Mathematics I

The primary goals of this course are to introduce the language and fundamental concepts of discrete mathematics, algebra, and real analysis of a single variable. In more detail, the course will cover the elementary theory of sets, relations, functions, and real-valued functions of a single real variable. A first introduction to the notion of abstract algebraic structures is provided using monoids, groups, and rings as examples. The basic properties of the ring of integers and the fields of rational, real, and complex numbers are discussed, focusing on the resolution of linear congruences and their algorithmic aspects. Elementary operations with complex numbers are introduced for solving first- and second-degree equations. Vector spaces are also treated, along with linear transformations and their matrix representation. The theory of linear algebra is then applied to the solution of systems of linear equations, highlighting their algorithmic aspects as well.

At the end of the course, students should be able to understand the basic mathematical formalism of sets, relations, and functions. They will know how to solve simple exercises concerning real-valued functions of a single real variable, such as calculating the domain and representing the graph of elementary functions. They will have acquired a foundational familiarity with the concept of abstract algebraic structures. They will have understood the fundamental properties of the ring of integers and the field of rational, real, and complex numbers. They will have acquired familiarity with the operations on complex numbers and be able to calculate powers and roots of complex numbers. They will be able to recognize and work with vector spaces and linear transformations. Finally, they will be capable of performing operations with matrices, associating them with linear systems, and using them to analyze their solvability.