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
Center Faculty of Mathematics
Call: First Semester
Teaching: Sin Docencia (En Extinción)
Enrolment: No Matriculable (Sólo Planes en Extinción)
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 the elementary properties of these objects and some basics of 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. Infinite Sets. (3 sessions)
5.1. Finite and Infinite Sets.
5.2. The Natural Numbers as Equipotency Classes of Finite Sets.
5.3. Principle of Induction. Operations and Order on Natural Numbers.
5.4. Countable and Uncountable Sets. Rational Numbers. The Diagonal Procedure and the Uncountability of R.
5.5. The Axiom of Choice and Zorn's Lemma.
6. Combinatorics. (3 sessions)
6.1. Variations. Variations with Repetition.
6.2. Factorial Numbers. Permutations. Permutations with Repetition.
6.3. Combinatorial Numbers. Combinations.
6.4. Combinations with Repetition.
6.5. Principle of Inclusion-Exclusion. Enumeration of the Surjective Maps.
6.6. The Tartaglia-Pascal´s triangle. The Newton´s Binomial.
7. Integer and Modular Arithmetic. (7 sessions)
7.1. Binary Operations.
7.2. Integer Numbers and structure of (Z,+). Properties of Z.
7.3. Divisibility. Prime Numbers and the Fundamental Theorem of Arithmetics.
7.4. Greatest Common Divisor and Least Common Multiple. Bézout'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, Ed. 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.
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? 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.
D. E. Ernts: An Introduction to Proof via Inquiry-Based Learning, AMS/MAA Textbooks Vol. 73, 2022.
H. Rademacher, O. Toeplitz: Números y Figuras. Alianza editorial, 1970.
To achieve the generic, specific and transversal competencies listed in the Report on the Degree in Mathematics from USC: CB1, CB2, CB3, CB4, CB5, CG1, CG2, CG3, CG4, CG5, CT1, CT2, CT3, CT5.
No teaching.
The assessment system will be a final written exam.
no teaching
Leovigildo Alonso Tarrio
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813159
- leo.alonso [at] usc.es
- Category
- Professor: University Lecturer
| 01.09.2026 10:00-14:00 | Grupo de examen | Classroom 06 |
| 06.19.2026 10:00-14:00 | Grupo de examen | Classroom 06 |