This subject is part of the basic knowledge module and one of the 6 math-related subjects in the Degree in Computer Science.



The subject is organized as follows: first the logic of both compound and quantified statements are studied, followed by set theory and relations. Then, counting and discrete probability are covered, and finally, graphs and trees are studied.



By taking this subject students will gain the ability to analyze, synthesize and reason in an abstract way in order to model and solve engineering problems.

Competences



- Specific competence

M01CM03: Ability to understand and master the basic concepts of discrete mathematics

(logic, algorithms and computational complexity) and their application to solve

engineering problems.



- General competences

G008: Knowledge of basic subjects and technologies that allows for learning and developing

new methods and technologies, and for facing new and different situations.

G009: Ability to solve problems with initiative, autonomy and creativity. Communicative

skills to describe knowledge, abilities and skills of the profession of computer

engineer.





Learning results (RA)



RA01: To properly use discrete mathematics' terminology in arguments and in practice.

RA02: To identify the most suitable method for solving mathematical problems in the engineering

field.

RA03: To apply discrete mathematics' methodology for solving mathematical problems in the

engineering field.

RA04: To properly use software tools for solving mathematical problems in the engineering field.

RA05: To describe the methods used and the results obtained in problem solving.

RA06: To analyze and interpret in a reasonable way the results obtained.

RA07: To conduct efficient bibliographic search that helps solve problems in the engineering

field.

RA08: To create reports describing in a coherent way knowledge, methods and results.

MATHEMATICAL LOGIC The ability to think abstractly is gained by learning how to use logically valid forms of argument and how to reason from definitions. The logic of both compound and quantified statements are studied.



SET THEORY Definitions and notation of set theory are introduced. Properties of sets and operations with sets are studied. Functions defined on general sets are also described as well as the composition of functions and the inverse function. On the other hand, operations with whole numbers are defined. Concretely, divisibility, prime numbers, maximum common divisor, minimum common multiple, Euclides algorithm, etc.



BINARY RELATIONS The concept of binary relation is described to study both order and equivalence relations. Particular emphasis is put on congruence modulo n, an equivalence relation.



COMBINATORICS AND DISCRETE PROBABILITY The addition and multiplication rules are introduced to study first the different counting techniques in combinatorics such as variations, permutations and combinations, and then, their direct applications to the calculation of discrete probabilities.



GRAPH THEORY The graph concept is introduced though specific examples and then, its basic properties are studied. Eulerian and Hamiltonian graphs are described. The role of graphs and the types of problems they can help solve in the engineering field are discussed. Specific problem solving with graphs is carried out.

Students will be provided with documentation covering all the topics of the subject and proposed exercises, some of which will be solved in class. The rest will have to be solved by students in an autonomous way.



The virtual platform 'egela' will be used to complement the theoretical and practical lessons.



The mathematical software 'Mathematica' will be used in the computer lessons.



In general, if the opposite is not indicated, during examinations in the UPV/EHU, students are not allowed to use books, notes, computers or any other telephonic or electronic devices.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 60
  • Realization of Practical Work (exercises, cases or problems) (%): 15
  • Team projects (problem solving, project design)) (%): 10
  • Computer exam (%): 15

Students might choose between continuous or final evaluation.



Continuous evaluation

1. The student's final score will be computed as the weighted sum of the grades obtained in the different tasks the student took part based on the following criteria:

- Final written exam: 60% of the final score

- Mid-term exam: 15%

- Group task: 10%

- Computer exam: 15%

2. To pass the subject it is mandatory to obtain at least 40% of the total score of the final written exam. If this minimum requirement is not met, the student will be graded 'FAILED'.

3. The student, who want to drop out of the continuous evaluation in order to enroll in the final evaluation, will have to complete and sign an application form and deliver it to the lecturer before the 11th week of the semester. The application form will be available on the virtual platform 'egela'.

4. If a student does not take the final written exam, the student will be graded 'NOT PRESENTED'.



Final evaluation

1. The student must drop out of the continuous evaluation in order to enroll in the final evaluation. This must be done by completing and signing an application form and delivering it to the lecturer before the 11th week of the semester. The application form will be available on the virtual platform 'egela'.

2. The final evaluation consists of a final exercise composed of the following tasks:

- Final written exam: 85%

- Computer exam: 15% (because of its exceptional nature, it will be carried out along the semester in the computer room, date and time designated by the lecturer)

To pass the subject (final score >=5) the student must obtain at least 40% of the total score of the final written exam and 40% of the total score of the computer exam.

3. If a student does not take the final written exam, the student will be graded 'NOT PRESENTED'.

1. The score obtained during the semester in the continuous evaluation, i.e. that corresponding to the mid-term exam, group task and computer exam, will be maintained if it is greater or equal to 50% of its total; the student keeping that score will be evaluated according to the continuous evaluation and applying the same weights to each one of the tasks. To pass the subject it is mandatory to obtain at least 40% of the total score of the final written exam. If this minimum requirement is not met, the student will be graded 'FAILED'.



2. The rest of the students will be evaluated according to the final evaluation. Those students that in the ordinary exams passed the computer exam will only be evaluated through a final written exam (85%). However, those students that failed the computer exam will be evaluated via a final exam (100%) which will also cover the contents related to the computer exam. To pass the subject (final score >=5) the student must obtain at least 40% of the total score of the final written exam and 40% of the total score of the computer exam.



3. If a student does not take the final exam, the student will be graded 'NOT PRESENTED'.

Students are not forced to use any specific material.

Students might get all the necessary material to prepare and study the subject in the virtual platform 'egela'. This material will be make available as soon as the semester starts.

On the other hand, additional information can be obtained from the sources listed in the bibliography section of this document.

Basic bibliography

S. S. Epp. Discrete mathematics with applications. Brooks Cole, 2010.



F. Garcia Merayo, G. Hernandez Peñalver, A. Nevot Luna. Problemas resueltos de matemática discreta. Thompson-Paraninfo, Madrid, 2003.



W.K. Grassmann, J.P. Tremblay. Matemática discreta y lógica. Prentice Hall Iberia, 1998.



R. P. Grimaldi. Matemática discreta y combinatoria: una introducción con aplicaciones. Addison-Wesley Iberoamericana, Argentina, 1997.



C. L. Liu. Elementos de matemáticas discretas. Mc.Graw-Hill, México, 1995.

In-depth bibliography

T. Veerarajan. Matemática discreta con teoría de grafos y combinatoria. McGraw-Hill Interamericana, 2008.

L. Lovász, J. Pelikán, K. Vesztergombi. Discrete mathematics. Elementary and beyond. Springer, Nueva York, 2003.

N. L. Biggs. Matemática discreta. Vicens Vives, Barcelona 1994.

C. L. Chang, R. C. T. Lee. Symbolic Logic and mechanical theorem proving. Academic Press, New York, 1973.

A. Gibbons. Algorithmic graph theory. Cambridge University Press, Cambridge, 1985.

A. Deaño. Introducción a la lógica formal. Alianza, Madrid 1980.

Web addresses

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2015/

https://ocw.mit.edu/courses/mathematics/18-310-principles-of-discrete-applied-mathematics-fall-2013/

https://ocw.mit.edu/courses/mathematics/18-315-combinatorial-theory-introduction-to-graph-theory-extremal-and-enumerative-combinatorics-spring-2005/

https://ocw.mit.edu/courses/mathematics/18-314-combinatorial-analysis-fall-2014/

https://mathworld.wolfram.com/

http://www.divulgamat.net/