Unit DISCRETE MATHEMATICS
 Course
 Informatics
 Studyunit Code
 GP004143
 Curriculum
 In all curricula
 Teacher
 Federico Alberto Rossi
 CFU
 12
 Course Regulation
 Coorte 2023
 Offered
 2023/24
 Type of studyunit
 Obbligatorio (Required)
 Type of learning activities
 Attività formativa integrata
DISCRETE MATHEMATICS  MOD. II
Code  GP004151 

CFU  6 
Teacher  Federico Alberto Rossi 
Teachers 

Hours 

Learning activities  Base 
Area  Formazione matematicofisica 
Academic discipline  MAT/03 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  Italian 
Contents  1. Vector Spaces 2. Matrices 3. Linear Systems 4. Linear Applications 5. Diagonalizability of matrices 6. Complements (Finite Fields. An introduction to Group theory. Permutation groups. Criptography) 
Reference texts  Recommended Textbooks: 1. M. Abate "Algebra Lineare" McGrawHill 2. M. Abate, C. de Fabritiis "Esercizi di Geometria" McGrawHill Other Textbooks: 1. E. Schlesinger "Algebra lineare e geometria" Zanichelli 2. L. Mauri E. Schlesinger "Esercizi di algebra lineare e geometria" Zanichelli 3. S. Lang "Algebra Lineare" BollatiBoringhieri 4. G. Catino, S. Mongodi "Esercizi svolti di geometria e algebra lineare" Esculapio 5. F. Bottacin, “Linear Algebra and Geometry”, Esculapio 
Educational objectives  The main objective of teaching is to provide students with basic knowledge of mathematics, particularly linear algebra, so that they can use mathematical tools in both theoretical computer science and computer applications. Particular focus is given to comprehension of arguments and rigor in the presentation of ideas and reasoning. Knowledge and Understanding: Mathematical understanding of the proposed topics and knowledge of both the theory carried out and fundamental examples. Mode of testing knowledge: Written examination. Skills: Be able to read and understand, independently, basic Linear Algebra texts. Connect similar arguments, find examples and counterexamples. Be able to understand and solve problems and exercises that are unfamiliar but clearly related to what has been done in theory and in lecture. Mode of testing skills: Written exam. Autonomy of judgment: The exposition of content and arguments will be carried out in a way that enhances the student's ability to recognize rigorous demonstrations, identify fallacious reasoning, and adopt optimal strategies for solving problems and exercises. Communication skills: The presentation of topics will be carried out in a way that will enable the acquisition of a good ability to communicate problems, ideas and solutions, both in written and oral form. 
Prerequisites  Knowledge of high school math concepts (e.g. Set theory. Functions and Applications. Equivalence relations and partitions. Binary operations. Complex numbers. Polynomials, division, roots and reducibility. Basic algebraic calculations.). Contents of the first module of Matematica Discreta. 
Teaching methods  The course is organized in classroom lectures on all course topics. Part of each lecture will be devoted to solving problems and exercises. 
Other information  Attendance is strongly recommended. For information on support services for students with disabilities and/or learning disability ("DSA") visit the university page: https://www.unipg.it/disabilitaedsa 
Learning verification modality  The exams are structured into several tests, as follows. 1) Theory test (multiplechoice test): n multiplechoice questions. Evaluation is done by assigning the following scores: +3 for a right answer, 1 for a wrong answer, 0 for a question left unanswered. A score of at least 3n/2 (i.e., 50%) must be obtained to pass the test. 2) Written exam, in which you have to solve some exercises (such as those done in the tutorials) in 120 minutes, justifying all the steps thoroughly. A score of not less than 15/30 is required to pass the test. The theory test and the written exam are held on the same day, one after the other. Consultation of books and notes is not allowed during the conduct of exam. The final grade (on a scale of 30) will be the weighted sum of the theoretical and written test grades, with weights of 1 and 3, respectively. The examination is passed if the final mark is not less than 18. An optional oral test may take place at the request of the lecturer or the student. For information on support services for students with disabilities and/or learning disability ("DSA") visit the university page: https://www.unipg.it/disabilitaedsa 
Extended program  1. Vector spaces: linear dependence, bases. 2. Matrices: operations, rank, invertibility, determinant. Elementary transformations and reduction to scale. 3. Systems of linear equations: basic results and RouchéCapelli and Cramer theorems. 4. Linear applications: associated matrix, properties. 5. Diagonalizability of matrices: eigenvalues, eigenvectors, algebraic and geometric multiplicity. 6. Linear Algebra in Finite Fields. 7. Groups. Finite groups. Lagrange Theorem and Euler's Theorem. Permutation Groups. 
DISCRETE MATHEMATICS  MOD. I
Code  A003099 

CFU  6 
Teacher  Federico Alberto Rossi 
Teachers 

Hours 

Learning activities  Base 
Area  Formazione matematicofisica 
Academic discipline  MAT/03 
Type of studyunit  Obbligatorio (Required) 
Language of instruction  Italian 
Contents  Basic Mathematics and Algebraic Structures.  Sets and mappings.  Equivalence relations, partitions.  Induction and recursivity.  Combinatorial Analysis. Ordered sets.  Integers: divisors, Euclid's division. Bezout's identity.  Congruence mod n. Chinese Remainder Theorem.  Polynomials.  Graphs 
Reference texts  Recommended Textbooks: 1. G.M. Piacentini Catteneo, "Matematica Discreta e applicazioni", Zanichelli Other Textbooks: 1. C. Delizia, P. Longobardi, M. Maj, C. Nicotera "Matematica Discreta", McGraw Hill 2. A. Facchini "Algebra e Matematica Discreta" Decibel  Zanichelli 3. K.H. Rosen, "Discrete Mathematics and Its Applications", McGraw Hill 
Educational objectives  The main objective of the teaching is to provide students with the basic knowledge of mathematics, in particular algebra and combinatorics, to be used both in the field of theoretical computer science and in field of computer applications. Particular attention is given to the understanding of arguments and rigor in the presentation of concepts and reasonings. Knowledge and understanding: Mathematical understanding of the proposed topics and knowledge of the theory carried out and of the fundamental examples. Methods of verification of knowledge: Written exam. Skills: Being able to independently read and understand basic texts of Algebra and Combinatorics. Connect arguments, find examples and counterexamples. Being able to understand and solve unknown but clearly related problems and exercises to what has been done in theory and in class. Assessment of skills: Written exam. Autonomy of judgment: The presentation of the contents and arguments will be carried out in so as to improve the student's ability to recognize rigorous demonstrations, of identify fallacious reasoning and adopt optimal strategies to solve problems and exercises. Communication skills: The presentation of the topics will be carried out in a way that allows the acquisition of a good ability to communicate problems, ideas and solutions, both in shape written and oral. 
Prerequisites  Knowledge of high school math concepts (e.g. Set theory. Functions and Applications. Equivalence relations and partitions. Binary operations. Complex numbers. Polynomials, division, roots and reducibility. Basic algebraic calculations.). 
Teaching methods  The course is organized in classroom lectures on all the topics of the course. Part of each lesson will be devoted to solving problems and exercises. 
Other information  Attendance is strongly recommended. For information on support services for students with disabilities and/or learning disability ("DSA") visit the university web page https://www.unipg.it/disabilitaedsa . 
Learning verification modality  The exams are structured as follows. 1) Theoretical test (multiple choice test): n multiple choice questions. The evaluation is done by assigning the following scores: +3 for a correct answer, 1 for an answer wrong, 0 for a question left unanswered. To pass the test it is necessary obtain a score of at least 3n 2 (i.e. 50%). 2) Written test, in which students have to solve some problems (similar to those viewed in the problems sessions) in 120 minutes. To pass the test it is necessary to obtain a score of not less than 15/30. The theoretical test and the written test take place on the same day, one after the other. It is not allowed to consult books and notes during the test written. The final mark will be the weighted sum of the marks of the theoretical and written tests, with the weights of 1 and 3 respectively. For information on support services for students with disabilities and / or SLD, visit the page http://www.unipg.it/disabilitaedsa 
Extended program  Basic Mathematics: set theory. Functions. Algebraic structures: Groups, Fields, Rings. 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. Polynomials. Polynomials over the rationals, the reals and over the complex numbers. Polynomials over finite fields. Graphs: Sketches of graph theory, subgraphs, paths, adjacency matrix. Groups. Finite groups. Lagrange Theorem and Euler's Theorem. Permutation Groups. 