Agostino, Alberto Policriti) Sigma (2008) Los sofismas. In: Sigma, blog of the Department of Mathematics of the Facultad Multidisciplinaria de Occidente at the University of El Salvador. (© gratisOA). To find out more Portal:Mathematics Portal:Philosophy And more: Outline of logic Category:Concepts in logic WikiProject Logic Logic alphabet Metamath. © Public domain (with some exceptions) Equational logic; for instance, chapter 5 (Equational Logic: Part 1) from Backhouse, Roland, Program Construction. The Correct Way, 2002. And even more: Index of logic articles List of logicians Sets, relations and functions Key concepts — Sets Main article: Set (mathematics) Main category: Set theory - 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 Main article: Binary relation Main category: Mathematical relations See also: Finitary relation - 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 Intransitive relation (es) Antitransitive relation Quasitransitive relation Euclidean relation Serial relation Trichotomous relation Semiconnex relation (Complete relation) Connex relation (Total, linear, or strong complete 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 Main articles: Function (mathematics) and Map (mathematics) Main category: Functions and mappings - 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)#Inverse image and Inverse function#Preimages) - Multisets Multiset Multiplicity (mathematics) — Paradoxes Russell's paradox (see also: Class (set theory)) Interesting number paradox Connections — Extensive systems Main article: Extensive system See also: Comparative system, Ordinal scale, and Proportional scale Worth-read article: Mosterín, Jesús (1978, January). La estructura de los conceptos científicos. Investigación y Ciencia, 16, 82-93. Worth-read chapter: Mosterín, Jesús (1993). Los conceptos científicos. In: Ulises Moulines, Carlos (ed.). La ciencia: estructura y desarrollo. Madrid:Trota, Consejo Superior de Investigaciones Científicas and Sociedad Estatal Quinto Centenario. Pages 15-30. ISBN 84-87699-72-3 — Entity-relationship model Main article: Entity-relationship model See also: Relational database, Relational model, Relational algebra, Relational calculus, and Codd's theorem Bibliography: theory and proposed and solved exercises In English: In Spanish: —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Sections 2.1, 2.2, 2.3, Chapter 9 and related exercises). McGraw-Hill, New York, New York, United States, 2012, ISBN 978-0-07-338309-5 A. M. Díaz Hernández. Material from «Curso 0: Matemáticas». Part: Conjuntos, correspondencias y relaciones. OCW UNED. (Theory and exercises). 2010. © CC BY-NC-ND. http://ocw.innova.uned.es/matematicas-industriales/contenidos/pdf/tema6.pdf P. J. Herrero Piñeyro. Material from «Topología de Espacios Métricos». Part: Conjuntos, aplicaciones y números. OCW UM. 2010. © CC BY-NC-SA. http://ocw.um.es/ciencias/topologia-de-espacios-metricos/material-de-clase-1/cap-0-conj-apl.pdf —¤— J. M. León Rojas (2009): Conjuntos, aplicaciones y relaciones (Notas incompletas de clase), v. 1.4.1-alpha2, rev. 23-02-2014. © CGL, CC0, CC BY. —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. 5th edition. (Sections 1.6, 1.7, 1.8, Chapter 7 and related exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, ISBN 84-481-4073-7 Software In English: In Spanish: WolframAlpha (Examples, Mathematics, Set Theory) Multimedia In English: In Spanish: — Sets Baugher, Greg A. "Section 2.1 The Basics of Sets" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Baugher, Greg A. "Section 2.2 Set Operations" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Krithivasan, Kamala. "Lecture 10 - Sets" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 12 - Set operations on strings over an alphabet" (Video). Indian Institute of Technology Madras. Simonson, Shai. "Sets" (Video). ArsDigita University. (Alternate URL) — Relations Krithivasan, Kamala. "Lecture 13 - Relations" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 18 - Special properties of relations" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 19 - Closure of relations" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 20 - Closure of relations (contd.)" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 21 - Order relations" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 22 - Order relations and equivalence relations" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 23 - Equivalence relations and partitions" (Video). Indian Institute of Technology Madras. Simonson, Shai. "Equivalence Relations and Partial Orders" (Video). ArsDigita University. (Otra URL) — Functions Baugher, Greg A. "Section 2.3 Functions" (Vídeo). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), EUA. Krithivasan, Kamala. "Lecture 24 - Functions" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 25 - Functions (Contd.)" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 26 - Functions (Contd.)" (Video). Indian Institute of Technology Madras. — Sets Jordán Lluch, Cristina. "Conjuntos y notación" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Diagramas de Venn e igualdad de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Subconjuntos de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Conjunto de partes de un conjunto" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Unión de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Intersección de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Diferencia de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Complementario de un conjunto" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Diferencia simétrica de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Propiedades booleanas de la teoría de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Teorema de inclusión - exclusión en teoría de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Simplificación de expresiones en teoría de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "¿Cómo justifico que una igualdad de conjuntos no es valida?" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Recubrimiento y partición de un conjunto" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Producto cartesiano de conjuntos" (Video). Universidad Politécnica de Valencia (UPV). Soto Espinosa, Jesús. "Conjuntos finitos" (Vídeo). Guadalupe, Murcia, Región de Murcia (ES-MC), España: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Ejercicios" (Vídeo). Guadalupe, Murcia, Región de Murcia (ES-MC), España: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). (Ejercicio 1). — Relations Jordán Lluch, Cristina. "¿Por qué estudiar relaciones binarias?" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Relaciones binarias. Notación" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Propiedades reflexiva y simétrica de relaciones binarias. Definiciones y caracterización" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Operaciones entre relaciones binarias: Unión, intersección, diferencia, complementario" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Operaciones entre relaciones binarias. Ejemplo en forma implícita" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Relaciones binarias. Representación gráfica" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Relaciones binarias. Representación matricial" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Inversa de una relacion binaria" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Relaciones binarias de equivalencia" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Ejemplos sencillos de relaciones binarias de equivalencia" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Cotas superiores e inferiores de una relacion binaria" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Maximos, minimos, maximales, minimales de una relacion binaria de orden" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Operaciones entre relaciones binarias: Ejemplos en contexto real" (Video). Universidad Politécnica de Valencia (UPV). — Functions Soto Espinosa, Jesús. "Aplicaciones entre conjuntos finitos" (Vídeo). Guadalupe, Murcia, Región de Murcia (ES-MC), España: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Ejercicios" (Vídeo). Guadalupe, Murcia, Región de Murcia (ES-MC), España: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). (Ejercicio 2). Soto Espinosa, Jesús. "Aplicaciones. Ejercicio 1" (Vídeo). Guadalupe, Murcia, Región de Murcia (ES-MC), España: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Aplicaciones. Ejercicio 2" (Vídeo). Guadalupe, Murcia, Región de Murcia (ES-MC), España: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Aplicaciones. Ejercicio 3" (Vídeo). Guadalupe, Murcia, Región de Murcia (ES-MC), España: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). To find out more Portal:Mathematics Transitive binary relations Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all $a,b$ and $S\neq \varnothing :$ ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$ Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$ A term's definition may require additional properties that are not listed in this table. Cardinality, induction and recursion Key concepts — Cardinality Main articles: Cardinality and Cardinal number Main category: Cardinal numbers - 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) Connections — Hypercomputability Main article: Hypercomputation Supertask Thomson's lamp Zeno machine See also: Transcomputational problem, Quantum computing, and Matrioshka brain Bibliography: theory and proposed and solved exercises In English: In Spanish: —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Sections 2.5, 5.1, 5.2, 5.3, Chapter 9 and related exercises). McGraw-Hill, New York, New York, United States, 2012, ISBN 978-0-07-338309-5 A. M. Díaz Hernández. Material from «Curso 0: Matemáticas». Part: Conjuntos, correspondencias y relaciones. OCW UNED. (Theory and exercises). 2010. © CC BY-NC-ND. http://ocw.innova.uned.es/matematicas-industriales/contenidos/pdf/tema6.pdf P. J. Herrero Piñeyro. Material from «Topología de Espacios Métricos». Part: Conjuntos, aplicaciones y números. OCW UM. 2010. © CC BY-NC-SA. http://ocw.um.es/ciencias/topologia-de-espacios-metricos/material-de-clase-1/cap-0-conj-apl.pdf —¤— J. M. León Rojas (2001): Cardinalidad (Notas incompletas de clase), v. 1.0-alpha2, rev. 24-02-2014. © CGL, CC0, CC BY. —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. 5th edition. (Sections 3.2.5, 3.3, 3.4, Chapter 7 and related exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, ISBN 84-481-4073-7 Software In English: In Spanish: WolframAlpha (Examples, Mathematics, Transfinite Numbers)) Multimedia In English: In Spanish: Baugher, Greg A. "Sections 5.1-5.2 Mathematical Induction and Strong Induction" (Vídeo). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), EUA. Baugher, Greg A. "Sections 5.3-5.4 Recursive Definitions and Recursive Algorithms" (Vídeo). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), EUA. Krithivasan, Kamala. "Lecture 11 - Induction" (Video). Indian Institute of Technology Madras. Simonson, Shai. "Lecture 5 - Diagonalization, functions and sums review" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Lecture 10 - Mathematical induction" (Video). ArsDigita University. (Another URL) Jordán Lluch, Cristina. "¿Qué se entiende por cardinal de un conjunto?" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Principio de inducción" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Ejercicios de inducción" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "La inducción como método para demostrar desigualdades" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Ejemplo de desigualdad demostrada por inducción" (Video). Universidad Politécnica de Valencia (UPV). Jordán Lluch, Cristina. "Una conocida igualdad demostrada por inducción" (Video). Universidad Politécnica de Valencia (UPV). To find out more 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 Continuum hypothesis. Encyclopedia of Mathematics. Disponible en: http://www.encyclopediaofmath.org/index.php?title=Continuum_hypothesis 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/. 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/ Portal:Mathematics And more: * Category:Set theory Transitive binary relations Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all $a,b$ and $S\neq \varnothing :$ ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$ Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$ A term's definition may require additional properties that are not listed in this table. Algebraic structures Key concepts — Algebraic structures Main article: Algebraic structure Main category: 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 Main article: Group theory Group Abelian group (commutative group) See also: Cyclic permutation (Cycle), Symmetric group, Permutation group, Isometry group, Stirling numbers of the first kind and List of small groups — 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 Connections — Cryptography Main article: Cryptography Main category: Cryptography Pairing-based cryptography — Category theory Main article: Category theory Live article: Baez, John (November 8, 2019). 'Qué es la teoría de categorías y cómo se ha convertido en tendencia'. El País (In: Café y Teoremas) (In Spanish). Olog — Coding theory Main article: Coding theory Linear code See also: Block code, Group code and Hamming code Bibliography: theory and proposed and solved exercises In English: In Spanish: —¤— Ben Brubaker. The Very Basics of Groups, Rings, and Fields. 2006. http://www-users.math.umn.edu/~brubaker/docs/152/152groups.pdf —¤— Peter J. Cameron. Introduction to Algebra. 2nd edition. Oxford University Press, New York, United States. 2008. ISBN: 978–0–19–856913–8. —¤— Peter J. Cameron. Notes on Algebraic Structures. 2006. http://www.maths.qmul.ac.uk/~pjc/notes/algstr.pdf Kenneth H. Rosen. Discrete Mathematics and its Applications. English Global Edition, 2013, 7th edition., ISBN-13: 978-0-07-131501-2 (adapted by: Kamala Krithivasan). (Chapter 12: «Algebraic Structures and Coding Theory»; slides: <http://highered.mheducation.com/sites/0071315012/student_view0/lecture_powerpoint_slides.html>). —¤— Máximo Anzola and José Caruncho. Problemas de Álgebra. Tomo 1. Conjuntos-Grupos. 3rd edition. Primer Ciclo, Madrid, Spain. (Chapter 9 «Operaciones», 24 solved exercises; Chapter 10 «Grupos-Estructura», 154 solved exercises), 1981. ISBN: 84-300-4073-0. —¤— Máximo Anzola and José Caruncho. Problemas de Álgebra. Tomo 2. Anillos - Polinomios - Ecuaciones. 3ªrd edition. Primer Ciclo, Madrid, Spain. (Chapter 1 «Estructura de anillo», 40 solved exercises; Chapter 4 «Estructura de cuerpo», 32 solved exercises), 1982. ISBN: 84-300-6417-6. Department of Algebra, University of Seville. Álgebra básica. Notas de teoría. http://www.algebra.us.es/Documentos/AB_1011_Teoria —¤— José García García and Manuel López Pellicer. Álgebra lineal y geometría. Curso teórico-práctico. 7th edition. Marfil, Alcoy, Spain. ISBN: 84-268-0269-9. J. M. León Rojas (2009): Estructuras algebraicas (Notas incompletas de clase), v. 1.4.2-alpha2, rev. 25-04-2014. © CGL, CC0, CC BY. J. M. León Rojas (2014): Estructuras mixtas (Notas incompletas de clase), v. 0.1.1-alpha0. © CGL, CC0, CC BY. Software In English: In Spanish: WolframAlpha (Examples, Mathematics, Finite Groups) Multimedia In English: In Spanish: — Algebras Krithivasan, Kamala. "Algebras" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Algebras (contd.)" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Algebras (contd.)" (Video). Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 40 - Lattices" (Video). Indian Institute of Technology Madras. — Groups Soto Espinosa, Jesús. "Grupos" (Video). Universidad Católica de Murcia (UCAM). Ovalle Rodríguez, Janinna. "El grupo simétrico" (Video). Universidad Abierta y a Distancia de México (UnADM). RTVE. "«La aventura del saber». Serie: «Más por menos». Capítulo: «La geometría se hace arte» (Script + presentation = Antonio Pérez) (Minute 5,22, the Alhambra, its mosaics and group theory; recommendation: see entire chapter)" (Video). Radio televisión española (RTVE). © gratisOA. — Examples of groups Soto Espinosa, Jesús. "Grupos. Ejemplo 1" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Grupos. Ejercicio 2" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Grupos. Ejercicio 3" (Video). Universidad Católica de Murcia (UCAM). Ovalle Rodríguez, Janinna. "El grupo aditivo de los enteros módulo n" (Video). Universidad Abierta y a Distancia de México (UnADM). Ovalle Rodríguez, Janinna. "Grupo de simetrías del cuadrado I. El grupo simétrico" (Video). Universidad Abierta y a Distancia de México (UnADM). Ovalle Rodríguez, Janinna. "Grupo de simetrías del cuadrado II. El grupo simétrico" (Video). Universidad Abierta y a Distancia de México (UnADM). Ovalle Rodríguez, Janinna. "Ciclos y órbitas" (Video). Universidad Abierta y a Distancia de México (UnADM). Ovalle Rodríguez, Janinna. "Grupos de permutaciones. El grupo simétrico" (Video). Universidad Abierta y a Distancia de México (UnADM). — Homomorphism of groups Soto Espinosa, Jesús. "Homomorfismo de grupos. Inyectividad. Sobreyectividad" (Video). Universidad Católica de Murcia (UCAM). — Rings Soto Espinosa, Jesús. "Anillo" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "El anillo $\mathbb {Z} _{n}$. Congruencias" (Video). Universidad Católica de Murcia (UCAM). — Integral domains Soto Espinosa, Jesús. "Dominio de integridad" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Unidad en un dominio de integridad" (Video). Universidad Católica de Murcia (UCAM). To find out more Portal:Mathematics And more: Multiplicative group of integers modulo n Theme 2.- Number theory Number theory Key concepts Main article: Number theory Main category: Number theory — Divisibility and modular arithmetic See also: Division (mathematics) and Category:Division (mathematics) 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ふぁい) Fermat-Euler theorem Fermat's little theorem Wilson's theorem Wolstenholme's theorem Table of congruences Josephus problem (first contact) — Applications of congruences Hash function Pseudorandom number - Cryptology Cryptology Cryptography RSA public-key cryptosystem RSA Factoring Challenge — Divisibility rules Numeral system Power residue Pascal's tape (es; fr) Divisibility rule — Diophantine equations Some useful background: Algebraic equation System of equations Vieta's formulas Ruffini's rule Diophantine equation System of linear Diophantine equations — Paradoxes Will Rogers phenomenon Bibliography: theory and proposed and solved exercises In English: In Spanish: —¤— Thomas Koshy. Elementary number theory with applications. Academic Press (an imprint of Elsevier Inc.), New York, United States, 2nd edition, 2007, ISBN: 978-0-12-372487-8 —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Chapter 4 and related exercises). McGraw-Hill, New York, New York, United States, 2012, ISBN 978-0-07-338309-5 Kenneth A. Rosen. Elementary number theory and its applications. Addison-Wesley, Reading, Massachusetts, United States, 1986, ISBN 0-201-06561 —¤— Máximo Anzola y José Caruncho. Problemas de Álgebra. Tomo 2. Anillos - Polinomios - Ecuaciones. 3ª edición. Primer Ciclo, Madrid, Spain. (Capítulo 7 «Divisibilidad en $\mathbb {N}$ y $\mathbb {Z}$», 94 ejercicios resueltos; Capítulo 8 «Ecuaciones diofánticas», 27 ejercicios resueltos; Capítulo 9 «Sistemas de numeración», 25 ejercicios resueltos), 1982. ISBN 84-300-6417-6. —¤— González Gutiérrez, Francisco José (2004a). "Apuntes de matemática discreta. 10. Divisibilidad. Algoritmo de la División" (PDF) (in Spanish). Cádiz, Andalusia (ES-AN), Spain: University of Cádiz (UCA). —¤— González Gutiérrez, Francisco José (2004b). "Apuntes de matemática discreta. 11. Teorema Fundamental de la Aritmética" (PDF) (in Spanish). Cádiz, Andalusia (ES-AN), Spain: University of Cádiz (UCA). —¤— González Gutiérrez, Francisco José (2004c). "Apuntes de matemática discreta. 12. Ecuaciones diofánticas" (PDF) (in Spanish). Cádiz, Andalusia (ES-AN), Spain: University of Cádiz (UCA). —¤— González Gutiérrez, Francisco José (2004d). "Apuntes de matemática discreta. 13. Clases de restos módulo m" (PDF) (in Spanish). Cádiz, Andalusia (ES-AN), Spain: University of Cádiz (UCA). —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. 5th edition. (Sections 2.4, 2.5, 2.6 and related exercises). McGraw-Hill/Interamericana de Spain, S.A.U., Aravaca (Madrid), Madrid, Community of Madrid (ES-MD), Spain, 2004, ISBN 84-481-4073-7 —¤— Sendra, J.; Martín, E.; Méndez, A.; Ortiz, C. (2011). "Tema 3: Ecuaciones diofánticas, congruencias y criterios de divisibilidad" (PDF) (in Spanish). Madrid, Community of Madrid (ES-MD), Spain: Technical University of Madrid (UPM). Software In English: In Spanish: WolframAlpha (Examples, Mathematics, Number Theory) WolframAlpha (RSA encryption) Multimedia In English: In Spanish: — Divisibility and modular arithmetic Baugher, Greg A. "Section 4.1 Divisibility and Modular Arithmetic" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. — Primes and GCD Baugher, Greg A. "Section 4.3 Primes and Greatest Common Divisors" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Simonson, Shai. "Euclid's algorithm" (Video). ArsDigita University. (Otra URL) Olson, Glenn R. "P1--Gear Problem and LCM" (Video). Maine East High School. Park Ridge, Illinois (US-IL), USA. — Solving congruences and their applications Baugher, Greg A. "Sections 4.4-4.5 Solving Congruences and Applications" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. — Diophantine equations Olavsku. "Number Theory 7 - Diophantine Equations" (Video). Olson, Glenn R. "N1--Introduction to Linear Diophantine Equations" (Video). Maine East High School. Park Ridge, Illinois (US-IL), USA. Olson, Glenn R. "N2--Solve Basic Linear Diophantine Equation" (Collection of videos). Maine East High School. Park Ridge, Illinois (US-IL), USA. Olson, Glenn R. "N3--Simplify and Solve Linear Diophantine Equation" (Collection of videos). Maine East High School. Park Ridge, Illinois (US-IL), USA. Olson, Glenn R. "N4--Solve Linear Diophantine Equation with Right Hand Side Not Equal to 1" (Collection of videos). Maine East High School. Park Ridge, Illinois (US-IL), USA. Olver, Brandon. "Linear Diophantine Equations - Ex 1" (Video). Ballarat Grammar. Wendouree, Ballarat, Victoria (AUえーゆー-VIC), Commonwealth of Australia. Olver, Brandon. "Linear Diophantine Equations - Ex 2" (Video). Ballarat Grammar. Wendouree, Ballarat, Victoria (AUえーゆー-VIC), Commonwealth of Australia. — Cryptography Baugher, Greg A. "Section 4.6 Cryptography" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Simonson, Shai. "Cryptography" (Video). ArsDigita University. (Otra URL) — Divisibility Soto Espinosa, Jesús. Divisibilidad. Ejemplo 1. UCAM Soto Espinosa, Jesús. Divisibilidad. Ejemplo 2. UCAM Soto Espinosa, Jesús. Divisibilidad. Ejemplo 3. UCAM Soto Espinosa, Jesús. Divisibilidad. Ejemplo 3+. UCAM Soto Espinosa, Jesús. Divisibilidad. Ejemplo 4. UCAM Soto Espinosa, Jesús. Divisibilidad. Ejemplo 5. UCAM Soto Espinosa, Jesús. Divisibilidad. Ejemplo 6 (inducción). UCAM Soto Espinosa, Jesús. Divisibilidad. Ejemplo 4 (inducción). UCAM Soto Espinosa, Jesús. "Algoritmo de la división" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. Algoritmo de la división. Ejemplo 1. UCAM Soto Espinosa, Jesús. Algoritmo de la división. Ejemplo 2. UCAM Soto Espinosa, Jesús. Algoritmo de la división. Ejercicio 3. UCAM — Primes and GCD Soto Espinosa, Jesús. "Números primos, ejemplo 1" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Números primos, ejemplo 2" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Números primos, ejemplo 3" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Números primos, ejemplo 5" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Infinitud de los números primos" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Teorema fundamental de la aritmética" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Máximo común divisor" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Máximo común divisor, ejemplo 1" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Máximo común divisor, ejemplo 2" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Máximo común divisor, ejemplo 3" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Máximo Común Divisor, ejemplo 4" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Algoritmo de Euclides" (Video). Universidad Católica de Murcia (UCAM). — Bézout's lemma Soto Espinosa, Jesús. "Identidad de Bézout" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Identidad de Bézout, ejemplo 1" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Identidad de Bézout, ejemplo 2" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). — Modular arithmetic. Euler's φふぁい function (totient function) Soto Espinosa, Jesús. "Función φふぁい de Euler" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Función φふぁい de Euler, propiedad 1" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Función φふぁい de Euler, propiedad 2" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). — Diophantine equations Hervás Jorge, Antonio. "Ecuaciones diofánticas" (Video). Universidad Politécnica de Valencia (UPV). Soto Espinosa, Jesús. "Ecuación diofántica" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación diofántica de dos variables" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación diofántica, Ejemplo 1" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación diofántica, Ejemplo 2" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación diofántica, Ejemplo 3" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación diofántica, Ejemplo 4" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación diofántica, Ejemplo 5" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación diofántica de tres variables, Ejercicio 1" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación diofántica de tres variables, Ejercicio 2" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Sistemas de ecuaciones diofánticas" (Video). Universidad Católica de Murcia (UCAM). — Congruences Congruencias. TurboOpenFing Soto Espinosa, Jesús. "Congruencias" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Congruencias: relación de equivalencia" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Congruencias. Propiedad 1" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Congruencias. Propiedad 2" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Congruencias. Propiedad 3" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Congruencias. Ejercicio 1" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Congruencias. Ejercicio 2" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Congruencias. Ejercicio 3" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Congruencias. Ejercicio 4" (Video). Universidad Católica de Murcia (UCAM). Ecuaciones Soto Espinosa, Jesús. "Ecuación de congruencias" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación de congruencias. Ejemplo 1" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación de congruencias. Ejemplo 2" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación de congruencias. Ejercicio 3" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación de congruencias. Ejercicio 4" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Ecuación de congruencias. Ejercicio 7" (Video). Universidad Católica de Murcia (UCAM). Sistemas Soto Espinosa, Jesús. "Sistemas de congruencias" (Video). Universidad Católica de Murcia (UCAM). — Congruences: Casting out nines Hervás Jorge, Antonio. "Congruencias de números enteros: La prueba del 9" (Video). Universidad Politécnica de Valencia (UPV). — False positives: Casting out elevens Hervás Jorge, Antonio. "Los falsos positivos: La prueba del 11" (Video). Universidad Politécnica de Valencia (UPV). — Power residues Soto Espinosa, Jesús. "Restos potenciales" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Restos potenciales. Ejercicio 1" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Restos potenciales. Ejercicio 2" (Video). Universidad Católica de Murcia (UCAM). — Divisibility rules Soto Espinosa, Jesús. "Criterios de divisibilidad" (Video). Universidad Católica de Murcia (UCAM). Soto Espinosa, Jesús. "Criterios de divisibilidad. Ejercicio 1" (Video). Universidad Católica de Murcia (UCAM). See also 1729 (number) 5040 (number) To find out more Portal:Mathematics And more: Divisibility Division algorithm (Algorithms for division) Primality Quadratic residue Quadratic reciprocity Primality test Pseudo-random number generation List of random number generators Cryptography Highly totient number Highly composite number Smooth number Rough number Semiprime Elliptic curve cryptography List of prime numbers List of numbers Theme 3.- Combinatorics Combinatorics Key concepts Main article: Combinatorics Main category: Combinatorics See also: Combinatorial principles — 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 Main article: Combinatorial proof Main category: Enumerative combinatorics 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 Partitions of a set Stirling numbers of the second kind Bell numbers Lah numbers · Catalan and Narayana numbers. Noncrossing partitions 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 Bibliography: theory and proposed and solved exercises In English: In Spanish: —¤— Kenneth P. Bogart. Combinatorics through guided discovery. 2004. https://math.dartmouth.edu/news-resources/electronic/kpbogart/ —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Chapters 6 and 8 and related exercises). McGraw-Hill, New York, New York, United States, 2012, ISBN 978-0-07-338309-5 —¤— Sedgewick, Robert (June 1977). "Permutation Generation Methods" (PDF). Computing Surveys. 9 (2): 137–164. © ARR. Máximo Anzola and José Caruncho. Problemas de Álgebra. Tomo 1. Conjuntos-Grupos. Primer Ciclo, Madrid, Spain. (Chapter 8 'Combinatoria', 31 solved problems), 1981. L. Barrios Calmaestra. Combinatoria. In: Proyecto Descartes. Ministry of Education, Government of Spain, 2007. (Open access). http://descartes.cnice.mec.es/materiales_didacticos/Combinatoria/combinatoria.htm M. Delgado Pineda. Material from «Curso 0: Matemáticas». Part: Combinatoria: Variaciones, Permutaciones y Combinaciones. Potencias de un binomio. OCW UNED. (Theory and exercises). 2010. (CC BY-NC-ND). http://ocw.innova.uned.es/matematicas-industriales/contenidos/pdf/tema5.pdf I. Espejo Miranda, F. Fernández Palacín, M. A. López Sánchez, M. Muñoz Márquez, A. M. Rodríguez Chía, A. Sánchez Navas and C. Valero Franco. Estadística Descriptiva y Probabilidad. Servicio de Publicaciones de la Universidad de Cádiz. (Appendix 1: Combinatoria). 2006. (GNU FDL). http://knuth.uca.es/repos/l_edyp/pdf/febrero06/lib_edyp.apendices.pdf —¤— Franco Brañas, José Ramón; Espinel Febles, María Candelaria; Almeida Benítez, Pedro Ramón (2008). Manual de combinatoria. Badajoz, Extremadura (ES-EX), España: @becedario. ISBN 978-84-96560-73-4. © ARR. —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. 5th edition. (Chapters 4 and 6 and related exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, ISBN 84-481-4073-7 Software In English: In Spanish: WolframAlpha (Examples, Mathematics, Combinatorics) Multimedia In English: In Spanish: Baugher, Greg A. "Section 6.1 The Basics of Counting" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Baugher, Greg A. "Sections 6.3 & 6.5 Permutations and Combinations" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Baugher, Greg A. "Section 6.6 Generating Permutations and Combinations" (Video). Department of Math/Science/Informatics, Penfield College, Mercer University. Georgia (US-GA), USA. Krithivasan, Kamala. "Lecture 27 - Pigeonhole Principle" (Video). Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India: Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 28 - Permutations and combinations" (Video). Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India: Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 29 - Permutations and Combinations (cont.)" (Video). Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India: Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 30 - Generating Functions" (Video). Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India: Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 31 - Generating Functions (cont.)" (Video). Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India: Indian Institute of Technology Madras. Simonson, Shai. "Lecture 11 - Combinations and permutations" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Lecture 12 - Counting" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Lecture 13 - Counting (cont.)" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Lecture 14 - Counting problems using combinations, distributions" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Lecture 15 - Counting problems using combinations, distributions (cont.)" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Lecture 16 - The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Recitation 1 - A combinatorial card trick" (Video). ArsDigita University. (Another URL) — Basic principles Soto Espinosa, Jesús. "Principios básicos de conteo" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Ejercicios" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). (Ejercicio 3). — Variations, permutations and combinations Soto Espinosa, Jesús. "Variaciones" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Variaciones con repetición" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Permutaciones" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Permutaciones, ejemplo 1" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Permutaciones circulares" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Permutaciones con repetición" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Combinaciones" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Combinaciones con repetición" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Ejercicios" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). — Binomial numbers Soto Espinosa, Jesús. "Número binomial" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Número binomial, ejercicio 2" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Número binomial, ejercicio 3" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Número binomial, ejercicio 5" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Número binomial, fórmula de Stifel" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Coeficiente Multinomial, ejercicio 1" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Combinatoria, ejemplo 6" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Combinatoria, ejemplo 7" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Combinatoria, ejemplo 8" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Combinatoria, ejemplo 9" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Combinatoria, ejemplo 10" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Combinatoria, ejemplo 11" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Teorema del binomio" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Fórmula de Leibniz" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Ejercicios" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). — Inclusion-exclusion principle Soto Espinosa, Jesús. "Principio de inclusión-exclusión" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Generalización del principio de inclusión-exclusión" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Principio de inclusión-exclusión - Ejemplo 1" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Desarreglos" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Contando desarreglos" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Ejercicios" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). — Partitions Soto Espinosa, Jesús. "Particiones. Número de Bell" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Número de Stirling de segunda clase" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). Soto Espinosa, Jesús. "Ejercicios" (Video). Guadalupe, Murcia, Region of Murcia (ES-MC), Spain: Escuela Politécnica Superior, Universidad Católica San Antonio de Murcia (UCAM). See also 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). To find out more Portal:Mathematics And more: Generating functions Examples of generating functions Theme 4.- Difference equations Finite difference equations (recurrence relations) Key concepts Main article: Recurrence relation Main category: Recurrence relations Some useful previous concepts: Recursive definition, Recursion and Recursion (computer science) — Linear difference equations Main article: Linear difference equation Recurrence relation (difference equation) Solving homogeneous linear recurrence relations with constant coefficients Solving non-homogeneous linear recurrence relations with constant coefficients — Linear discrete dynamical systems - Population dynamics Population dynamics - Linear discrete dynamical models Dynamical system - BIDE models Matrix population models - Markov chains Markov chain — Solving equations numerically Main article: Root-finding algorithm Main category: Root-finding algorithms See also: Numerical analysis Fixed-point iteration Secant method Connections — Computational complexity Main article: Computational complexity theory - Analysis of algorithms Main article: Analysis of algorithms Bibliography: theory and proposed and solved exercises En español: En inglés: Richard Johnsonbaugh. Discrete mathematics. 6th edition. (Chapter 7 and corresponding exercises). Prentice Hall Inc., New York, New York, United States, 2005, ISBN 0-13-117686-2 —¤— Eric Lehman, F. Thomson Leighton, Albert R. Meyer. Mathematics for Computer Science. (Chapter V). 2017 (2 May). CC BY-SA. https://courses.csail.mit.edu/6.042/spring17/mcs.pdf —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Chapters 5 and 8 and corresponding exercises). McGraw-Hill, Nueva York, Nueva York, Estados Unidos, 2012, ISBN 978-0-07-338309-5 Wikibooks contributors. Discrete Mathematics/Recursion. Wikibooks. https://en.wikibooks.org/wiki/Discrete_Mathematics/Recursion Javier Cobos Gavala. Apuntes de introducción a la matemática discreta. (E.T.S. de ingeniería informática, University of Seville). http://ma1.eii.us.es/material/IMD_ii_Ap.pdf Colaboradores de Wikilibros. Matemática Discreta/Relaciones de Recurrencia. Wikilibros. https://es.wikibooks.org/wiki/Matem%C3%A1tica_Discreta/Relaciones_de_Recurrencia —¤— Richard Johnsonbaugh. Matemáticas discretas. 6th edition. (Chapter 7 and corresponding exercises). Pearson Educación de México, S.A. de C.V., Naucalpan de Juárez, Edo. de México, México, 2005, ISBN 970-26-0637-3 Alfredo Méndez Alonso, Eduardo Martín Novo, Carmen Ortiz Martínez and Juana Sendra Pons. Tema 4: Relaciones de recurrencia. OCW-UPM, 2011. CC BY-NC-SA. http://ocw.upm.es/matematica-aplicada/matematica-discreta/contenidos/material-de-clase/tema-4 —¤— Navas Ureña, Juan; Esteban Ruiz, Francisco José; Quesada Teruel, José María (2009a). "Tema 9. Ecuaciones y sistemas en diferencias" (PDF). Modelos matemáticos en biología. Teoría (PDF). Jaén, Andalucía (ES-AN), España: Universidad de Jaén. Departamento de Matemáticas. © TDR. —¤— Navas Ureña, Juan; Esteban Ruiz, Francisco José; Quesada Teruel, José María (2009b). "Tema 2. Modelos discretos II" (PDF). Modelos matemáticos en biología. Ejercicios resueltos y propuestos (PDF). Jaén, Andalucía (ES-AN), España: Universidad de Jaén. Departamento de Matemáticas. © TDR. —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. 5th edition. (Chapters 4 and 6 and corresponding exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, ISBN 84-481-4073-7 Enrique Vílchez Quesada. Resolución de relaciones de recurrencia lineales no homogéneas con coeficientes constantes a través de valores y vectores propios. Uniciencia, 24, 2010, pp. 121-132. https://dialnet.unirioja.es/servlet/articulo?codigo=5381351 Álex Zeballos. Solución a Ecuaciones de Recurrencia Online - Usando Wolfram Alpha. http://wolframalpha0.blogspot.com.es/2012/01/solucion-ecuaciones-de-recurrencia.html Software In English: In Spanish: WolframAlpha (Examples, Mathematics, Recurrences) Multimedia In English: In Spanish: Krithivasan, Kamala. "Lecture 32 - Recurrence relations" (Video). Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India: Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 33 - Recurrence relations (cont.)" (Video). Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India: Indian Institute of Technology Madras. Krithivasan, Kamala. "Lecture 34 - Recurrence relations (cont.)" (Video). Adyar, Chennay (formerly Madras), Tamil Nadu (IN-TN), India: Indian Institute of Technology Madras. Simonson, Shai. "Lecture 6 - Basic arithmetic and geometric sums, closed forms" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Lecture 7 - Chinese rings puzzle" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Lecture 8 - Solving recurrence equations" (Video). ArsDigita University. (Another URL) Simonson, Shai. "Lecture 9 - Solving recurrence equations (cont.)" (Video). ArsDigita University. (Another URL) Rodríguez Álvarez, María José. "Método de la ecuación característica para recurrencias homogéneas lineales de segundo orden con coeficientes constantes (raíces reales)" (Video). Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática. Valencia, Valencian Community (ES-VC), Spain: Universidad Politécnica de Valencia (UPV). (Otra URL) Rodríguez Álvarez, María José. "Recurrencias lineales de segundo orden con coeficientes constantes no homogénea (constante)" (Video). Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática. Valencia, Valencian Community (ES-VC), Spain: Universidad Politécnica de Valencia (UPV). (Otra URL) Rodríguez Álvarez, María José. "Recurrencias no homogéneas lineales de segundo orden con coeficientes constantes" (Video). Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática. Valencia, Valencian Community (ES-VC), Spain: Universidad Politécnica de Valencia (UPV). (Otra URL) Rodríguez Álvarez, María José. "Resolución de recurrencias con Mathematica" (Video). Departamento de Matemática Aplicada, Escuela Técnica Superior de Ingeniería Informática. Valencia, Valencian Community (ES-VC), Spain: Universidad Politécnica de Valencia (UPV). (Otra URL) To find out more Integer sequences List of integer sequences in the OEIS that have their own English Wikipedia entries Index to OEIS: Section Recurrent Sequencies Recursion (computer science) Exponential factorial Ackermann function McCarthy 91 function Tower of Hanoi Josephus problem Appendices Graphs Key concepts 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 Bibliography: theory and proposed and solved exercises In English: In Spanish: —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Chapters 10 and 11 and corresponding exercises). McGraw-Hill, New York, New York, United States, 2012, ISBN 978-0-07-338309-5 —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. 5th edition. (Chapters 8 and 9 and corresponding exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, ISBN 84-481-4073-7