ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician, English
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Algebra
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
This is a course on the fundamentals of mathematics and provides preparation for the other subjects of math studies. Students will develop good habits of understanding, communicating and writing mathematics. Techniques of reasoning will be discussed mainly from discrete mathematics. The methods will be applied to solve many interesting problems. One could say that this is a course about understanding and thinking carefully, not about computation or memorizing rules.
The course explores themes involving numbers, sets, and functions. With elementary properties of these objects and some basics on propositional logic, we move on to study induction and cardinality. In discrete mathematics, we consider techniques of counting. The study of natural numbers includes properties of divisibility and modular arithmetic.
1. Introduction to the Mathematical Logic. (2 sessions)
1.1. Necessity and Importance of the Logic Language: Paralogisms.
1.2. Propositional Logic: Atomic and Molecular Propositions.
1.3. Truth Tables. Tautologies and Contradictions.
1.4. The Process of Deduction. Reasoning and Formal Proofs in the Propositional Calculus.
2. Sets. (4 sessions)
2.1. Sets and Elements. Subsets: The Power Set.
2.2. Graphic Representations: Venn Diagrams.
2.3. Operations with Sets: Properties. The Boolean Algebra of the Power Set.
2.4. Coverings and Partitions. Disjoint Union and Cartesian Product.
3. Maps. 4 sessions)
3.1. Concept. Graph of a Map: Examples.
3.2. Types of Maps: Injections, Surjections and Bijections.
3.3. Maps Composition: Properties. Inverse Map.
3.4. Extensions of a Map to the Power Set.
4. Relations. (6 sessions)
4.1. Notion of Relation. Composition of Relations. Inverse Relation.
4.2. Graphic Representations.
4.3. Binary Relations in a Set: Properties. Induced Relation.
4.4. Equivalence Relations: Equivalence Classes: Properties. Quotient Set. Partitions.
4.5. Canonical Factorization of a Map.
4.6. Order Relations: Graphic Representations: Hasse Diagrams (Trees). Total and Partial Order. Distinguished Elements in an Ordered Set. Chains, Lattices and Well-ordered Sets.
5. Combinatorics. (3 sessions)
5.1. Variations. Variations with Repetition.
5.2. Factorial Numbers. Permutations. Permutations with Repetition.
5.3. Combinatorial Numbers. Combinations.
5.4. Combinations with Repetition.
5.5. Principle of Inclusion-Exclusion. Enumeration of the Surjective Maps.
5.6. The Tartaglia-Pascal´s triangle. The Newton´s Binomial.
6. Infinite Sets. (4 sessions)
6.1. Finite and Infinite Sets.
6.2. The Natural Numbers as Equipotency Classes of Finite Sets .
6.3. Principle of Induction. Operations and Order on Natural Numbers.
6.4. Countable and Uncountable Sets. Rational Numbers. The Diagonal Procedure and the Uncountability of R.
6.5. The Axiom of Choice and Zorn's Lemma.
7. Integer and Modular Arithmetic. (5 sessions)
7.1. Binary Operations.
7.2. The Set of Integer Numbers. Properties of Z.
7.3. Divisibility. Prime Numbers and the Fundamental Theorem of Arithmetics.
7.4. Greatest Common Divisor and Least Common Multiple. Bezout's Theorem.
7.5. Euclidean Algorithm. The Extended Euclidean Algorithm.
7.6. Modular Arithmetics. The Rings Z/(n). Congruence. Units Modulo n. The Euler-Fermat Theorem.
7.7. Diophantine Equations. Resolution of Linear Diophantine Equations.
7.8. Relatively Prime Integers: The Chinese Remainder Theorem.
7.9. Polynomials in one Variable.
Basic bibliography:
F. Aguado, F. Gago, M. Ladra, G. Pérez, C. Vidal, A. M. Vieites; Problemas resueltos de Combinatoria. Laboratorio de Sagemath, Ediciones Paraninfo, S.A., 2018.
J.P. D’Angelo, D. B. West; Mathematical Thinking: Problem-Solving and Proofs, 2ª ed., Prentice Hall, 2000.
V. Fernández Laguna: Teoría básica de conjuntos, Anaya, 2004.
M. A. Goberna, V. Jornet, R. Puente, M. Rodríguez; Álgebra y Fundamentos: una Introducción, Ariel, 2000.
K. H. Rosen; Matemática Discreta y sus Aplicaciones, 5ª ed., McGraw-Hill, 2004.
Complementary bibliography:
M. Anzola, J. Caruncho; Problemas de Álgebra (Conjuntos-Estructuras), BUMAR, 1982.
R. Courant, H. Robbins; What Is Mathematics? An Elementary Approach to Ideas and Methods, 1941
(2ª ed., rev. por Ian Stewart, Oxford University Press, 1996).
Tr.: ¿Qué es la Matemática?, FCE, 2003.
T. S. Blyth, E. F. Robertson; Sets, Relations and Mappings, Cambridge University Press, 1984.
H. Rademacher, O. Toeplitz; Números y Figuras. Alianza editorial, 1970.
To contribute to achieving the generic, specific and transversal competentes listed in the Report on the Degree in Mathematics from USC and, in particular CE1, CE7, CE6, CD8, CB1, CB2, CB4, CB5, CG2, CG5, CT1, CT3 and CT5
The weekly distribution of the subject will be the next: 2 hours of theory, 1 hour of problems and 1 hour of tutorial (the last one eventually with computer).
The blackboard classes in big group devote the exposition of the fundamental contents of the subject, with theory, resolution of problems and presentation of some exercises.
The blackboard classes in reduced group will deal with complementary aspects of the subject, realization of problems and exercises and corrections by the teacher.
In the tutorials in reduced group the fundamental leading role will be on the students, that must present exercises and expositions of some matter related to the subject.
In the tutorials in much reduced group the teacher will make a personalized tracking of the learning of the students.
During the semester the students may be asked to hand in written exercises in class. The combined marking of these activities will represent the 25% of the final grade. The remainder 75% will result from the final written exam.
The written exam will consists of theory and theoretical-practical questions and exercises.
The same applies for the extra opportunity in July.
It will be considered to be "No presentado" the student who does not attend neither one of the two final examinations.
Attendance at classes:
Lecture classes: 28 hours.
Interactive problem classes in small groups: 14 hours.
Interactive laboratory classes in small groups: 14 hours.
Hours of tutorials in very small groups: 2 hours.
Total presence hours: 56
Personal work hours:
42 hours of autonomous study, individually or in group.
40 hours of solving/writing exercises, conclusions or other works.
Total workload: 140 hours.
The student must attend classes regularly, and should work individually or collectively each and every one of homework problems proposed in class. They may ask for help on office hours as difficulties arise.
Plan de continxencia:
Escenario 2: distanciamento
Metodoloxía da ensinanza
Dado que a docencia presencial convivirá coa virtual e lle corresponde ao centro definir as fórmulas de convivencia de ambas modalidades de docencia, unha vez coñecidas estas utilizaranse os medios telemáticos Campus virtual da USC, Microsoft Teams, ou doutro tipo que nos proporcionen as autoridades académicas e levarase a cabo de xeito síncrono como asíncrono tanto as explicacións dos contidos como as cuestións prácticas da materia.
As titorías serán preferentemente virtuais.
As canles previstas co alumnado no caso telemático son o Campus virtual da USC, correo electrónico e Microsoft Teams.
Sistema de avaliación da aprendizaxe
Cando os procedementos non se poidan facer de maneira presencial realizaranse de modo telemático.
No caso de que non se poida celebrar un exame final presencial, para o cómputo da cualificación final, tanto na primeira oportunidade como na segunda, aplicarase a seguinte fórmula:
F= 0.4*C+0.6*E, F = cualificación final, C= avaliación continua, E = exame final
Escenario 3: peche das instalacións
Metodoloxía da ensinanza
A docencia será completamente de carácter virtual, con mecanismos síncronos ou asíncronos.
As titorías serán exclusivamente virtuais.
As canles previstas co alumnado neste caso son o Campus virtual da USC, correo electrónico e Microsoft Teams.
Sistema de avaliación da aprendizaxe
Realizaranse todos os procedementos de modo telemático.
No caso de que non se poida celebrar un exame final presencial, para o cómputo da cualificación final, tanto na primeira oportunidade como na segunda, aplicarase a seguinte fórmula:
F= 0.4*C+0.6*E, F = cualificación final, C= avaliación continua, E = exame final
Leovigildo Alonso Tarrio
Coordinador/a- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813159
- leo.alonso [at] usc.es
- Category
- Professor: University Lecturer
Antonio Garcia Rodicio
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813144
- a.rodicio [at] usc.es
- Category
- Professor: University Professor
Manuel Eulogio Ladra Gonzalez
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813138
- manuel.ladra [at] usc.es
- Category
- Professor: University Professor
José Javier Majadas Soto
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813168
- j.majadas [at] usc.es
- Category
- Professor: University Professor
Beatriz Álvarez Díaz
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813175
- beatriz.alvarez.diaz [at] rai.usc.es
- Category
- Ministry Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 09 |
| 13:00-14:00 | Grupo /CLE_02 | Spanish | Classroom 08 |
| Tuesday | |||
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 07 |
| 13:00-14:00 | Grupo /CLE_02 | Spanish | Classroom 08 |
| Wednesday | |||
| 11:00-12:00 | Grupo /CLIL_01 | Galician, Spanish | Classroom 06 |
| 12:00-13:00 | Grupo /CLIL_03 | Spanish, Galician | Computer room 4 |
| 13:00-14:00 | Grupo /CLIL_06 | Spanish | Classroom 03 |
| Thursday | |||
| 11:00-12:00 | Grupo /CLIL_02 | Galician, Spanish | Classroom 02 |
| 12:00-13:00 | Grupo /CLIL_05 | Spanish | Graduation Hall |
| 13:00-14:00 | Grupo /CLIL_04 | Galician, Spanish | Classroom 03 |
| Friday | |||
| 11:00-12:00 | Grupo /CLIS_04 | Spanish | Classroom 06 |
| 11:00-12:00 | Grupo /CLIS_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| 12:00-13:00 | Grupo /CLIS_02 | Spanish | Classroom 03 |
| 12:00-13:00 | Grupo /CLIS_03 | Spanish | Ramón María Aller Ulloa Main Hall |
| 01.20.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 02 |
| 01.20.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |
| 01.20.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 01.20.2021 10:00-14:00 | Grupo /CLE_01 | Ramón María Aller Ulloa Main Hall |
| 06.18.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |