Foundations of Mathematics II
Code: 107927 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| Mathematics | FB | 1 |
Contact
- Name:
- Carlos Broto Blanco
- Email:
- carles.broto@uab.cat
Teachers
- Francisco Perera Domenech
- Gabriel Martínez De Cestafe Pumares
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
Practice and skill in manipulating algebraic expressions are required. No specific prior mathematical knowledge is required to follow the course, but a minimum achievement of the skills and results of the subject "Fundamentals of Mathematics I" is recommended. However, the desire to understand reasoning in depth and to have a critical sense of the mathematical statements of others and, above all, of one's own is essential.
Objectives and Contextualisation
At the beginning of the course, we will emphasize the logical structure of mathematics and the axiomatic method. We will delve into set theory, equivalence relations, counting elements and cardinality.
In the second part of the course we will visit integers, their quotients and polynomials with the perspective and tools of the first part, we will see beautiful demonstrations of well-known facts such as that there are infinite prime numbers or that there is a greatest common divisor of two numbers and we will also see that in polynomials we find analogous results.
We hope that the theorems and demonstrations of the course will contribute to the student acquiring adequate training that will allow him to begin to make demonstrations for himself, to be critical of mathematical statements and, above all, combative when faced with problems.
Learning Outcomes
- CM06 (Competence) Discriminate between statements of results and their proofs to identify situations in which a counterexample is needed.
- CM07 (Competence) Construct proofs that respect the rules of propositional logic and mathematical induction.
- KM11 (Knowledge) Identify the basic principles of classical logic, as well as their relationship to the use of sets.
- KM12 (Knowledge) Describe some basic axiomatic systems in set theory, modular arithmetic, and polynomial arithmetic.
- KM13 (Knowledge) Describe the basic algorithms for factoring and solving Diophantine equations.
- KM14 (Knowledge) Describe the processes for solving Diophantine equations and calculating polynomial roots.
- SM11 (Skill) Use the axiomatic method in the construction of the hierarchy of numbers, and in particular, in the justification of the introduction of complex numbers.
Content
1. Mathematical Logic
2. Set Theory
3. Arithmetic
4. Congruences
5. Polynomials
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Problem sessions | 14 | 0.56 | CM06, CM07, KM13, KM14, SM11, CM06 |
| Theory classes | 30 | 1.2 | CM06, CM07, KM11, KM12, KM13, KM14, SM11, CM06 |
| Type: Supervised | |||
| Seminars | 6 | 0.24 | CM06, CM07, KM13, KM14, SM11, CM06 |
| Type: Autonomous | |||
| Study of theory and solving exercises | 88 | 3.52 | CM06, CM07, KM11, KM12, KM13, KM14, SM11, CM06 |
The methodology and training activities are adapted to the training objectives: introducing the mathematical language, learning to use it correctly, seeing proofs (and finding them, and writing them correctly!) and proof methods. To achieve these objectives, it is important that the student understands the theory but also, and even more, it is important that he/she tries to do the exercises.
In the problem classes, the exercises from the lists that the student will have previously worked on on his own will be discussed and solved on the board.
In the seminar sessions, the teacher will provide materials with exercises to practice discovering and writing proofs. Students must ask as many questions as necessary and finally the teacher will explain the resolution of the most representative exercises.
It must be clear that the correct assimilation of the syllabus of this subject requires dedication and continuous and sustained work on the part of the student. It is highly recommended to consult the bibliography as part of this independent work.
Note: 15 minutes of a class will be reserved, within the calendar established by the center/degree, for students to complete the surveys to evaluate the performance of the teaching staff and to evaluate the subject/module.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Final exam | 40% | 3 | 0.12 | CM06, CM07, KM11, KM12, KM13, KM14, SM11 |
| Midterm exam | 20% | 3 | 0.12 | CM06, CM07, KM11, KM12, KM13 |
| Retake exam | 60% | 3 | 0.12 | CM06, CM07, KM11, KM12, KM13, KM14, SM11 |
| Seminars | 25% | 3 | 0.12 | CM06, CM07, KM11, KM12, KM13, KM14, SM11 |
| Submission of solved problems | 15% | 0 | 0 | CM06, CM07, KM11, KM12, KM13, KM14, SM11 |
The course evaluation is continuous. The grade is obtained with the following activities:
1) Submission of solved exercises. The weight of these submissions in the final grade is 15%.
2) Activities assessable in seminars. The weight of these activities in the final grade is 25%.
3) Midterm exam. 20% of the grade.
4) Final exam. 40% of the grade.
In order to pass the subject without a retake exam, the average of the midterm and final must be at least 3.5.
Those students who have not passed the subject (and only these) may take a retake exam, the grade of which will replace that of sections 3) and 4). Activities 1) and 2) are not retakeable.
The "non-evaluable" grade will be awarded to those who have only participated in assessable activities with a total weight of less than 50%.
Single assessment:
1. The three types of assessment are maintained: exams, submission of exercises and seminars, with the same weight in the final grade and the same recovery.
2. The "exams" type of assessment will consist of a written exam on the entire course content.
3. The "submission of exercises" type of assessment will consist of the resolution, in oral presentation, of an exercise that, throughout the course, has been worked on in the classroom.
4. The "seminars" type of assessment will consist of an oral presentation on the topics covered in the course seminars.
5. All previous assessment activities will be carried out on the same day as the final exam of the continuous assessment.
Bibliography
J. Aguadé, Matemàtiques: comenceu per aquí. DDD Dipòsit Digital de Documents de la UAB, 2024. https://ddd.uab.cat/record/299307
M. Aigner i G. M. Ziegler, Proofs from The Book. Springer Verlag, 1999.
R. Antoine, R. Camps i J. Moncasi. Introducció a l'àlgebra abstracta amb elements de matemàtica discreta. Manuals de la UAB, Servei de Publicacions de la UAB, núm. 46, Bellaterra, 2007.
A. Cupillari, The nuts and bolts of proofs. Elsevier Academic Press, 2005.
P.J. Eccles, An introduction to mathematical reasoning, numbers, sets and functions. Cambridge University Press, Cambridge, 2007.
D.C. Ernst, An Introduction to Proof via Inquiry-Based Learning. Northern Arizona University 2017
P.R. Halmos. Naive set theory. Springer-Verlag, 1974
A. Reventós, Temes diversos de Fonaments de les Matemàtiques. Apunts.
Software
Sage
Groups and Languages
Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (PAUL) Classroom practices | 1 | Catalan | second semester | morning-mixed |
| (PAUL) Classroom practices | 2 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 1 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 2 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 3 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 4 | Catalan | second semester | morning-mixed |
| (TE) Theory | 1 | Catalan | second semester | morning-mixed |