This course establishes the foundations of Analysis and Algebra.
Current mathematics is constructed as follows:
We start from a small number of statements, called axioms, assumed to be true at
a priori (and which we therefore do not seek to demonstrate).
We then define the notion of demonstration (by deciding, for example,
what is an implication, an equivalence...)
We finally decide to qualify as true any statement obtained at the end of
demonstration and we call such a (true) statement a “theorem”.
From the axioms, we therefore obtain theorems which come little by little
enrich mathematical theory.
The first course consists of introducing some notions of mathematical logic, this is the minimum you need to have to establish a solid foundation and to study the rest.
The second establishes the notion of set and real function with real variable.
The third aims to study the basic analytical properties of a real function with a real variable, such as the study of limits, continuity, differentiability...
The fourth course introduces new elementary functions such as cosine and hyperbolic sine functions and their inverses.
The fifth course focuses on the notion of limited development.
We end the module by introducing the basics of linear algebra by defining vector spaces and linear applications.
