DISCRETE MATHEMATICS (Matematica Discreta)
Credits: 12 (Annual course)
Teachers: Salvatore Milici and Gaetano Quattrocchi


Syllabus:
* Sets and structures
  - Sets and related operations. Cardinality of a set.
  - Application. Equivalence relations and partial orders.
  - Binary algebraic operations.
  - Algebraic structures: groups, fields.
* Matrices
  - Matrices. Operations on matrices.
  - Determinants and their properties. Rank of a matrix.
* Linear systems
  - Linear systems and row reduced matrices
  - Computation of the inverse matrix
  - Cramer and Rouché-Capelli Theorems
* Vectors and Linear geometry on the plane
  - Applied vectors. Decomposition theorem. Scalar product.
    Vector product. Mixed product. Free vectors.
  - Lines on the plane and their equations; parallelist and orthogonality;
    lines intersection; homogeneous coordinates; sheaves of lines.
  - Planar isometries: translation, rtation, reflexion, glissoreflexion.
    Planar similarities.
* Linear Geometry in the space
  - Planes and lines in the space and their equations; parallelism and
    orthogonality
  - Planes intersection; Intersection between a plane and a line;
    Lines intersecion; homogeneous coordinates.
  - sheaves of planes
* Vectorial spaces
  - Definitions. Subspaces. Generators. Linear dependency.
  - bases. Dimension. Ordered bases.
* Linear applications
  - Definition and examples. Properties of linear applications
  - Rank of a linear application. Similar matrices.
  - Change of basis. Formulae for component trasformation.
* Linear applications and matrices
  - Matrices associated to a linear application. Similar matrices.
  - Study of a linear application.
* Autovalues and autovectors
  - Autovalues and autovectors. The characteristic polynomial.
  - Search of autovalues and their associated autospaces.
  - Simple endomorphism.
  - Diagonalizable matrices. Matrices similarity.
* Number Theory
  - First and second induction principle.
  - Division Theorem. GCD and LCD.
  - Prime factors decomposition.
  - Number system in base B.
* Congruences
  - Congruences, Congruences equations.
  - Congruence systems and chinese theorem. Small Fermat Theorem and
    the RSA code.
* Combinatorial calculus
  - rules of pruduct and sum.
  - Permutations, combinations, dispositions (with and without repetition).
  - Stifel Formula and Vandermonde identity.
  - Newton binomial and Leibnitz Formula.
  - Stirling number of second species. Distrubution of n cobbles in k holes.
  - Diophantine equations.
  - Inclusion and exclusion principles. Euler Function.
  - Discrete probability.
* Graphs
  - Definition, terminology, examples and applications.
  - trees, graph representations.
  - Connected graphs. Eulerian and Hamiltonian paths.
  - Shortest path problem.
  - Planar graphs.
  - Colorations.