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. (1 session)
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. (3 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. (7 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, CE6, CE7, CE8, CB1, CB2, CB4, CB5, CG2, CG5, CT1, CT2, CT3, CT4 and CT5
Scenario 1: adapted normality
The weekly distribution of the subject will be the next: 2 hours of lectures, 1 hour of seminar class and 1 hour of laboratory
(the last one eventually with computer).
The lecture 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 seminar classes in reduced group will deal with complementary aspects of the subject, realization of problems and exercises and corrections by the teacher.
In the laboratories 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.
In all scenarios, in cases of fraudulent performance of exercises or tests, the provisions of the Regulations on the Evaluation of Students' Academic Performance and the Review of Grades will apply.
Scenario 1: adapted normality
During the semester the students may be asked to hand in written exercises in class. The combined marking of these activities will be part of the qualification.
For the calculation of the final mark (F) the continuous evaluation (C) and the final exam mark (E) will be taken into account and the following formula will be applied:
F= max (E, 0.25*C+0.75*E)
The same applies for the extra opportunity in July.
The written exam will consists of theory and theoretical-practical questions and exercises.
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:
Autonomous study, individually or in group: 42 hours.
Solving/writing exercises, conclusions or other works: 40 hours.
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.
In accordance with the "Contingency plan for the development of teaching in the academic year 2021-22", approved by the Governing Council of the University of Santiago de Compostela on April 30, 2021, we include the adaptations corresponding for the sections on teaching methodology and on assessment system planned for scenarios 2 and 3:
Contingency plan
Scenario 2: Distancing
Teaching methodology
Given that face-to-face teaching will coexist with virtual teaching, and that it is up to the center to define the coexistence formulas of both teaching modalities, once these are known, the telematic means Campus Virtual of the USC, Microsoft Teams, or any other type provided by the academic authorities, will be conducted in a synchronous as well as an asynchronous manner, both in terms of content explanations and practical issues.
Tutorials will preferably be virtual.
The channels planned with the students in the telematic case are the Campus Virtual of the USC and Microsoft Teams.
Assessment system
When the procedures cannot be done in person, they will be carried out telematically.
If a face-to-face final exam cannot be held, for the computation of the final grade, both in the first opportunity and in the second, the following formula will be applied:
F= 0.4*C+0.6*E, F: Final mark, C: Continuous evaluation, E: Exam mark
Scenario 3: Lockdown
Teaching methodology
The teaching will be completely virtual, with synchronous or asynchronous mechanisms.
The tutorials will be exclusively virtual.
The channels planned with the students, in this case, are the Campus Virtual of the USC and Microsoft Teams
Assessment system
All procedures will be performed telematically.
If a face-to-face final exam cannot be held, for the computation of the final grade, both in the first opportunity and in the second, the following formula will be applied:
F= 0.4*C+0.6*E
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
| Monday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| 11:00-12:00 | Grupo /CLE_02 | Spanish | Classroom 03 |
| Tuesday | |||
| 09:00-10:00 | Grupo /CLIS_03 | Spanish | Classroom 07 |
| 11:00-12:00 | Grupo /CLIS_04 | Spanish | Classroom 07 |
| 12:00-13:00 | Grupo /CLIS_02 | Spanish | Classroom 07 |
| 13:00-14:00 | Grupo /CLIS_01 | Spanish | Classroom 02 |
| Wednesday | |||
| 11:00-12:00 | Grupo /CLE_02 | Spanish | Classroom 02 |
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| 12:00-13:00 | Grupo /CLIL_04 | Spanish | Classroom 09 |
| 13:00-14:00 | Grupo /CLIL_06 | Spanish | Classroom 09 |
| Thursday | |||
| 09:00-10:00 | Grupo /CLIL_05 | Spanish | Classroom 08 |
| 11:00-12:00 | Grupo /CLIL_03 | Spanish | Classroom 08 |
| 12:00-13:00 | Grupo /CLIL_01 | Spanish | Classroom 09 |
| Friday | |||
| 12:00-13:00 | Grupo /CLIL_02 | Spanish | Classroom 07 |
| 01.24.2022 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 06.21.2022 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |