Unit DISCRETE MATHEMATICS
 Course
 Informatics
 Studyunit Code
 GP004143
 Curriculum
 In all curricula
 Teacher
 Daniele Bartoli
 CFU
 12
 Course Regulation
 Coorte 2021
 Offered
 2021/22
 Type of studyunit
 Obbligatorio (Required)
 Type of learning activities
 Attività formativa integrata
DISCRETE MATHEMATICS  MOD. I
Code  GP004150 

CFU  6 
Teacher  Daniele Bartoli 
Teachers 

Hours 

Learning activities  Base 
Area  Formazione matematicofisica 
Academic discipline  MAT/02 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  Italian 
Contents  Sets and mappings. Equivalence relations, partitions. Induction. Combinatorial Analysis. Ordered sets. Algebraic Structures. Integers: divisors, Euclid's division. Bezout's identity. Congruence mod n. Chinese Remainder Theorem. An introduction to Graph theory: degree, walks, connectedness. Polynomials. Finite Fields. 
Reference texts  G.M. Piacentini Catteneo, "Matematica Discreta e applicazioni", Zanichelli 
Educational objectives  The module is the first course in Discrete Mathematics. The main goal of the course is to provide students with the basics of discrete mathematics that will be useful for studying computer science subjects in the subsequent years of the course. The main knowledge gained will be: Familiarity with the language of propositional logic Familiarity with different types of proofs Familiarity with modular arithmetic Familiarity with the language of graph theory The main skills (ie the ability to apply their knowledge) will be:  Performing simple proof by induction  Solving equations and linear systems in modular arithmetic  Solving counting problems in combinatorics  Factoring polynomials defined over finite fields  Recognizing the main properties of a graph 
Prerequisites  None 
Teaching methods  The course consists of classroom lectures on all topics of the course. In each lesson about half of the time will be devoted to solving problems and exercises 
Other information  http://www.unipg.it/disabilitaedsa 
Learning verification modality  The final mark is obtained summing the marks of 4 partial exams (30 minutes each) on  Induction and relations  Combinatorial calculation  Systems of equations in Zn  Graphs a final exam which consists of 6 exercises and a theoretical exercise.¿Each of the 4 partial exams can be repeated once per week and only the best score for each type is considered. Each of the partial exams gives between 0 and 10 points, the final exam from 0 to 60. Also, at most five extra points are given to the students for their active participation. Before the final exam the student must pass a test consisting of 10 questions about definitions and statements of theorems. The student must answer correctly at 8 questions at least in 20 minutes and the questions will be taken from an online database of roughly 70 questions available in advance for the students.¿The final mark is the scaled between 0 and 30. 
Extended program  Sets, inclusion, set operations. Power set. Complement. De Morgan's laws. Mappings. Injective and surjective mappings. Bijections. Inverse mapping. Product of mappings. Binary relations. Equivalence relations. Partitions. Order relations. Total and partial order. Natural numbers: order and operations. Divisibility. Prime numbers. Induction. Finite cardinals. Combinatorial Analysis, Newton's binomial theorem. Definitions of semigroup, monoid, ring, fields. Cancellative and invertible elements. Definition of group, ring, field. Ring of integers. Divisibility. Euclidean division. Greatest common divisor and least common multiple. Euclidean Algorithm. Bézout's identity. Congruence modulo n. Rings of congruence classes: 0divisors, invertible elements, modular inversion. Solution of linear congruence equations. Chinese Remainder Theorem. Graphs, subgraphs, isomorphisms of graphs. Degree. Adjacency matrix. Walks and their classification. Eulerian paths and circuits.Graph coloring, bipartite graphs, complete bipartite graphs. Polynomials. Polynomials over the rationals, the reals and over the complex numbers. Finite Fields. 
DISCRETE MATHEMATICS  MOD. II
Code  GP004151 

CFU  6 
Teacher  Daniele Bartoli 
Teachers 

Hours 

Learning activities  Base 
Area  Formazione matematicofisica 
Academic discipline  MAT/03 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  ITALIAN 
Contents  Matrices. Vector spaces and linear maps. Eigenvectors and eigenvalues. Permutations 
Reference texts  M.C. Vipera, Matematica discreta, MargiacchiGaleno editore Serge Lang, Introduction to Linear Algebra, Springer 
Educational objectives  The students should become familiar with the basic concepts and notions of algebra and linear algebra in order to use them either in theoretical informatic or in applications. 
Prerequisites  The course is held in the first semester of the first year. There are not any prerequisites with the exception of some interest in mathematical investigations 
Teaching methods  The course consists of classroom lectures on all topics of the course.Some time will be devoted to solving problems and exercises 
Other information  http://www.unipg.it/disabilitaedsa 
Learning verification modality  The final mark is obtained summing the marks of 4 partial exams (30 minutes each) on  Linear Systems,  Matrices,  Homomorphisms,  Diagonalization of Matrices and a final exam which consists of 5 exercises and a theoretical exercise.¿Each of the 4 partial exams can be repeated once per week and only the best score for each type is considered. Each of the partial exams gives between 0 and 10 points, the final exam from 0 to 60. Also, at most five extra points are given to the students for their active participation. Before the final exam the student must pass a test consisting of 10 questions about definitions and statements of theorems. The student must answer correctly at 8 questions at least in 20 minutes and the questions will be taken from an online database of roughly 70 questions availlable in advance for the students.¿The final mark is the scaled between 0 and 30. 
Extended program  Vector spaces. Matrices over a field. Linear systems: elimination theory, Cramer theorem.Linear maps. Linear maps and matrices. Eigenvectors and eigenvalues. Groups. Permutations. 