DISCRETE MATHEMATICS

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 *.

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