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Epistemowikia 
Epistemowikia:Proyecto de aprendizaje/Discrete and numerical mathematics

Fundamentals

Numbers and numericals

Counting, recounting and discrete nonwild guessing

Visualizing relationships: graphs, trees and networks
 
[TI] = Transversal issue. 
The learning community of the course 'Further Mathematics' at the School of Technology (Escuela Politécnica, EPCC) of the University of Extremadura, in Cáceres, hopes to contribute to the English Wikipedia through this university project.
Contenido
 1 Rationale
 2 Work schedule
 3 Starting on Epistemowikia
 4 On the English Wikipedia
 5 Selfreport
 6 Learning university project on discrete mathematics and numerical analysis: learning paths on Wikipedia
 6.1 Considerations
 6.2 Examples of exam (selfassessment mock exams) and some solutions
 6.3 Logic
 6.4 Cardinality
 6.5 Number theory
 6.5.1 Divisibility
 6.5.2 Primes
 6.5.3 Diophantine equations, systems of Diophantine equations and on their solutions
 6.5.4 Modular arithmetic
 6.5.5 Congruence equations, systems of congruence equations and on their solutions
 6.5.6 Cryptography
 6.5.7 Bibliography: theory and proposed and solved exercises
 6.5.8 See also
 6.6 Numerical analysis
 6.7 Combinatorial theory
 6.8 Recurrence relations
 6.9 Algebraic structures
 6.10 Graphs, trees and networks
 6.11 Complimentary knowledge pills
 6.12 Minihackatones (miniencuentros intensivos de aprendizaje en colaboración) / Minihackathons (intensive collaborative learning meetings)
 7 Templates
 8 See also
 9 External links
Rationale
It is our aim with this project to contribute to that the socalled 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 nonwild guessing» and «Visualizing relationships: graphs, trees and networks».
Each of these contributions would focus on one or more of the following activities:
 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: Expanding and improving articles.
 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).  Adding examples and case studies.
 Adding theoretical and practical applications and cases of use, specially in the field of Science, Technology and Society [STS].
 Adding notes, references, bibliography, inner links, external links and multimedia content (photos, illustrations, videos).
 Conceptual correction.
 Style correction.
 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).  Translating articles.
(Respect what the community of the English Wikipedia indicates on the help page about translation).
 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 20162017.
 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 selfreport 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 selfreport 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 20162017.
 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.
 If you want to collaborate, you have to join the Epistemowikia.
 First, create an account.
 After doing that, your email must be confirmed. To do that, please log in with your username and password and go to your preferences page and follow the directions from there. If you do this, your edits will be attributed to your username, along with other benefits.
 You could modify, in accordance with your wishes, your preferences.
 On your personal preferences page, you must activate the following three options:
 [X] Enable email from other users
 [X] Email me when a page or a file on my watchlist is changed
 [X] Email me when my user talk page is changed
 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 AMFM  Forum AMFM
 Epistemowikia:Plan de aprendizaje/Ampliación de Matemáticas  Further Mathematics/Actividades/Foros/MetodologíaEvaluación  MethodologyAssessment
 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
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.  Join the project on this dedicated page.
 Include your major contributions to the project on this common dedicated page.
 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':
http://cala.unex.es/cala/epistemowikia/index.php/Usuario:myusername/Logbook  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'):
http://cala.unex.es/cala/epistemowikia/index.php/Usuario:myusername/sandbox  Important:
 On Epistemowikia, mathematical formulas are represented using MathML (through a thirdparty 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).
 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.
 After doing that, your email must be confirmed. To do that, please log in with your username and password and go to your preferences page and follow the directions from there. If you do this, your edits will be attributed to your username, along with other benefits.
 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
 Here is your watchlist.
 Join the project on this dedicated page.
 Include your major contributions to the project on this common dedicated page.
 Any person can watch the list of all your contributions easily on this page. 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': https://en.wikipedia.org/wiki/Usuario:myusername/Logbook (precisely, this one is the page that will host your selfreport).
 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
Selfreport
In all cases, every and each student must continually write, keeping up with the developing of their work, a selfreport, 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
Considerations
Examples of exam (selfassessment mock exams) 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.
Solution: 
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 , and , such that , and . (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, , of the sequence using the Newton's divided differences interpolation polynomial.
Solution: 
Because of custom and maybe tradition we begin considering , and so we start dealing this question with and then adjust to match what was required. Let be . Then, the table of divided differences is:
where: The interpolating polynomial is: as well as in recurrence form: Thus:
In summary, starting with , the general term is , and adjusting to match the beginning as required, the general term is: Sol.: . 
Question 4. (2.5 points)
In baseten (decimal numeral system), find the digits such that the number be divisible by .
Solution:  
. A number is divisible by precisely if the sum of all its digits is divisible by : this is: Moreover: then: We have to find out what differences satisfy the fact of belonging to : so there are possible cases: A number is divisible by precisely if the sum of its digits at even places minus the sum of its digits at odd places is divisible by : this is: Moreover: then: We have to find out what differences satisfy the fact of belonging to : so there are possible cases: Therefore, there are possible cases:
So there are three possible solutions: . Thus, the possible numbers are: , which are divisibles by . Their quotients are: . 
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 twoelement asc, explaining why they are so and assuming that we know the asc which elements are the most usual connectives .
Solution: 

Question 2. (2,5 points).
Proof by definition that is an infinite set.
Solution: 
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 , defined by . Let us prove that it is a bijective mapping. In effect:

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 and 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.
Solution: 
Following the steps of RSA algorithm:
Let us answer now the two sections of the question.

Question 4. (2,5 points).
One company spent euros in buying electronic devices, some of them ground breaking and providing maximum performance. Smartphones were euros each, tablets were euros each and laptops were euros each. How many of each device did they buy,knowing that they bought at least one device of each type? Solve this question using the theory of:
 a) diophantine equations;
 b) congruence equations.
Solución:  
Once translated the information from the wording into a system of linear equations and simplifying the latter:

Part 2: Themes 3 and 4
Example of exam, 1
Maximum time: 2 hours.
Question 1. (2.5 points)
Let be the set of decimal digits, that is, . Using combinatorial reasoning, calculate:
 a) The number of subsets of which elements are all primes.
 b) The number of subsets of having a prime number of elements.
Solution: 

Question 2. (2.5 points)
A group of twelve people visit a museum. Everybody is wearing a woolen overcoat. Upon entering, they leave their coats in the attended cloakroom. On leaving, the cloakroom attendant puts the twelve coats on the counter. Each person in the group picks out one at random, completely absentminded because of a very interesting discussion. Using combinatorial reasoning, calculate in how many ways can the coats be chosen by them so that none of them get their own coat back.
Solution: 
This involves finding the number of derangements of objects. Instead of calculating for , we are going to do for the general case of having objects. Let denote the objects themselves. Let be the set of all permutations of the objects and let be the set of all derangements that have fixed elements. Then, the set of all derangements is:
Let us see it:
Let us note that in the case of , that is, , when we subtract those which fix the , then we are subtracting once those which fix both the and the and when we subtract those that fix the we are subtracting again those which fix both the and the . Thus, we have to add them one time. If we follow this reasoning, the number of permutations (derangements) for which no number is in its original place is:
Thus, if , there are:
derangements. 
Question 3. (2.5 points)
Let the following be the definition of the sum of two natural numbers and :
Prove that the solution of this recurrence is .
Solution: 
Let us note that is alien to recursion. Thus, in an easier but equivalent way, denoting by , we get a linear non homogeneous recurrence relation with constant coefficients and with a constant function as the function on the RHS of the equation:

Question 4. (2.5 points)
Let
be a binary relation defined on the set , for every two elements and in , where is the figure of the units of the usual product between two natural numbers (for example, ).
 a) Find out theCayley table for the operation on .
 b) Is an abelian group? (You can reason using the Cayley table).
Solution: 

Example of exam, 2
Maximum time: 2 hours.
Question 1. (2.5 points)
An urn contains seven balls numbered one through seven. They are randomly chosen, one by one and without reposition until the urn is empty. As they are removed from the urn, we write their figures down from left to right on a first out, first writen basis. Using combinatorial reasoning calculate how many numbers thus formed start and end with an even digit.
Solution: 
There are positions for the figures. At both ends, hypotheses imply even figure. There are three even figures between and : , and . Being guided by the distribution of objects into recipients models, consider these even figures (distinguishable boxes) and the two ends (distinguishable objects) of the sevendigit number, on an underlying injective mapping (at most, one end by each figure, as there are not two balls with the same figure). For each one of these cases at the ends (each one of the variations) we have to take into account all the possibilities for the intermediate positions. The number of these possibilities is given by the permutations of elements (one new abstraction as distinguishable objects [the intermediate positions] being distributed into distinguishable recipients [the figures , and and the even figure that is at none of the ends of the sevendigit number thus formed], this time on an underlying bijective mapping). Applying the rule of product:
Sol.: numbers. 
Question 2. (2.5 points)
A secret ballot is made in a meeting of seventeen people. Two people have cast invalid ballots, three have cast blank ballots, five have cast dissenting votes and seven have cast assenting votes. Using combinatorial reasoning calculate in how many ways this could have occured.
Solution: 
Let us use the occupancy model for the non ordered distribution of balls into boxes; balls and boxes representing ballots and people, respectively. Consider the persons (distinguishable boxes) and the assenting ballots (indistinguishable balls), on an underlying injective mapping (at most, one ballot by each person) — alternatively, we could consider the number of subsets of elements from a set of elements —. In any case, there are ways of distributing the assenting ballots into the boxes. For each one of these cases (each one of the combinations), there are empty boxes left. Now, using a similar reasoning, there are ways of distributing the dissenting ballots into the boxes, with empty boxes remaining for each one of these cases. Similarly, there are ways of distributing the blank ballots into the boxes, with empty boxes remaining for each one of the cases. So, lastly, there are ways in which invalid ballots can be placed into the boxes. Applying the rule of product:
Sol.: In ways. 
Question 3. (2.5 points)
Let and be the numbers of malicious software belonging to two malware types, in the day , that coexist in a certain insecure wide area network (WAN) under malware evolution daily control. Let us assume that the original memberships were of and and that the coexistence evolution is as follows:
 every day, the growth in malware type is the sum of the triple of the growth in type on the previous day and the growth in type also on the previous day plus seven new malware (that were classified as type ),
 and also every day, the growth in malware type is the result of subtracting the growth in type on the previous day from the growth in type on the previous day, plus three new malware (that were classified as type ).
Find out and solve the system of recurrence equations of the evolution of the malware.
Solution: 
Let us analyze the evolutions of the two types of malware on their own growths (that these evolutions had to be calculated in terms of the populations or not is not specified by the wording and, on the other side, calculating them on their growths is easier because the order of the recurrence relation has decreased in one time unit). Let and denote the growths from time to time , i.e. and . The system of linear recurrence equations that correponds to this situation is:

Question 4. (2.5 points)

+—+ +—+ D <———> C +—+ +—+ \ ⋀ \  \  \  ╶┘  +—+ +—+ A <———> B +—+ +—+ 
Solución: 

Everything: Themes 1, 2, 3 and 4
Example of exam 1
Maximum time: 2 hours.
Question 1. (2.5 points).
On the island of knights and knaves there are two types of inhabitants, `knights' who always tell the truth and `knaves' who always lie. It is assumed that every inhabitant is either a knight or a knave. There were two inhabitants, and , standing together in the front yard of a house. You passed by and asked them, `Are you knights or knaves?'
 a) answered, `If is a knight then I am a knave.' Can it be determined whether and were knights or knaves? (1.25 p.)
 b) Afterward, said, `Don't believe ; he's lying.' With this new information, can it be determined whether and were knights or knaves? (1.25 p.)
Solution: 
Let us use for ` is a knight'  therefore, is an abbreviation for ` is a knave' .

Example of exam 2
Maximum time: 2 hours.
Question 2. (2.5 points).
Knowing that (integers) is a countable set and that the countable union of countable sets is countable, prove that (rationals) is countable.
Solution: 
is a countable set as it can be expressed by the countable union , where every is countable, since , defined by and is a biyection. Note that the set is the set of all the rational numbers that have the same denominator . 
Question 3. (2.5 points).
Use congruence relation theory to respond.
 a) Prove that, for any , is divisible by . (1.25 p.)
 b) Calculate the remainder of (for any ), when it is divided by . (1.25 p.)
Solution: 
We use congruence relation theory.
(i) Because congruence relations are symmetric and transitive. 
Question 5. (2.5 points).
Use a combinatorial reasoning to respond.
 a) A number is palindrome if it reads the same from left to right and from right to left. In base ten, how many sevendigit numbers are palindromes? (1.25 p.)
 b) Let us assume a sided polygon network (gon network). Calculate , the number of nodes (vertices) of the network, knowing that the number of line segments (sides + diagonals) is . (1.25 p.)
Solution: 

Cardinality 
To find out more: Bibliography: theory and proposed and solved exercisesIn English:
In Spanish:
See also

Number theory 
DivisibilityPrimes
Diophantine equations, systems of Diophantine equations and on their solutionsModular arithmetic
Congruence equations, systems of congruence equations and on their solutions
CryptographyTo find out more:
Bibliography: theory and proposed and solved exercisesIn English:
In Spanish:
See also
Plantilla:Number theory Plantilla:Divisor classes Plantilla:Classes of natural numbers Plantilla:Number theoretic algorithms Plantilla:Prime number classes Plantilla:Prime number conjectures 
Numerical analysis 
To find out more: Bibliography: theory and proposed and solved exercisesIn English:
In Spanish:

Combinatorial theory 
To find out more:
Bibliography: theory and proposed and solved exercisesIn English:
In Spanish:
See also

Recurrence relations 
Para saber más:
Bibliography: theory and proposed and solved exercisesIn English:
In Spanish:

Algebraic structures 
To find out more: Bibliography: theory and proposed and solved exercisesIn English:
In Spanish:

Graphs, trees and networks 
To find out more: Bibliography: theory and proposed and solved exercisesIn English:
In Spanish:

Complimentary knowledge pills 
Conjectures, open problems and imagination
HistoryParadoxes 
Minihackatones (miniencuentros intensivos de aprendizaje en colaboración) / Minihackathons (intensive collaborative learning meetings) 

Templates
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.
See also
 Epistemowikia:Proyecto de aprendizaje/Discrete and numerical mathematics/Participants and major contributions (participants and major contributions to this project) (backup/mirror copy on Epistemowikia, some internal/external links swapped to external/internal)
 Epistemowikia:Proyecto de aprendizaje/Matemática discreta y numérica (sister project for the Spanish Wikipedia) (backup/mirror copy on Epistemowikia, some internal/external links swapped to external/internal)
 Epistemowikia:Plan de aprendizaje/Ampliación de Matemáticas  Further Mathematics (mother course, only on Epistemowikia for the time being)
External links
 Wikipedia:School and university projects/Discrete and numerical mathematics (this project on the English Wikipedia)
 Wikipedia:School and university projects/Discrete and numerical mathematics/Participants (participants and main contributions to this project on the English Wikipedia)
 Proyecto educativo «Matemática discreta y numérica» (sister project on the Spanish Wikipedia)