The course is an introduction to the fundamental notions of Discrete Mathematics, one of the fastest growing areas of modern mathematics. These notions are nowadays more and more used in the development, analysis and comprehension of mathematical models in every area of engeneering. Among the many topics that go under the name of discrete mathematics, in the course we consider set theory, enumerative combinatorics, graph theory, elementary number theory and modular arithmetics, and abstract algebra.

At the end of the course, students who have mastered contents and methodology, will be able to express the fundamental concepts of Discrete Mathematics through definitions, theorems, examples and counterexamples, to justify the effectiveness of mathematical procedures used to solve the problems presented in the course and to properly use the mathematical language while arguing (Dublin Descriptor 1, or DD1). 

Regarding the application of the acquired knowledge and understanding, students will be able to:
• use formal rules for computations on symbols, independently from the meaning of the symbols (DD2);
• solve equations over the integers or over the integers modulo m (DD2);
• compute the number of elements of sets defined by a property (DD2);
• analyze graphs (DD2).

SET THEORY: Basic notions of set theory. Natural numbers and the induction principle. 

ENUMERATIVE COMBINATORICS: Cardinality of a set. Basic counting principles. Principle of inclusion and exclusion. Formulas of Sylvester and Da Silva. Binomial numbers, Stiefel's formula and other identities on binomial numbers. Permutations and selections from a set. Derangements. Binomial theorem. Stirling numbers of the second kind. Partitions of a natural number. Ferrers diagrams. The Tower of Hanoi problem. Formal power series. Partial fractions. Generating functions. Linear homogeneous recursions. Fibonacci and Lucas numbers. Closed form of the Fibonacci numbers.

GRAPHS: Basic definitions. Planarity of graphs. Euler's formula. Bipartite graphs. Eulerian graphs. Hamiltonian graphs. Trees. Binary trees. Spanning trees. Vertex colorings. Chromatic number. Edge colorings. Chromatic index. K\"{o}nig's theorem. Matchings in bipartite graphs. Hall's theorem. Adjacency matrix, incidence matrix with respect to an orientation, Laplacian matrix. Theorem of Poincaré. Spectrum of a graph.

ELEMENTARY NUMBER THEORY AND MODULAR ARITHMETIC: Integers. Divisibility and prime numbers. Euler's function. Congruences. Integers modulo m. Euler's theorem. Fermat's little theorem. Chinese remainder theorem. Wilson's primality test.  

ABSTRACT ALGEBRA: Elements of group, rings and fields. Finite fields.

• SDG4 - QUALITY EDUCATION

Students are required to have basic knowledge of mathematical analysis and linear algebra.

• Written exam mandatory, without midterm assessments
• Oral exam optional (student's choice)
• Project(s) / Assignment(s) optional (student's choice), individual, without periodic reviews, with final presentation/discussion

The aim of the exam is to verify if every student has got the following expertises:

1. ability of applying theoretical results to solve exercises (DD2),

2. knowledge of definitions (DD1),

3. results on the topics of the course (DD1 and DD2),

4. proofs of the main mathematical theorems presented in the course (DD1). 

The final evaluation consists of an oral exam on the topics presented in the course, part of which is devoted to the solution of exercises.