The subject of Discrete Mathematics is part of the basic training module, in the second semester of the first year of the Degree in Computer Engineering in Management and Information Systems and in the second semester of the second year of the Double Degree in Business Administration and Management and in Computer Engineering Management and Information Systems.



It is about providing the student with mathematical tools and means necessary to understand and apply techniques that are useful for the exercise of the profession for which the degree qualifies and, on the other hand, to prepare him for said activity, providing him with skills such as rigor, creative capacity, abstract reasoning, clarity and precision in making judgments and the ability to analyze and synthesize.

Read and interpret mathematical texts related to Discrete Mathematics. Provide a minimum capacity for abstraction, concretion, concision, imagination, intuition, reasoning, criticism, objectivity, synthesis and precision. Develop the capacity for logical reasoning. Develop the ability to understand and/or build mathematical models to solve Computer Science problems as well as design some algorithms and evaluate the complexity of the discrete and combinatorial problems that appear.



Acquire basic knowledge of Number Theory, both Integer Arithmetic and Modular Arithmetic and its application in different fields of Computer Science. Raise awareness of the difficulty in factoring large numbers and provide the information necessary for the development of cryptographic codes such as the RSA code.



Analyze the topic of cardinality in sets including counting techniques in finite sets. Be able to count in complicated sets. Solve combinatorial type problems. Acquire basic knowledge of graphs, paths and cycles, connectivity, trees and optimization. Develop the ability of graph theory to model and solve problems in everyday life.

Topic 1: Mathematical Logic. Propositional logic. Syntax and Semantics. Reasoning techniques. Predicate logic.

Topic 2: Counting techniques. Cardinal of finite sets. Addition principle. Principle of boxes. Combinatorics. Principle of inclusion and exclusion.

Topic 3: Integer arithmetic. Integer arithmetic. Divisibility. Greatest common divisor. Euclid's algorithm. Bezout's identity. Prime numbers. Prime numbers to each other. Least common multiple. Euler function.

Topic 4: Modular arithmetic. Consistencies. Modular arithmetic. Invertible elements. Theorems of Euler and Fermat. Linear congruences. The RSA code.

Topic 5: Graphs: Eulerian and Hamiltonian Graphs. Basic definitions. Isomorphic graphs. Paths and cycles. Connected graphs. Eulerian graphs. Hamiltonian graphs.

Topic 6: Graphs: Trees and maps. Trees. Directed graphs. Optimization and graphs. Planar graphs and maps. Graph coloring.

The methodology used in this subject, in order to achieve learning and the acquisition of skills by students, will be carried out through various teaching methods, among which we highlight the expository method and the resolution of exercises and problems.



In the face-to-face sessions of the expository method, the teacher will develop, in a clear and accessible way, the theoretical contents of the different topics, showing their need and relationship between them. An attempt will be made to stimulate student participation by asking questions during the theoretical presentation. Optionally, depending on the difficulty of the topics to be discussed, students may be required to previously read the theoretical material to discuss its understanding and use in practical cases in class.



In the face-to-face sessions of practical work in the classroom, problems will be solved by applying theoretical knowledge. Debate and student participation will be promoted. To do this, you will have a list of problems that you must solve on your own for later discussion in the classroom.



There will be a virtual classroom on the eGela platform that allows permanent contact between teachers and students. In it, material and information on the subject will be available.



Students will have the possibility of attending personalized tutorials with the teacher, during designated hours, which can be consulted in GAUR or on the School's website. During these hours any academic issue related to the subject can be discussed.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 100

According to the regulations governing student evaluation (BOPV 3/13/2017), the evaluation is continuous and if you want to use the final evaluation system, you must inform the teaching staff responsible for the subject of the waiver of continuous evaluation in the First 9 weeks of the semester. This will be done through the link enabled for this purpose on the eGela page of the subject.



I) CONTINUOUS EVALUATION SYSTEM

The final grade will be divided into two parts:

The activities or tasks carried out throughout the semester will account for 50% of the final grade with release of the same in case of obtaining a grade greater than or equal to 50% and the written exam to be taken in the official exam period of the call. Ordinary course set by the center will account for 50% of the final grade.



To obtain a grade in the ordinary call it will be necessary to have taken the written exam of the ordinary call, otherwise it will appear as not presented. The final grade will be the sum of the previous grades.



II) FINAL EVALUATION SYSTEM

In the final evaluation system, 100% of the grade will correspond to the written exam to be taken during the exam period of the call. In this case, it will not be required to approve each part separately.



In either of the two evaluation systems, whoever does not take the final written exam will obtain the final grade "Not Presented" regardless of whether or not they have completed the rest of the tasks.

The UPV/EHU has approved the "Protocol on Academic Ethics and Prevention of Dishonest or Fraudulent Practices in Assessment Tests and Academic Work at the UPV/EHU". In point 3.3, it says: "In general terms, and unless otherwise indicated, during the development of an evaluation test at the UPV/EHU, the use of books, notes or notes, as well as devices, will be prohibited. or telephone, electronic, computer, or other devices, by the students. At the time of the test, they may be used.

indicate, if necessary, the places where unauthorized materials can be deposited, so that they remain out of the reach of students."



In the evaluation tests of this subject, only the use of a calculator is allowed.

In the extraordinary call, 100% of the grade will correspond to a final test that will consist of a written exam to be taken during the exam period of the call. In this call, the parts approved during the course will not be kept and it will not be required to approve each part separately.



Anyone who does not take the final written exam will obtain the final grade of "Not Presented".



The UPV/EHU has approved the "Protocol on Academic Ethics and Prevention of Dishonest or Fraudulent Practices in Assessment Tests and Academic Work at the UPV/EHU". In point 3.3, it says: "In general terms, and unless otherwise indicated, during the development of an evaluation test at the UPV/EHU, the use of books, notes or notes, as well as devices, will be prohibited. At the time of the test, they may be used”.



In the evaluation tests of this subject, only the use of a calculator is allowed.

Through the eGela platform, students will be provided with all information related to the subject: program, presentations, problem lists, etc. and permanent communication will be established with them through the bulletin board (for teacher communications) and the doubt forum (for student queries). This page will be the base tool in case teaching and/or evaluation cannot be carried out in person.

Basic bibliography

CIRRE TORRES, F.J. "Matem¿ca Discreta" (Colecci¿ase Universitaria)Ed. Anaya



VEERARAJAN, T. "Matem¿cas Discretas. Con teor¿de gr¿cas y combinatoria" Ed. Mc. Graw Hill



GRIMALDI, R.P. "Matem¿ca Discreta y Combinatoria" Ed. Addison Wesley Iberoamericana



GARC¿ MERAYO, F. "Matem¿ca Discreta" Ed. Thomson

In-depth bibliography

ROSEN, K.H. "Matemática Discreta y sus aplicaciones"
Ed. Mc. Graw Hill

BIGGS, N.L. "Matemática Discreta"
d. Vicens-Vives

GRASSMANN, W.K. y TREMBLAY, J.P. "Matemática Discreta y Lógica"
Ed. Prentice Hall

Libros de problemas:

GARCÍA MERAYO, F.; HERNÁNDEZ PEÑALVER, G.; NEVOT LUNA, A ·"Problemas resueltos de Matemática Discreta"
Ed. Thomson.

GARCÍA, C., LÓPEZ, J.M. y PUIGJANNER, D. "Matemática Discreta. Problemas y ejercicios resueltos"
Ed. Prentice Hall

LIPSCHUTZ, SEYMOUR; LIPSON, MARC "2000 Problemas resueltos de Matemática Discreta"
Ed. Mc. Graw Hill

Otros libros de profundización:

LIU, C.L."Elementos de Matemáticas Discretas"
Ed. Mc-Graw-Hill.

CHARTRAND, G. y OELLERMANN,O. "Applied and Algorithmic Graph Theory"
Ed. Mc-Graw-Hill.

ANDERSON, I. "Introducción a la Combinatoria"
Ed. Vicens-Vives.

JONES, G.A. y JONES, J.M. "Elementary Number Theory"
Ed Springer Verlag.

KNUTT, D.E. "El arte de programar ordenadores" (Vol 1. Algoritmos fundamentales)
Ed. Reverté.

ALBERTSON, M.O. y HUTCHINSON, J.P. "Discrete Mathematics with algorithms"
Ed. John Wiley.

HORTALÁ GONZALEZ, T. y otros. "Lógica Matemática para Informáticos. Ejercicios resueltos."
Ed. Pearson. Prentice Hall.

Web addresses

http://mathworld.wolfram.com/
http://www.mersenne.org/
http://www.divulgamat.net/