Last year's content

Last year (course 2015-2016) we had some fantastic teachers who eagerly provided us with interesting classes on very diverse topics.

What follows is a recompilation of the notes and exercises they made available after their classes.

Lectures by Diego Chicharro

Elementary Number Theory

A short review of divisibility, plus some interesting problems.

Content

  • The basis representation theorem.
  • Euclid's division lemma.
  • Basic concepts on divisibility.
  • Prime numbers.
  • The Fundamental Theorem of Arithmetic (FTA).

Lecture notes

Exercises Solutions

Lectures by Jaime Sevilla

Set Theory

A gentle introduction to axiomatic set theory.

Content

  • An introduction to formal mathematics: the axiomatic schema, naive Set Theory and Russell's Paradox.
  • The axioms of Set Theory.
  • Construction of mathematical objects in Set Theory: tuples, functions, families.
  • Peano Arithmetic in Set Theory.
  • Equivalence. Finite and infinite sets.

Lecture notes & exercises

Equinumerability

Basic results about countable and uncountable sets.

Content

  • Schröder-Bernstein's theorem.
  • Countability.
  • Uncountability.

Lecture notes & exercises

Computability

The basic definitions and results regarding the study of what can and cannot be computed by an algorithm.

Content

  • Turing Machines.
  • Uncomputability. The diagonal function and the halting problem.
  • The productivity function.

Lecture notes & exercises

Turing Machine Compiler