Mathematical Language, Sets and Numbers

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 contribute to achieving the generic, specific and transversal competencies 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.

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 lecture classes in big groups devote the exposition of the fundamental contents of the subject, with theory, resolution of problems and presentation of some exercises.
The seminar classes in a reduced group will deal with complementary aspects of the subject, the realization of problems and exercises and corrections by the teacher.
In the laboratories in a 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 a much-reduced group, the teacher will make a personalized tracking of the learning of the students.

The assessment system will be the same for the groups of the subject.
During the semester, the students may be asked to hand in written exercises in class. The continuous evaluation will consist of the individual resolution of assignments (one or two in the course) and tests (one or two in the course).

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 to the extra opportunity in July.

The written exam will consist of theory and theoretical-practical questions and exercises.

Students who do not attend either of the two final examinations will be considered "Not Presented".

In cases of fraudulent performance of exercises or tests, the provisions of the Regulations on the Evaluation of Students' Academic Performance and Revision of Grades shall apply.

Attendance at classes:
- Lecture classes: 28 hours.
- Interactive seminar classes: 14 hours.
- Interactive laboratory classes: 14 hours.
Hours of tutorials in very small groups: 2 hours.
Total presence hours: 58

Personal work hours:
Autonomous study, individually or in group: 47 hours.
Solving/writing exercises, conclusions or other works: 45 hours.

Total workload: 150 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.