ECTS credits ECTS credits: 9
ECTS Hours Rules/Memories Hours of tutorials: 2 Expository Class: 56 Interactive Classroom: 28 Total: 86
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Algebra, Geometry and Topology
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
This is a course in the fundamentals of mathematics and provides preparation for the other subjects in the mathematics major. Students will develop good habits of understanding, communicating and writing mathematics. Methods and techniques of reasoning will be worked on. The methods will be applied to solve various interesting problems. It could be said that this is a course about understanding and thinking, not about calculating and memorizing rules.
The program explores topics involving numbers, sets and functions. With elementary properties of these, it moves on to induction and cardinality. The study of natural numbers includes the properties of divisibility and modular arithmetic.
1. Introduction to mathematical logic (2 hours of lectures).
2. Sets. (4 expository hours).
2.1. Sets and elements. Subsets: Parts of a set.
2.2. Operations with sets: Properties. Boolean algebra of the parts of a set.
2.3. Covering and partition. Disjoint union and Cartesian product.
3. Applications (5 expository hours)
3.1. Concept of map. Graph of an map: Examples.
3.2. Injective, surjective and bijective applications.
3.3. Composition of applications; properties; inverse application.
3.4. Extensions of an application to the power set.
4. Relationships (6 expository hours)
4.1. Notion of relation. Composition of relations. Inverse relation.
4.2. Graphical representations. Binary relations in a set; properties.
4.3. Induced relation.
4.4. Equivalence relations: Equivalence classes: Properties. Equivalence relations: Equivalence classes: Properties. Example: rational numbers.
4.5. Canonical factorization of an application.
4.6. Order relations: Graphical representations: Hasse diagrams (trees). Total and partial order. Salient elements of an ordered set. Chains, lattices and well-ordered sets.
5. Infinite sets (6 expository hours).
5.1. Finite and infinite sets.
5.2. Principle of induction. Operations and order in N.
5.3. Cardinality. Cantor Bernstein's theorem. Order relation.
5.4. Numerable and non-numerable sets. Numerability of Q and non-numerability of R.
5.5. Cardinality of unions, products, the set of parts, etc.
5.6. The axiom of choice and Zorn's lemma. Application 1: the order relation between cardinals is of total order. Application 2: cardinality of AxA when A is an infinite set.
6. Combinatorics (4 expository hours).
6.1. Number of applications and injective applications between finite sets.
6.2. Permutations. Permutations with repetition.
6.3. Combinatorial numbers. Combinations. Newton's binomial. Combinations with repetition.
6.4. Principle of inclusion-exclusion. Number of surjective applications between finite sets.
7. Algebraic structures (3 expository hours).
7.1. Groups, homomorphisms and isomorphisms of groups.
7.2. Cyclic groups.
7.3. Symmetric groups. Sign of a permutation.
8. The ring of integers (5 expository hours).
8.1. Rings and ideals; homomorphisms and isomorphisms of rings.
8.2. Integers and structure of (Z,+). Properties of Z.
8.3. Divisibility. Prime numbers. Greatest common divisor and least common multiple.
8.4. Bézout's identity. Fundamental theorem of arithmetic.
8.5. Euclid's algorithm. Extended Euclid's algorithm.
9. Modular arithmetic (8 expository hours)
9.1. Congruences. Z/(n) rings. Groups of units modulo n.
9.2. Euler's fi function and the Euler-Fermat theorem.
9.3. Introduction to the diophantine equations. Solving linear diophantine equations.
9.4. The Chinese remainder theorem. Multiplicativity of Euler's phi function.
10. Arithmetic of polynomials in one variable (4 expository hours).
10.1. Univariate polynomials over a field. Degree of a polynomial.
10.2. Divisibility, units and irreducible polynomials.
10.3. Euclidean division and Bezout's theorem.
11. Operations and relations between subsets of R^n. (9 expository hours).
11.1 Real numbers, properties.
11.2 Inner product, norm and distance.
11.3 Applications: direct, reciprocal and inverse images in R^n. Examples.
11.4 Intersections, unions, relations and quotient sets in R^n. Examples.
Basic bibliography:
J.P. D’Angelo, D. B. West: Mathematical Thinking: Problem-Solving and Proofs, 2ª ed., Prentice Hall, 2000.
M. A. Goberna, V. Jornet, R. Puente, M. Rodríguez: Álgebra y Fundamentos: una Introducción, Ariel, 2000.
Frans Keune: Elements of Higher Mathematics. Learning Mathematics through Numbers
Radboud University Press, 2024.
Available at https://books.radbouduniversitypress.nl/index.php/rup/catalog/book/elem…
( https://doi.org/10.54195/PEAQ9203 )
Complementary bibliography:
M. Anzola, J. Caruncho: Problemas de Álgebra (Conjuntos-Estructuras), 1982.
E. D. Bloch: Proofs and Fundamentals A First Course in Abstract Mathematics, Springer, 2011.
T. S. Blyth, E. F. Robertson: Sets, Relations and Mappings, Cambridge University Press, 1984.
R. Courant, H. Robbins: What Is Mathematics? 1941 (2ª ed., rev. por Ian Stewart, Oxford University Press, 1996). Existen traducciones al español.
K. Houston: How to Think Like a Mathematician, Cambridge University Press , 2009.
H. Rademacher, O. Toeplitz: Números y Figuras. Alianza editorial, 1970.
Knowledge: Con01, Con02, Con03, Con04, Con05.
Skills: H/D01, H/D02, H/D03, H/D04, H/D06, H/D07, H/D08.
Competences: Comp01, Comp02, Comp03, Comp04.
The weekly distribution of the course will be as follows:
4 hours of lectures and 2 hours of laboratory.
The expository classes in a large group will be dedicated to the exposition of the fundamental contents of the to the exposition of the fundamental contents of the discipline, with the exposition of the theory, examples, resolution of problems and presentation of some exercises.
The laboratory classes will deal with complementary aspects of the subject, problems and exercises and resolution of general doubts, where generally the main role will be played by the students, who will have to present exercises and expositions related to the subject.
The assestment system will be coordinated for the two groups of the subject. Continuous evaluation combined with a final test is foreseen as an evaluation criterion. This final test will be held on the date set by the Faculty of Mathematics for this purpose and will be the same for all students of the subject. The continuous evaluation will consist of tests (one or two in the course) that may or may not coincide for the different groups, but will be coordinated and similar. The realization of problems in the laboratory hours may also be evaluated.
Final test will consist of:
Written resolution of theoretical questions and written resolution of practical questions: 35%-65%.
Statement of theorems, development of demonstrations and resolution of problems/exercises: 35%-65%.
For the computation of the final grade (N) the continuous evaluation (C) and the grade of the final exam (F) will be taken into account and the following formula will be applied:
N= max (F, 0,20*C+0,80*F)
This same formula will also be applied in the extraordinary period of July.
A student who does not attend either of the two corresponding opportunities will be considered “not presented”. For cases of fraudulent performance of exercises or tests, the provisions of the Regulations for the Evaluation of Students' Academic Performance and Review of Grades will apply.
Classroom hours:
- Lectures: 56 hours.
- Interactive laboratory classes: 28 hours.
Small group tutorials: 2 hours.
Personal work of the student: 139 hours
The objective is not to solve problems, but to study thoroughly the theory before tackling the proposed problems. The proposed problems will serve two purposes: to check that the theory has been understood and to develop the ability to solve new problems based on the theory.
Leovigildo Alonso Tarrio
Coordinador/a- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813159
- leo.alonso [at] usc.es
- Category
- Professor: University Lecturer
María Elena Vázquez Abal
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813143
- elena.vazquez.abal [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
Miguel Dominguez Vazquez
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813156
- miguel.dominguez [at] usc.es
- Category
- Professor: University Lecturer
| Monday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_02 | Spanish | Classroom 03 |
| 12:00-13:00 | Grupo /CLE_01 | Galician, Spanish | Classroom 02 |
| Tuesday | |||
| 10:00-11:00 | Grupo /CLE_01 | Spanish, Galician | Classroom 02 |
| 11:00-12:00 | Grupo /CLE_02 | Spanish | Classroom 03 |
| 12:00-13:00 | Grupo /CLIL_05 | Spanish | Classroom 09 |
| 13:00-14:00 | Grupo /CLIL_06 | Spanish | Classroom 07 |
| Wednesday | |||
| 09:00-10:00 | Grupo /CLE_01 | Galician, Spanish | Classroom 03 |
| 10:00-11:00 | Grupo /CLIL_03 | Galician, Spanish | Classroom 08 |
| 10:00-11:00 | Grupo /CLIL_04 | Spanish | Classroom 09 |
| 11:00-12:00 | Grupo /CLIL_01 | Galician, Spanish | Classroom 05 |
| 12:00-13:00 | Grupo /CLIL_05 | Spanish | Classroom 09 |
| Thursday | |||
| 10:00-11:00 | Grupo /CLIL_02 | Galician, Spanish | Classroom 01 |
| 10:00-11:00 | Grupo /CLIL_06 | Spanish | Classroom 08 |
| 11:00-12:00 | Grupo /CLE_01 | Galician, Spanish | Classroom 02 |
| 11:00-12:00 | Grupo /CLIL_04 | Spanish | Classroom 09 |
| 13:00-14:00 | Grupo /CLE_02 | Spanish | Classroom 03 |
| Friday | |||
| 09:00-10:00 | Grupo /CLIL_01 | Spanish, Galician | Classroom 01 |
| 10:00-11:00 | Grupo /CLE_02 | Spanish | Classroom 03 |
| 10:00-11:00 | Grupo /CLIL_02 | Spanish, Galician | Classroom 07 |
| 12:00-13:00 | Grupo /CLIL_03 | Galician, Spanish | Classroom 01 |
| 01.09.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 06.19.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |