# DISCRETE MATHEMATICS

**Academic Year 2016/2017**- 1° Year

## Learning Objectives

1) Knowledge and understanding: The aim of the course is to give the basics of linear algebra, analytic geometry, number theory and combinatorics that are useful to interpret and describe problems in computer science.

2) Learning skills: The aim of the course is to provide the study method to the students, the forma mentis and the logical rigor that will be needed in order to solve autonomously new problems that may arise during a work activity.

## Detailed Course Content

PART A

1. Preliminaries. Sets and applications *. Relations: equivalence relations and partial order *. Cardinality of a set *. Binary algebraic operations *. Algebraic structures: groups, fields *.

2. Number Theory. * Natural numbers. * Integers. Induction principles *. Division theorem *. Greatest common divisor (M.C.D.) and least common multiple (m.c.m.) *. Numbering systems * * .Congruences. Equations of congruences *. Systems of congruences and Chinese remainder theorem *. Fermat's theorem *. * Applications to cryptography.

3. Combinatorics *. Product Rule and the summation rule *. Permutations, combinations, arrangements (with and without repetition) *. Formula of Stifel and identity Vandermonde *. Binomial of Newton *. Number of distributions of marbles in the polls k *. Stirling number of second kind *. Principle of inclusion and exclusion *. Graphs: definitions and examples. Representations of a graph *. Trees *.

4. Discrete Probability. Definition of probability *. uniform probability and its properties *. * conditional probability. Stochastic independence *.

5. Calculation of matrix algebra and linear systems *. Matrices. Matrix operations *. Matrices considerable *. linear systems *. Calculating the inverse matrix *. Determinant of a square matrix and its properties *. Rank of a matrix *. Cramer's theorem and Rouche-Capelli *.

PART B

1. Vector Calculus. Applied vectors *. Decomposition theorem *. Scalar product and cross product *. Mixed product *. Free vectors *.

2. Linear geometry in the plane. Lines in the plane and their equations *. Parallelism and squareness *. Intersection between plane and lines *. Homogeneous coordinates in the plane *. Bundles of straight *.

3. Isometries *. Translation, rotation around a point *. Reflection *.

4. Linear geometry in space. Planes and lines in space and their equations *. Parallelism and squareness *. Intersection between planes, between a plane and a straight line and between lines *. homogeneous coordinates in space *. Improper points and lines in space *. Bundles of plans *.

5. Vector spaces. Definition of vector space *. Subspaces and transactions between them *. Subspace sum *. Linear independence and linear dependence *. Bases of a vector space *. Dimension of a vector space *. Ordered basis of a vector space *.

6. Linear applications. Linear Application Definition *. Kernel and image of a linear map *. Properties of linear applications *. Rank of a linear map *. Basic changes *. Transformation formulas of the components *. Matrix associated to a linear map *. * Similar matrix. Eigenvalues and eigenvectors *. * characteristic polynomial. Research of the eigenvalues and eigenspaces associated with them *. * simple endomorphisms. * diagonalizable matrices. Similarity between matrices *.

## Textbook Information

- Appunti in rete alla pagina web https://andreascapellato.wordpress.com/didattica-2/
- S. Giuffrida, A. Ragusa, Corso di Algebra Lineare, Il Cigno Galileo Galilei Roma.
- G. Paxia, Lezioni di Geometria, Cooperativa Universitaria Libraria Catanese.
- K.H. Rosen, Discrete Mathematics and Its Applications, Mc Graw Hill.