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# Programme of Course "Matematica Discreta"

The course is composed by the following modules: "Matematica Discreta I"   "Matematica Discreta II"

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### Type of course unit:

Bachelor Degree in Computer Science curriculum General: Compulsory

### Semester:

Module Matematica Discreta I: 2° semester
Module Matematica Discreta II: 2° semester

### Number of credits:

Bachelor Degree in Computer Science: 12 (workload 300 hours)

## 1. Course Objectives

Module Matematica Discreta I: The goal of this course is to expose the main concrete techniques in linear algebra (matrices, systems, determinants, vector spaces and linear maps) and to show the first strategies in abstract algebra.
Module Matematica Discreta II: LOGIC: The goal of this Module is to provide  the motivations, definitions and techniques in support of the usefulness of logic in the effective and efficient modeling of data and knowledge.  This Module is an introduction to mathematical logic and covers elementary discrete mathematics for computer science.  On successful completion of this module, the student should understand the fundamental concepts of mathematical logic and should be aware of potential applications in computing, including the limitations of algorithms. GEOMETRY: The goals of this Module are to introduce students to the terminology and theorems of plane and solid geometry, and to apply algebraic, spatial, and logical reasoning to solve geometry problems. This Module covers the fundamental concepts of Linear Algebra and its role in describing geometric settings. On successful completion of this module, the student will develop spatial sense, visualize and represent geometric figures , explore transformations of geometric figures, understand and apply geometric properties and relationships, synthesize geometric concepts into algebraic, functional, and problem-solving activities.

## 2. Course Contents and learning outcomes (Dublin Descriptors)

Topics of the course include:

### Module Matematica Discreta I

• Sets: functions, equivalence relations, products, elementary combinatorics.
• Permutations.
• Groups: subgroups, quotients, isomporphism theorems, factor groups, permutation groups, cyclic groups.
• Arithmetic: divisibility theory in the ring of integers and of polinomials over a field.
• Congruences. Chinese remainder theorem.
• Rings: subrings, ideals, quotients, isomorphism theorem, ring of polynomials, domains, euclidean rings, PID, UFD.
• Fields: simple field extensions, finite fields.
• Matrices and systems of linear equations: Gauss reduction, determinants.
• Vectors, vector spaces, independence, bases.
• inner product, cross product.
• Eigenvalues, eigenvectors. Diagonalization and canonical forms of matrices.
• Application: systems of differential equations.

### Module Matematica Discreta II

• LOGIC Propositional Logic: Logical formulae, valuations, truth tables, logical equivalence of formulae, satisfaction and logical implication.  Deductive Logic: Formal axiom schemes, the structure of formal proofs, Sequent Calculus, Natural Deduction, the Deduction Theorem, and connections between truth and proof (the Soundness and Completeness Theorems).
• GEOMETRY Euclidean plane geometry, angles, radians, notion of geometric place, properties of triangles, parallelograms, circles, symmetry and similarity, transformations in the plane, Cartesian coordinates and equations of simple geometric places, elements of trigonometry, elements of spatial Euclidean geometry, volumes.

On successful completion of this course, the student should

### Module Matematica Discreta I

• being aware of the main structures in Linear Algebra and Abstract Algebra.

• demonstrate skill in mathematical reasoning, manipulation and calculation, demonstrate capacity for finding rigorous proofs of small problems; demonstrate skill in mathematical reasoning, manipulation and calculation by synthesizing geometric concepts into algebraic, functional, and problem-solving activities; demonstrate capacity to deduce properties of, and relationships between, figures from given assumptions and from using transformations.

### Module Matematica Discreta II

• On successful completion of this module, the student should
•  have profound knowledge of basic techniques in set theory;
•  have knowledge and understanding of logical and deductive arguments;
•  have profound knowledge of basic techniques in Linear Algebra;

•  have knowledge and understanding of logical and deductive arguments;
•  have knowledge and understanding of geometric relationships within the axiomatic structure of Euclidean geometry;
•  understand and apply geometric properties and relationships;
• demonstrate capacity for finding rigorous proofs of small problems;
• understand and explain the meaning of complex statements using mathematical notation and language;
• understand the fundamental concepts of mathematical logic and should be aware of potential applications in computing.
• demonstrate skill in mathematical reasoning, manipulation and calculation by synthesizing geometric concepts into algebraic, functional, and problem-solving activities;
• understand and explain the meaning of complex statements using mathematical notation and language;
• understand and explain the relation of geometry to algebra and trigonometry by using the Cartesian coordinate and recognize geometric relationships in the world;
• ability to read and understand other books/papers using notions learnt by the course and undertsnd their applications.

## 3. Course Prerequisites

Module Matematica Discreta I: Set Theory (language of set theory, the notion of function, graphs of fundamental functions, concept of sufficient and necessary condition), Numerical Structures (natural numbers, prime numbers, numerical fractions, rational numbers, basics of real numbers, inequalities, absolute value, powers and roots); Elementary algebra ( polynomials and operations on polynomials, identity, first- and second-degree equations).
Module Matematica Discreta II: For LOGIC the student must have the basic mathematical notions and methods as acquired in the secondary Schools. For GEOMETRY the student must know:  Set Theory (language of set theory, the notion of function, graphs of fundamental functions, concept of sufficient and necessary condition), Numerical Structures (natural numbers, prime numbers, numerical fractions, rational numbers, basics of real numbers, inequalities, absolute value, powers and roots); Elementary algebra ( polynomials and operations on polynomials, identity, first- and second-degree equations); Algebraic Structures (Groups, homeomorphisms, rings); Linear Algebra: Linear systems, matrices, matrix operations, vectors and vector spaces, elementary operations on vectors, linear independence, bases, rank of a matrix linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.

## 4. Teaching methods and language

Module Matematica Discreta I: Lectures and exercises
Module Matematica Discreta II: Lectures and Exercices

Language:Italian

### Reference textbooks

• W.K. Nicholson, Algebra lineare. McGraw-Hll.
• B. Scimemi, Algebretta. Decibel.
• B. Scimemi , Gruppi. Decibel.

## 5. Assessment Methods

Module Matematica Discreta I: Written exam and oral discussion of the written exam.
Module Matematica Discreta II: Oral and written exam

Course information last updated on: 28 agosto 2016, 11:49