Discrete mathematics

A.Y. 2021/2022
Overall hours
MAT/01 MAT/02 MAT/03 MAT/04 MAT/05 MAT/06 MAT/07 MAT/08 MAT/09
Learning objectives
The objectives of the course include the basic notions of mathematical reasoning and their associated formalisms, with a particular focus in discrete mathematics (set theory, algebraic structures, linear algebra and geometry).
Expected learning outcomes
The ability of formalizing mathematical notions and reasonings, mastering basic notions of set theory and algebraic structures, knowing and properly applying the foundamentals of linear algebra and affine geometry.
Course syllabus and organization

Single session

Lesson period
First semester
More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation.
Course syllabus
The course will cover the following topics:
- Basic operations between sets;
- Relations and their fundamental properties: transitivity, reflexivity, symmetry.
- Fundamental sets of numbers: natural, integers and rational numbers.
- Induction principle.
- Congruences, Chinese Remainder Theorem.
- Groups, homomorphisms between groups. Permutation groups.
- Fields and rings: definitions, examples, fundamental properties.
- Vectors, operations between vectors. Applications might include geometry in space (if time permits).
- Vector spaces: linear dependence, generators, bases, dimension, Grassman formula.
- Matrices: operations between matrices, relationship between matrices and linear systems, Gauss-Jordan method. Relationship with homomorphisms, search for eigenvalues ​​and eigenvectors, diagonalizability.
- Geometry: planes, lines, parallel, orthogonality and interserctions in dimensions 2 and 3.
Prerequisites for admission
Basic Logic and Mathematics skills, including properties and operations on numbers (integers, rationals, reals), powers, roots and their properties, polynomials (operations, factoring, divisions with remainder, principle of identity), equations and algebraic inequalities of first and second degree or reducible to them, resolution of simple linear systems, literal calculus, ability to distinguish between hypotheses and theses.
Teaching methods
Frontal lessons.
If the course has to be teached from remote, lessons will be given synchronously from remote using the Zoom platform and lecture notes will be shared through the Ariel platform.
Teaching Resources
We will make notes available on the Ariel platform. There are plenty of books on the subject available on sale.
Assessment methods and Criteria
During the course, we will assign some optional homework, which will not be evaluated but used to test the level of understanding achieved by the class.
The exam will consist of a written test in which students will have to solve exercises.
If the emergency situation allows it, two intermediate tests will also be carried out.
Students who obtain a positive assessment in both intermediate tests are exempted from the exam.
MAT/01 - MATHEMATICAL LOGIC - University credits: 0
MAT/02 - ALGEBRA - University credits: 0
MAT/03 - GEOMETRY - University credits: 0
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0
MAT/06 - PROBABILITY AND STATISTICS - University credits: 0
MAT/07 - MATHEMATICAL PHYSICS - University credits: 0
MAT/08 - NUMERICAL ANALYSIS - University credits: 0
MAT/09 - OPERATIONS RESEARCH - University credits: 0
Practicals: 24 hours
Lessons: 32 hours