Tiziana DURANTE | GEOMETRY, ALGEBRA AND LOGIC
Tiziana DURANTE GEOMETRY, ALGEBRA AND LOGIC
cod. 0612700114
GEOMETRY, ALGEBRA AND LOGIC
| 0612700114 | |
| DIPARTIMENTO DI INGEGNERIA DELL'INFORMAZIONE ED ELETTRICA E MATEMATICA APPLICATA | |
| EQF6 | |
| COMPUTER ENGINEERING | |
| 2019/2020 |
| OBBLIGATORIO | |
| YEAR OF COURSE 1 | |
| YEAR OF DIDACTIC SYSTEM 2017 | |
| SECONDO SEMESTRE |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/03 | 5 | 40 | LESSONS | |
| MAT/03 | 4 | 32 | EXERCISES |
| Objectives | |
|---|---|
| KNOWLEDGE AND UNDERSTANDING CAPABILITIES. KNOWLEDGE AND UNDERSTANDING OF THE TERMINOLOGY, OF THE FUNDAMENTAL CONCEPTS AND OF THE PROVING METHODOLOGIES TYPICAL OF THE FIELDS OF GEOMETRY AND CALCULUS, WITH PARTICULAR REFERENCE TO THE TOPICS LISTED IN THE FOLLOWING. MATRICES AND LINEAR SYSTEMS. VECTOR AND EUCLIDEAN SPACES. EIGENVALUES AND DIAGONALIZATION. 2D AND 3D ANALYTIC GEOMETRY. LOGIC OF PROPOSITION. SETS AND BOOLE’S ALGEBRA. LOGIC OF PREDICATES. APPLIED KNOWLEDGE AND UNDERSTANDING CAPABILITIES THE OBTAINED ABILITIES ARE THE FOLLOWING: •TO BE ABLE TO APPLY THE STUDIED DEFINITIONS, THEOREMS AND RULES FOR SOLVING PROBLEMS. •TO BE ABLE TO USE STRUCTURES AND TOOLS FROM LINEAR ALGEBRA FOR MANAGING MATHEMATICAL PROBLEMS. •TO BE ABLE TO MAKE TREATMENTS WITH 2D AND 3D OBJECTS FROM THE ALGEBRAIC AND GEOMETRIC POINT OF VIEW. •TO BE ABLE TO COORDINATE VARIOUS SEMIOTIC REPRESENTATIONS (ALGEBRAIC, FIGURAL, VERBAL) OF A MATHEMATICAL OBJECT. •TO BE ABLE TO MANAGE THE RULES OF LOGIC OF PROPOSITIONS AND PREDICATES. |
| Prerequisites | |
|---|---|
| FOR A PROFITABLE ACHIEVEMENT OF THE EDUCATIONAL GOALS THE STUDENT IS REQUIRED TO MASTER KNOWLEDGE CONCERNING BASIC MATHEMATIC. |
| Contents | |
|---|---|
| ALGEBRAIC STRUCTURES DEFINITIONS: OPERATIONS AND PROPERTIES. GROUPS, RINGS, FIELDS. (HOURS LECT/EXE 1/1) LOGIC OF PROPOSITIONS SINTAX: FUNDAMENTAL OPERATORS. SEMANTICS: TABLES OF TRUTH, VALIDITY AND CONSEQUENCE. CALCULUS OF NATURAL DEDUCTION: THEOREMS OF DEDUCTION, CORRECTNESS AND COMPLETENESS, FORMAL SYSTEMS. (HOURS LECT/EXE 3/4) SETS AND BOOLE ALGEBRAS: ALGEBRA OF SETS. BOOLE ALGEBRA. DISJUNCTIVE AND CONJUCTIVE NORMAL FORMS. (HOURS LECT/EXE 3/4) LOGIC OF PREDICATES PREDICATIVE LANGUAGES: ALPHABET, TERMS, FORMULAS, FREE AND BOUNDED VARIABLES, QUANTIFIERS AND PROOFS. (HOURS LECT/EXE 3/4) MATRICES: DEFINITIONS AND PROPERTIES. DETERMINANTS: LAPLACE THEOREM. RANK OF A MATRIX. KRONECKERS’ RANK THEOREM. THE INVERSE OF A MATRIX. (HOURS LECT/EXE 3/4) LINEAR SYSTEMS: DEFINITION, ASSOCIATE MATRICES, COMPATIBILITY, NUMBER OF SOLUTIONS. SOLVING LINEAR SYSTEMS: ROUCHÉ-CAPELLI AND CRAMER THEOREMS, GAUSS ELIMINATION METHOD. NULL SPACE OF A MATRIX. LINEAR SYSTEMS WITH PARAMETERS (HOURS LECT/EXE 3/4). VECTOR SPACES: THE STRUCTURE OF A VECTOR SPACE. VECTOR SUBSPACE. LINEAR DEPENDENCE AND INDEPENDENCE. BASIS AND COMPONENTS. STEINITZ'S LEMMA AND BASIS THEOREM. DIMENSION. INTERSECTION AND SUM OF VECTOR SUBSPACES. GRASSMAN THEOREM. (HOURS LECT/EXE 3/5). EUCLIDEAN SPACES: DEFINITION OF SCALAR PRODUCT. DEFINITION OF REAL EUCLIDEAN VECTOR SPACE. NORM. CAUCHY-SCHWARZ INEQUALITY. ANGLE. ORTHOGONAL VECTOR AND SUBSPACE. ORTHONORMAL BASIS AND COMPONENTS. ORTHOGONAL PROJECTION. GRAM-SCHMIDT PROCEDURE. (HOURS LECT/EXE 2/3) LINEAR APPLICATIONS: DEFINITION OF LINEAR APPLICATION (HOMOMORPHISMS), ENDO-, EPI-, MONO-MORPHISMS. KERNEL AND IMAGE. DIMENSION THEOREM. (HOURS LECT/EXE 2/3). DIAGONALIZATION: EIGENVALUES AND EIGENVECTORS: DEFINITIONS, CHARACTERISTIC POLYNOMIAL AND EQUATION. EIGENSPACES AND RELATED PROPERTIES. ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZATION AND ORTHOGONAL ONE. MAIN THEOREM OF DIAGONALIZATION. SPECTRAL THEOREM. (HOURS LECT/EXE 3/3) 2D GEOMETRY: CARTESIAN COORDINATES IN THE PLANE. EQUATION OF LINE (CARTESIAN, PARAMETRIC, SYMMETRIC). PARALLELISM AND ORTHOGONALITY BETWEEN LINES. LINEAR APPLICATION IN 2D (ROTATIONS, DILATATIONS, COMPRESSIONS, DEFORMATIONS). CONICS: CLASSIFICATION AND CANONICAL FORM (HOURS LECT/EXE 2/3) 3D GEOMETRY: CARTESIAN COORDINATES IN THE SPACE. VECTOR PRODUCT AND MIXED PRODUCT. EQUATION OF PLANE (CARTESIAN, PARAMETRIC). EQUATION OF LINE (CARTESIAN, PARAMETRIC, SYMMETRIC). BUNDLES AND STARS OF PLANES. PARALLELISM AND ORTHOGONALITY IN THE SPACE. NON COPLANAR LINES. (HOURS LECT/EXE 2/4) TOTAL HOURS : 30 HOURS OF LECTURES AND 42 HOURS OF EXERCISES. |
| Teaching Methods | |
|---|---|
| THE COURSE COVERS THEORETICAL LECTURES, DEVOTED TO THE FACE-TO-FACE DELIVERY OF ALL THE COURSE CONTENTS, AND CLASSROOM PRACTICE DEVOTED TO PROVIDE THE STUDENTS WITH THE MAIN TOOLS NEEDED TO PROBLEM-SOLVING ACTIVITIES RELATED TO THE COURSE CONTENTS. |
| Verification of learning | |
|---|---|
| THE EXAM AIMS TO ASSESS: KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS OF THE COURSE; THE MASTERY OF THE MATHEMATICAL LANGUAGE IN THE WRITTEN AND ORAL EXAM; THE CAPABILITY TO PROVE THEOREMS; THE ABILITY TO SOLVE EXERCISES; THE CAPABILITY TO IDENTIFY AND APPLY THE MOST SUITABLE AND EFFICIENT SOLVING METHODS FOR AN EXERCISE; THE CAPABILITY TO APPLY THE ACQUIRED KNOWLEDGE IN SOLVING EXERCISES SHOWN DURING THE LECTURES. THE EXAM TEST AIMS TO ASSESS: KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS SHOWN DURING THE COURSE; MASTERY OF THE MATHEMATICAL LANGUAGE IN WRITTEN AND ORAL TESTS; CAPABILITY OF PROVING THEOREMS; ABILITY OF SOLVING EXERCISES; CAPABILITY OF INDIVIDUATE AND APPLY THE MOST APPROPRIATE AND EFFICIENT METHODS IN SOLVING AN EXERCISE; CAPABILITY TO APPLY ACQUIRED KNOWLEDGE IN SOLVING EXERCISES NON SHOWN DURING THE COURSE. THE EXAM CONSISTS IN A WRITTEN TEST AND AN ORAL INTERVIEW. THE WRITTEN TEST LASTS AT LEAST 120 MINUTES AND IT AIMS TO ASSESS THE CAPABILITIES TO CORRECTLY APPLY THEORETICAL KNOWLEDGE AND TO UNDERSTAND THE PROPOSED PROBLEMS. THE WRITTEN TEST IS MANDATORY TO ACCESS THE ORAL EXAM AND IT CONSISTS IN SOLVING TYPICAL EXERCISES SHOWN DURING THE COURSE (EXAMPLES CAN BE FOUND ON THE WEBSITE OF THE DIDACTICAL COUNCIL) AND THE ASSESSMENT WILL TAKE INTO ACCOUNT THE SOLVING PROBLEM APPROACH TO THE PROPOSED PROBLEMS AND THE CLARITY AND THE COMPLETENESS OF THE PRESENTATION. IN CASE OF SUCCESS IN THE WRITTEN TEST, A GRADE WILL BE ASSIGNED IN TERMS OF QUALITATIVE SLOTS. THE TEST IS TAKEN BEFORE THE ORAL EXAM AND IT SUCCESSES IF THE SCORE IS GREATER OR EQUAL TO THE ESTABLISHED MINIMUM LEVEL. THE ORAL INTERVIEW MAINLY AIMS TO ASSESS THE LEVEL OF KNOWLEDGE OF ALL THE TOPICS OF THE COURSE, AND IT FOCUSES ON DEFINITIONS, STATEMENTS AND PROOFS OF THEOREMS, UNDERSTANDING OF EXERCISES SOLVING PROCEDURES. THE FINAL GRADE, EXPRESSED AS PART OF 30 WITH POSSIBLE LAUDE, IS DETERMINED STARTING FROM THE GRADE OF THE WRITTEN TEST REGULATING IT IN EXCESS OR IN ACCORDING TO THE ORAL INTERVIEW. |
| Texts | |
|---|---|
| G. ALBANO, LA PROVA SCRITTA DI GEOMETRIA: TRA TEORIA E PRATICA, MAGGIOLI (2013). G. ALBANO, C. D’APICE, S. SALERNO, ALGEBRA LINEARE, CUES (2002). G.LOLLI, LOGICA MATEMATICA, DISPENSE ON LINE HTTP://HOMEPAGE.SNS.IT/LOLLI/DISPENSE07.HTM ASPERTI, A.&A.CIABATTONI, LOGICA A INFORMATICA, MCGRAW-HILL (1997). DIDACTICAL MATERIALS ON E-LEARNING PLATFORM. |
| More Information | |
|---|---|
| THE FREQUENCY IS MANDOTORY. THE LANGUAGE IS THE ITALIAN. |
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