Mathematics I

The aim of the course is to introduce students to the language and the first notions of discrete mathematics and algebra. In more detail, the course tackles the elementary theory of sets, relations, and functions. It hints at the concept of abstract algebraic structure focusing on the examples of monoids, groups, and rings. It discusses the basic properties of the ring of integers, as well as those of the fields of rational, real and complex numbers. Special attention is paid to the solution of linear congruences and its algorithmic aspects. The course then introduces vector spaces together with linear transformations and their representation by matrices. Finally, the theory is applied to the solution of systems of linear equations, emphasising once again the algorithmic aspects involved.

Upon course completion, students are expected to understand the basic mathematical formalism of sets, relations, and functions. They are also expected to have an initial degree of familiarity with the concept of abstract algebraic structure. They are expected to understand the basic properties of the ring of integers and of the fields of rational, real, and complex numbers. They are expected to recognise and manipulate vector spaces and their linear transformations. Finally, they are expected to be able to operate with matrices, associating them to systems of linear equations in order to discuss their solvability.