Discrete Mathematics

Teacher(s)

Saerens Marco;

Language

French

Prerequisites

This course assumes that the student already masters notions of algebra covered by the course LINFO1112

The prerequisite(s) for this Teaching Unit (Unité d’enseignement – UE) for the programmes/courses that offer this Teaching Unit are specified at the end of this sheet.

Main themes

Set theory

• Set notations and operations
• Binary relations between sets: applications and link with functions in analysis
• Cardinality of a set (finite and infinite) and notion of inclusion-exclusion
• Equivalence, equivalence classes

Logic

• Introduction to the logic of the proposals
• Introduction to the logic of predicates
• Prove methods
• Mathematical induction
• Notions of Boolean Algebra

Introduction to number theory

• Natural integer numbers, principle of recurrence, prime numbers, etc.
• Euclidean division, representation in a base, modulo arithmetic, representation of the integers in the computer
• Gcd, Euclid's algorithm
• Basic notions of cryptography

Combinatorial mathematics

• counting
• permutations
• arrangements
• Recurrence relations
• Solutions of recurrence equations

Introduction to graph theory

• Oriented and non-oriented graphs and their matrix representations
• Bipartite graphs and matching problems
• Paths on a graph and Eulerian / Hamiltonian circuits
• Planar graphs and coloring
• Problems of shorter path
• Ranking of the nodes of a graph: PageRank

Learning outcomes

At the end of this learning unit, the student is able to :
1	Given the learning outcomes of the "Bachelor in Engineering" program, this course contributes to the development, acquisition and evaluation of the following learning outcomes: S1.I1, S1.G1 S2.2 Students who have successfully completed this course will be able to: Use the terminology of functions, relationships and together well and perform related operations when the context requires it Explain the basic structure of the main proof techniques (direct proof, counterexample, proof by the absurd, induction, recurrence) Apply the various proof techniques in a convincing way by selecting the most adapted to the problem posed Analyze a problem to determine the underlying recurrence relationships Calculate counts, permutations, arrangements on sets as part of an application. Modeling various real-world problems encountered in computer science using the appropriate forms of graphs Explain the problem of the shortest path in a graph and apply classical algorithms to solve this problem

Teaching methods

About 30 hours of lectures, on-site or remotely depending on the situation.
A mandatory project/case study on the implementation and the application of some algorithms.

Evaluation methods

• A mandatory project/case study that counts for 2 to 4 out of 20 points (specified at the beginning of the semester). If the project report is not done (no report submitted), the student will get a 0/3 for this project.
• A written exam organized in session counting for 18 to 16 out of 20 points (specified at the beginning of the semester). Organized on-site or remotely, depending on the situation.

Concernant le projet/cas d'étude obligatoire et l'utilisation d'IA de type Chat GPT, assurez-vous que "En soumettant un travail pour évaluation, vous affirmez : (i) qu'il reflète fidèlement le phénomène étudié, et pour cela vous devez avoir vérifié les faits, surtout s'ils sont prétendus par une IA générative (dont vous devez mentionner explicitement l’utilisation en tant qu’outil de soutien à la réalisation de votre travail) ; (ii) avoir respecté toutes les exigences spécifiques du travail qui vous est confié, notamment les exigences pour la transparence et la documentation de la démarche scientifique mise en œuvre. Si l'une de ces affirmations n'est pas vraie, que ce soit intentionnellement ou par négligence, vous êtes en défaut de votre engagement déontologique vis-à-vis de la connaissance produite dans le cadre de votre travail, et éventuellement d’autres aspects de l’intégrité académique, ce qui constitue une faute académique et sera considéré comme tel".

Online resources

On Moodle

Bibliography

Rosen K., Discrete mathematics and its applications, 8th edition, 2019. Mc Graw Hill.

Teaching materials

• Slide du cours
• Textbook "Mathématiques discrètes" de K. Rosen

Faculty or entity

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