## Learning objectives

The goal of the course is to provide first year students in Mathematics and Physics a gentle introduction to mathematical thinking and a correct use of the mathematical language. The first week of the course will focus on the basic notions of mathematics such as functions between sets and functions in a real variable. During this time, the students are strongly encouraged to self-assess their knowledge on these topics and, if necessary, act swiftly to fill the gaps, for instance by taking advantage of the tutoring service offered by the Department.

During the second and third week, the course will revolve around basic notions of combinatorics, probability, and discrete mathematics.

## Prerequisites

## Course unit content

Overview of the basic notions of mathematics, focusing on functions between sets and functions in one or more real variables. Cardinalities, partial and total orders, foundations of mathematics. Basics of combinatorics, probability, and discrete mathematics.

## Full programme

Lecture 1 - 4h.

Logic and set theory.

Lecture 2 – 5h.

Functions. Definitions. Examples. Functions in one real variable. Operation on functions. Graphs.

Lecture 3 – 5h.

Equations and inequalities (of both rational and trigonometric type). Relations and pre-images.

Lecture 4 – 5h.

Implicit functions. Conic sections.

Lectures 5 - 5h.

Combinatorics and proofs. Permutations and combinations. Sums over multiple indices.

End of the course for the students in Physics.

Lecture 6 - 4h.

Equivalence relations. Modular arithmetics.

Lecture 7 - 4h

Cardinalities. Partial and total orders.

Lecture 8 – 4h

Introduction to Probability.

Lectures 9 - 4h.

Conditional probability and Bayes theorem.

Lecture 10 – 4h

Binomial probability. Expected value.

Lezione 11 - 4h. Exercise session.

## Bibliography

Lecture notes will be provided on Elly together with exercises and solutions.

Additional material (optional):

• S. Lang, Basic Mathematics, 1970.

• G. Prodi, Analisi Matematica (Cap. 0), 1972.

• E. Acerbi, G. Buttazzo, Matematica Preuniversitaria di Base, 2003.

• S. Ross, Probabilit`a e statistica per l’ingegneria e le scienze, 2013.

• F.G. Alessio, C. de Fabritiis, C. Marcelli, P. Montecchiari, Matematica zero, 2016.

## Teaching methods

The program of the course will be divided in "one-a-day" topics. This means that a given topic is discussed entirely from both a theoretical and practical point of view in the span of a morning and an afternoon session.

## Assessment methods and criteria

There will be a final written exam with open-ended questions.

## Other information