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# Epistemowikia:Proyecto de aprendizaje/Discrete and numerical mathematics

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 Discrete and numerical mathematics Free learning project. (Free as defined in Freedomdefined).

Fundamentals

• Logic
• Sets[TI], relations[TI] and functions[TI]
• Cardinality
• Sequences[TI] and sums[TI]
• Matrices[TI]

Numbers and numericals

• Algorithms[TI]
• Number theory
• Induction[TI] and recursion[TI]
• Verification[TI]
• Computability[TI]
• Numerical Algebra and Calculus

Counting, recounting and discrete non-wild guessing

• Combinatorial theory
• Discrete probability[TI]
• Recurrence relations
• Stochastic processes[TI]

Visualizing relationships: graphs, trees and networks

• Algebraic structures[TI]
• Graphs
• Trees
• Networks
• Game theory[TI]
• Optimization[TI]
[TI] = Transversal issue.
The Polytechnic School of the University of Extremadura at the Campus of Cáceres, Extremadura, Spain.

The learning community of the course 'Further Mathematics' at the Escuela Politécnica (University of Extremadura), in Cáceres, hopes to contribute to the English Wikipedia through this university project.

## Rationale

It is our aim with this project to contribute to that the so-called evaluation or assessment stops from being a mere measuring instrument of the extent to which students have assimilated the contents received, and it becomes a learning experience, being able of bringing to light, capacities, abilities, attitudes and values that have been intrinsic, reinforced or acquired during the learning process. At the same time, it puts to the test their understanding skills, their abilities to practical work and their creativity, reinforcing the capacity of assuming social responsibilities and the interpersonal and civic competences. In particular, the main elements linked to learning that are strengthened are:

• Peer learning
• Sharing and giving
• Research skills
• Technical and communication skills
• Cooperating and collaborating
• Critical thinking
• Collaborative problem solving
• Individual responsibility
• Socialization
• Working and processing in teams
• Knowledge transfer to society

## Work schedule

### Type and minimum number of contributions

Each student has to do a minimum of four major contributions to the English Wikipedia. These contributions must have to do with discrete or numerical mathematics and students must do at least one contribution per each header topic: «Fundamentals», «Numbers and numericals», «Counting, recounting and discrete non-wild guessing» and «Visualizing relationships: graphs, trees and networks».

Each of these contributions would focus on one or more of the following activities:

1. Contributing to existing articles.
After relevant articles to be worked have been found, student would focus their contribution on one or more of the following activities and targets:
1. Expanding and improving articles.
2. Critical analysis of existing articles (on the talk page of the article).
(Respect what the community of the English Wikipedia indicates on the help page about using talk pages and in the talk page guidelines).
3. Adding examples and case studies.
4. Adding theoretical and practical applications and cases of use, specially in the field of Science, Technology and Society [STS].
6. Conceptual correction.
7. Style correction.
8. Improving an article to be exported to the English Wikipedia and getting it to be highlighted.
(Read what the community of the English Wikipedia considers a good article and a featured article).
9. Translating articles.
(Respect what the community of the English Wikipedia indicates on the help page about translation).
2. Creating new articles.
After the topic choice and the related English Wikipedia contents review have been made, student could choose to create a new article.

In all cases, respect what the community of the English Wikipedia indicates on the page about writing better articles — that, specifically, includes direct links to the help page on editing (how to edit a page) and to the manual of style —.

### Due dates

• Monday, February 6, 2017: Beginning date of the academic component in the 2nd semester of the academic year 2016-2017.
• Thursday, February 9, 2017: You should have joined the Epistemowikia, if not yet, and become familiar with it, specially with edition — for this to happen you can take its help page (in Spanish) as a starting point —.
• Tuesday, February 21, 2017: Due date for having joined the English Wikipedia, if not yet, and for having chosen the articles of which you become responsible (two highly recommended starting points to address this issue would be the Logic portal and the Thinking portal).
However, from the moment it is clear to you, you can join the list at the English Wikipedia and start to work there.
• Thursday, April 6, 2017: You should have continually published the public part of a first self-report about what you have developed up to that moment, on your logbook and on the project contributions page.
• Thursday, May 11, 2017: You should have continually published the public part of your self-report including all you have done, on your logbook page and on the project contributions page.
• Friday, May 19, 2017: Ending date of the academic component in the 2nd semester of the academic year 2016-2017.
• Important: It is worth noting that Epistemowikia and Wikipedia are public, free and open wiki sites, therefore out of that date range the project stays open for ever.

### Dynamic obligations

While the project is active, each student is required to:

• develope all their contributions in accordance with the quality criteria of the English Wikipedia;
• monitor the comments and the edits to their articles;
• update their logbook as they work on their contributions;
• collaborate with the rest of the students by reading and reviewing their work and helping them as much as possible.

## Starting on Epistemowikia

We use Epistemowikia to develop our major contributions until they reach the minimum standard required by the English Wikipedia.

Some of the following lines are shared by both Epistemowikia and Wikipedia, and what is more, both sites are similar in many of them.

1. If you want to collaborate, you have to join the Epistemowikia.
1. First, create an account.
3. On your personal preferences page, you must activate the following three options:
[X] Enable email from other users
4. [X] Email me when a page or a file on my watchlist is changed
[X] Email me when my user talk page is changed
5. Here is your watchlist. You must include the course forums in it. You can, for instance, edit your raw watchlist including the following page names:
Epistemowikia:Plan de aprendizaje/Ampliación de Matemáticas - Further Mathematics/Actividades/Foros/Noticias, Actualidad y Ocio - News, Current Events and Leisure
Epistemowikia:Plan de aprendizaje/Ampliación de Matemáticas - Further Mathematics/Actividades/Foros/Cafetería - Cafe
Epistemowikia:Plan de aprendizaje/Ampliación de Matemáticas - Further Mathematics/Actividades/Foros/Foro AM-FM - Forum AM-FM
Epistemowikia:Plan de aprendizaje/Ampliación de Matemáticas - Further Mathematics/Actividades/Foros/Metodología-Evaluación - Methodology-Assessment
Epistemowikia:Plan de aprendizaje/Ampliación de Matemáticas - Further Mathematics/Actividades/Foros/Trabajos Fin de Asignatura - Term papers
Epistemowikia:Proyecto de aprendizaje/Matemática discreta y numérica
Epistemowikia:Proyecto de aprendizaje/Discrete and numerical mathematics
Thus, as 'Email me when a page or a file on my watchlist is changed' option is checked, you will be emailed every time a message is posted to any of these forums.
Note: Observe that, at any time, you can include a page into your watchlist simply checking the star that appears near the 'View history' tab.
6. Join the project on this dedicated page.
7. Include your major contributions to the project on this common dedicated page.
8. Any person can watch the list of all your contributions easily on Special:Contributions. Although anyone interested could follow your dedication to the project, you should highlight your major contributions on a personal dedicated page. This must be a subpage of your user page, that you will use as a public, free/libre and open logbook in which you will write down every and each of your contributions (that is, what is finished, what is being depveloped and what still has not started, including reference dates). You can easily create your logbook page, just browse to the following address, replace 'myusername' with your user name and push 'Enter':
9. Although there is a test page, it seems reasonable that you have a personal one (similar again, browse to the following address, replace 'myusername' with your user name and push 'Enter'):
10. Important:
1. On Epistemowikia, mathematical formulas are represented using MathML (through a third-party Mathoid server). If, at some point, you see an error message instead of the formula, browse to your appearance preferences page where you can check the PNG option instead (the first option).
2. If you make a minor edition, please check that option before saving the page. Doing this you indicate the community that the change made is not an essential change. On the other side, it generates a smaller load on the data base.

## On the English Wikipedia

All of the above, although they may lead to parallel but different contributions to Epistemowikia (for example, because of the social point of view), has served as a training ground for effective contributing to the English Wikipedia.

• Create an account, if you do not have one yet.
• You could modify, in accordance with your wishes, your preferences. For instance, on your personal preferences page, you must activate the following three options:
• [X] Enable email from other users
• [X] Email me when a page on my watchlist is changed
• Join the project on this dedicated page.
• Include your major contributions to the project on this common dedicated page.
• Your test page is one subpage of your user page, called Sandbox.
• If you make a minor edition, please check that option before saving the page. Doing this you indicate the community that the change made is not an essential change. On the other side, it generates a smaller load on the data base.
• If you wish to create an article and you are not sure of how to proceed, you should use the article wizard.
• Remember, the project main page on the English Wikipedia is: Wikipedia:School and university projects/Discrete and numerical mathematics

## Self-report

In all cases, every and each student must continually write, keeping up with the developing of their work, a self-report, on their logbook, about the whole of their contribution to the project and justify the relationship with the four heading topics. A simple format could be:

Private part:

Student (name and surname): __________
Username on the English Wikipedia (username and URL locator of their user page): __________
Comprehensive assessment of all the work done: __________

Public, free/libre & open part (it will be a copy on Epistemowikia): __________

Username on the English Wikipedia (username and URL locator of their user page): __________
Major contributions to the English Wikipedia (page titles and their URL locators): __________
Other contributions to the English Wikipedia: __________
Comprehensive summary of all the work done, justifying their relationship with the four heading topics: __________

## Learning university project on discrete mathematics and numerical analysis: learning paths on Wikipedia

### Examples of exam and some solutions

Important note: In an exam, both theoretical (including proofs of theorems) and practical questions about all topics worked out in class may be asked. In no case, the concreteness of the questions within the context of these examples, implies a course content cut.

#### Part 1: Themes 1 and 2

##### Example of exam, 1

Maximum time: 2 hours.

Question 1. (2.5 points)
With the help of propositional logic, prove that the following argument is valid or not. "This program will compile whenever we have declared the variables. Mind you, we will declare the variables precisely if we do not forget to do it. It turns out that the program has not compiled. Therefore, we have forgotten to declare the variables."
Important: Do not do it using truth tables.

 Solución: You can check the complete solution through semantic tableaux of a typical example in this document (in Spanish, for the time being).

Question 2. (2.5 points)

• (a) Propose three sets ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$, such that ${\displaystyle A\in B}$, ${\displaystyle B\in C}$ and ${\displaystyle A\notin C}$. (0,5 points).
• (b) According to a survey of a certain group of students, they said that, if they had to decide between two courses, equally interesting because of their contents, they prefer that one for which the time they dedicate to study it is the lowest and for which they foresee the best results in exams. In case of equality of study times and of exam results forecasts, they are indifferent to them. Study the properties of this binary relation. (2 points).

Question 3. (2.5 points)
Find a possible general formula for computing the nth term, that is, ${\displaystyle a_{n}}$, of the sequence ${\displaystyle a_{1}=1,a_{2}=2,a_{3}=4,a_{4}=7}$ using the Newton's divided differences interpolation polynomial.

Question 4. (2.5 points)
In base-ten (decimal numeral system), find the digits ${\displaystyle x,y}$ such that the number ${\displaystyle 12xy567}$ be divisible by ${\displaystyle 33}$.

Solución:
${\displaystyle 33=3\times 11}$.

A number is divisible by ${\displaystyle 3}$ precisely if the sum of all its digits is divisible by ${\displaystyle 3}$:

${\displaystyle 3\,\vert \,12xy567\Leftrightarrow 3\,\vert \,(7+6+5+y+x+2+1)}$
this is:
{\displaystyle {\begin{aligned}3\,\vert \,12xy567&\Leftrightarrow 3\,\vert \,(21+x+y)\\&\Leftrightarrow 21+x+y={\dot {3}}\\&\Leftrightarrow x+y={\dot {3}}-21\end{aligned}}}
Moreover:
${\displaystyle x,y{\mbox{ are digits in base-ten }}\Rightarrow 0\leqslant x,y\leqslant 9}$
then:
${\displaystyle 0\leqslant x+y\leqslant 18}$
We have to find out what differences ${\displaystyle {\dot {3}}-21}$ satisfy the fact of belonging to ${\displaystyle \left[0,18\right]}$:
${\displaystyle {\dot {3}}-21=\left\lbrace \ldots ,0(=21-21),3(=24-21),6(=27-21),9(=30-21),12(=33-21),15(=36-21),18(=39-21),\ldots \right\rbrace }$
so there are ${\displaystyle 7}$ possible cases:
${\displaystyle x+y=0\veebar x+y=3\veebar x+y=6\veebar x+y=9\veebar x+y=12\veebar x+y=15\veebar x+y=18}$

A number is divisible by ${\displaystyle 11}$ precisely if the sum of its digits at even places minus the sum of its digits at odd places is divisible by ${\displaystyle 11}$:

${\displaystyle 11\,\vert \,12xy567\Leftrightarrow 11\,\vert \,((7+5+x+1)-(6+y+2))}$
this is:
{\displaystyle {\begin{aligned}11\,\vert \,12xy567&\Leftrightarrow 11\,\vert \,(5+x-y)\\&\Leftrightarrow 5+x-y={\dot {11}}\\&\Leftrightarrow x-y={\dot {11}}-5\end{aligned}}}
Moreover:
${\displaystyle x,y{\mbox{ are digits in base-ten }}\Rightarrow 0\leqslant x,y\leqslant 9}$
then:
${\displaystyle -9\leqslant x-y\leqslant 9}$
We have to find out what differences ${\displaystyle {\dot {11}}-5}$ satisfy the fact of belonging to ${\displaystyle \left[-9,9\right]}$:
${\displaystyle {\dot {11}}-5=\left\lbrace \ldots ,-5(=0-5),6(=11-5),\ldots \right\rbrace }$
so there are ${\displaystyle 2}$ possible cases:
${\displaystyle x-y=-5\veebar x-y=6}$

Therefore, there are ${\displaystyle 7\times 2=14}$ possible cases:

Table of possible cases
Λ ${\displaystyle x+y=0}$ ${\displaystyle x+y=3}$ ${\displaystyle x+y=6}$ ${\displaystyle x+y=9}$ ${\displaystyle x+y=12}$ ${\displaystyle x+y=15}$ ${\displaystyle x+y=18}$
${\displaystyle x-y=-5}$ ${\displaystyle 2x=-5}$
${\displaystyle x={\frac {-5}{2}}}$
${\displaystyle 2x=-2}$
${\displaystyle x=-1}$
${\displaystyle 2x=1}$
${\displaystyle x={\frac {1}{2}}}$
${\displaystyle 2x=4}$
${\displaystyle x=2}$
${\displaystyle y=7}$
${\displaystyle 2x=7}$
${\displaystyle x={\frac {7}{2}}}$
${\displaystyle 2x=10}$
${\displaystyle x=5}$
${\displaystyle y=10}$
${\displaystyle 2x=13}$
${\displaystyle x={\frac {13}{2}}}$
No: ${\displaystyle x}$ is not a digit
in base-ten
No: ${\displaystyle x}$ is not a digit
in base-ten
No: ${\displaystyle x}$ is not a digit
in base-ten
Yes: ${\displaystyle x,y}$ are digits
in base-ten
No: ${\displaystyle x}$ is not a digit
in base-ten
No: ${\displaystyle y}$ is not a digit
in base-ten
No: ${\displaystyle x}$ is not a digit
in base-ten
${\displaystyle x-y=6}$ ${\displaystyle 2x=6}$
${\displaystyle x=3}$
${\displaystyle y=-3}$
${\displaystyle 2x=9}$
${\displaystyle x={\frac {9}{2}}}$
${\displaystyle 2x=12}$
${\displaystyle x=6}$
${\displaystyle y=0}$
${\displaystyle 2x=15}$
${\displaystyle x={\frac {15}{2}}}$
${\displaystyle 2x=18}$
${\displaystyle x=9}$
${\displaystyle y=3}$
${\displaystyle 2x=21}$
${\displaystyle x={\frac {21}{2}}}$
${\displaystyle 2x=24}$
${\displaystyle x=12}$
No: ${\displaystyle y}$ is not a digit
in base-ten
No: ${\displaystyle x}$ is not a digit
in base-ten
Yes: ${\displaystyle x,y}$ are digits
in base-ten
No: ${\displaystyle x}$ is not a digit
in base-ten
Yes: ${\displaystyle x,y}$ are digits
in base-ten
No: ${\displaystyle x}$ is not a digit
in base-ten
No: ${\displaystyle x}$ is not a digit
in base-ten

So there are three possible solutions: ${\displaystyle {\binom {x}{y}}\in \left\lbrace {\binom {2}{7}},{\binom {6}{0}},{\binom {9}{3}}\right\rbrace }$.

Thus, the possible numbers are: ${\displaystyle 1227567,1260567,1293567}$,

which are divisibles by ${\displaystyle 33}$. Their quotients are: ${\displaystyle 37199,38199,39199}$.

##### Example of exam, 2

Maximum time: 2 hours.

Question 1. (2,5 points).

• a) Define adequate set of connectives (asc), also called completely expressive or functionally complete set of connectives.
• b) Provide two examples of two-element asc, explaining why they are so and assuming that we know the asc which elements are the most usual connectives ${\displaystyle \left\lbrace \neg ,\wedge ,\vee ,\rightarrow ,\leftrightarrow \right\rbrace }$.
 Solución: a) In Propositional Logic, an adequate set of connectives (asc) is any set of connectives such that every logical connective can be represented as an expression involving only those belonging to the asc. b) As it is said in the wording, we assume that we know that the set of the most usual connectives, ${\displaystyle \lbrace \neg ,\wedge ,\vee ,\rightarrow ,\leftrightarrow \rbrace }$ is an asc. Two two-element asc are the sets ${\displaystyle \lbrace \neg ,\wedge \rbrace }$ and ${\displaystyle \lbrace \neg ,\vee \rbrace }$. In effect, we only have to check, for each two-element set, that the missing most usual connectives may be represented only with the ones in the set: {\displaystyle {\begin{aligned}\lbrace \neg ,\wedge \rbrace :p\vee q&\equiv \neg (\neg p\wedge \neg q)\\p\rightarrow q&\equiv \neg (p\wedge \neg q)\\p\leftrightarrow q&\equiv \neg (p\wedge \neg q)\wedge \neg (\neg p\wedge q)\\\lbrace \neg ,\vee \rbrace :p\wedge q&\equiv \neg (\neg p\vee \neg q)\\p\rightarrow q&\equiv \neg p\vee q\\p\leftrightarrow q&\equiv \neg (\neg (\neg p\vee q)\vee \neg (p\vee \neg q))\end{aligned}}}

Question 2. (2,5 points).
Proof by definition that ${\displaystyle \mathbb {N} }$ is an infinite set.

 Solución: A set is infinite precisely if there exists a bijection between it and one of its proper subsets (definition by Dedekind). As an example, consider ${\displaystyle f:\mathbb {N} \rightarrow \mathbb {N} \setminus \{0\}}$, defined by ${\displaystyle n\mapsto f(n)=n+1}$. Let us prove that it is a bijective mapping. In effect: ${\displaystyle f}$ is a mapping ${\displaystyle \Leftrightarrow \left(\forall x\in \mathbb {N} \right)\left(\exists y\in \mathbb {N} \setminus \{0\}\right)\left(f(x)=y\right)\wedge \left(\forall x,x'\in \mathbb {N} \right)\left(x=x'\rightarrow f(x)=f(x')\right)}$, which is trivial, as if ${\displaystyle x\in \mathbb {N} }$ is given and because of the definition of ${\displaystyle f}$, there exists ${\displaystyle y_{x}=x+1\in \mathbb {N} \setminus \{0\}}$, this ${\displaystyle y_{x}}$ being unique for each ${\displaystyle x}$, that is, that if ${\displaystyle x=x'}$, then, because of the definition of ${\displaystyle f}$, ${\displaystyle f(x)=x+1=y_{x}=y_{x'}=x'+1=f(x')}$; ${\displaystyle f}$ is injective ${\displaystyle \Leftrightarrow \left(\forall x,x'\in \mathbb {N} \right)\left(f(x)=f(x')\rightarrow x=x'\right)}$, which is trivial because of the definition of ${\displaystyle f}$, as if ${\displaystyle f(x)=f(x')}$, that is, if ${\displaystyle x+1=x'+1}$, then, ${\displaystyle x=x'}$; ${\displaystyle f}$ is surjective ${\displaystyle \Leftrightarrow \left(\forall y\in \mathbb {N} \setminus \{0\}\right)\left(\exists x\in \mathbb {N} \right)\left(f(x)=y\right)}$, which is also trivial due to the definition of ${\displaystyle f}$, as if ${\displaystyle y}$ is given, then ${\displaystyle x=y-1}$ satisfies ${\displaystyle f(x)=f(y-1)=(y-1)+1=y}$.

Question 3. (2,5 points).
Abigail wants to send Balbina the most simple call message: eh. They can only send numbers. Abigail and Balbina use the letters' position in the alphabet to code them (thus, Abigail codes e as 06 and h as 08). They use RSA to encrypt their messages. If Abigail choose ${\displaystyle p=3}$ and ${\displaystyle q=7}$ as the ground primes for RSA:

• a) imagine you are Abigail and obtain the encrypted message that you have to send to Balbina;
• b) imagine you are Balbina and decrypt the encrypted message that Abigail has sent to you.

Question 4. (2,5 points).
One company spent ${\displaystyle 100000}$ euros in buying ${\displaystyle 100}$ electronic devices, some of them ground breaking and providing maximum performance. Smartphones were ${\displaystyle 50}$ euros each, tablets were ${\displaystyle 1000}$ euros each and laptops were ${\displaystyle 5000}$ euros each. How many of each device did they buy? Solve this question using the theory of:

• a) diophantine equations;
• b) congruence equations.

#### Part 2: Themes 3 and 4

##### Example of exam, 1

Maximum time: 2 hours.

Question 1. (2.5 points)
...

Question 2. (2.5 points)
...

Question 3. (2.5 points)
...

Question 4. (2.5 points)
...

### Logic

To find out more:

1. The Logic Portal
2. The Thinking Portal
3. And more:

#### Bibliography: theory and proposed and solved exercises

In English:

• —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Chapter 1 and related exercises). McGraw-Hill, New York, New York, United States, 2012, ISBN 978-0-07-338309-5

In Spanish:

• —¤— Kenneth A. Rosen. Matemática discreta y sus aplicaciones. 5th edition. (Sections 1.1, 1.2, 1.3, 1.4, 1.5, 3.1 and related exercises). McGraw-Hill/Interamericana de España, S.A.U., Aravaca (Madrid), Madrid, Spain, 2004, ISBN 84-481-4073-7

#### Software

In English:

• KDEM.DA.HD.703-01.pdf (download) (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)

In Spanish:

---

### Cardinality

To find out more:

#### Bibliography: theory and proposed and solved exercises

In English:

• —¤— Kenneth A. Rosen. Discrete mathematics and its applications. 7th edition. (Sections 2.1, 2.2, 2.3, 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

In Spanish:

### Number theory

#### Cryptography

To find out more:

1. The Number Theory Portal
2. The Cryptography Portal
3. And more:
1. Divisibility
2. Primality
3. Pseudo-random number generation
4. Cryptography
5. List of notable numbers

#### Bibliography: theory and proposed and solved exercises

In English:

• —¤— 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

In Spanish:

### Numerical analysis

To find out more:

In English:

In Spanish:

### Combinatorial theory

To find out more:

#### Bibliography: theory and proposed and solved exercises

In English:

• —¤— 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

In Spanish:

• 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
• —¤— José Ramón Franco Brañas, María Candelaria Espinel Febles and Pedro Ramón Almeida Benítez. Manual de combinatoria. @becedario, Badajoz, Spain, 2008. ISBN: 978-84-96560-73-4
• —¤— 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

### Recurrence relations

To find out more:

### Algebraic structures

To find out more:

### Graphs, trees and networks

To find out more:

### Complimentary knowledge pills

#### History

The template {{learning project}} has to be added to the talk page of every article that has been created or changed on Epistemowikia as part of the learning project. When you move your articles to Wikipedia in English you have to change it to the {{Educational assignment}} template.