COMPLEMENTI DI MATEMATICA PER LA SICUREZZA

Diplomatic, International and Global Security Studies COMPLEMENTI DI MATEMATICA PER LA SICUREZZA

1212500046
DEPARTMENT OF MANAGEMENT & INNOVATION SYSTEMS
EQF6
DIPLOMATIC, INTERNATIONAL AND GLOBAL SECURITY STUDIES
2022/2023



YEAR OF COURSE 2
YEAR OF DIDACTIC SYSTEM 2019
AUTUMN SEMESTER
CFUHOURSACTIVITY
642LESSONS
Objectives
THE COURSE AIMS TO PROVIDE SOME BASIC ELEMENTS OF MATHEMATICAL LOGIC AND MATHEMATICAL MODELS TO ANALYZE SAFETY ISSUES.
STUDENTS SHOULD BECOME FAMILIAR WITH PROPOSITIONAL CALCULUS, FIRST ORDER LOGIC, PROOF AND VERIFICATION TECHNIQUES, MATHEMATICAL MODELS AND APPLICATIONS TO SAFETY.

KNOWLEDGE AND UNDERSTANDING
THE COURSE AIMS AT THE ACQUISITION OF THE FOLLOWING ELEMENTS: SET THEORY, PROPOSITIONAL CALCULUS, FIRST ORDER LOGIC, MATHEMATICAL MODELS.
THE SPECIFIC TRAINING OBJECTIVES OF TEACHING ESSENTIALLY CONSIST IN ACQUIRING:
- BASIC SKILLS REGARDING THE FORMALIZATION OF MATHEMATICAL LANGUAGE
- KNOWLEDGE OF THE SYNTAX OF PROPOSITIONAL AND FIRST ORDER LOGIC
- KNOWLEDGE OF THE SEMANTICS OF PROPOSITIONAL AND FIRST ORDER LOGIC
- SKILLS IN MODELING PROCEDURES FOR REAL PROBLEMS THAT LEAD TO VARIOUS TYPES OF MATHEMATICAL PROBLEMS.
APPLYING KNOWLEDGE AND UNDERSTANDING
STUDENTS WILL BE ABLE TO APPLY THE ACQUIRED SKILLS
-TO RECOGNIZE AND PROVIDE EXAMPLES OF SATISFIABLE TAUTOLOGIES, CONTRADICTIONS AND ENSEMBLES
-REPORT PROPOSITIONAL AND FIRST ORDER FORMULAS IN NORMAL FORM
-FORMALIZE MATHEMATICAL STATEMENTS IN THE LANGUAGE OF THE FIRST ORDER.
MAKING JUDGEMENTS
THE STUDENT WILL BE ABLE TO EVALUATE, PROVE OR REFUTE SIMPLE ASSERTIONS.

COMMUNICATION SKILLS
BEING ABLE TO EXPLAIN VERBALLY A TOPIC OF THE COURSE.

LEARNING SKILLS
BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS.
Prerequisites
PREREQUISITES
FOR THE SUCCESSFUL ACHIEVEMENT OF THE SET OBJECTIVES AND, IN PARTICULAR, FOR AN ADEQUATE UNDERSTANDING OF THE CONTENTS OF THE TEACHING, BASIC KNOWLEDGE OF MATHEMATICS, ALGEBRA AND SET THEORY ARE PARTICULARLY USEFUL AND THEREFORE REQUIRED.

MANDATORY PREPARATORY TEACHINGS
METHODS AND TECHNIQUES OF MATHEMATICS.
Contents
SET THEORY
RECALL ON SETS. CARDINALITY OF A SET. THE CONTINUOUS AND THE COUNTABLE. PEANO–DEDEKIND AXIOMS FOR NATURAL NUMBERS. EXAMPLES OF PARADOXES. SYSTEM OF AXIOMS THAT OVERCOME PARADOXES. CANTOR–SCHROEDER–BERNSTEIN THEOREM. THE AXIOM OF CHOICE. THE CONTINUUM HYPOTHESIS.
(LECTURES/EXERCISE HOURS 4/4)

PROPOSITIONAL LOGIC
THE LANGUAGE OF PROPOSITIONAL LOGIC. SEMANTICS AND SYNTAX. THE TABLES OF TRUTH. LOGICAL CONNECTIVES. FORMAL METHODS OF DEMONSTRATION. PROBLEM OF SATISFIABILITY. REDUCTION IN NORMAL CONJUNCTIVE OR DISJUNCTIVE FORM. THE SAT PROBLEM. RESOLUTION METHOD. INTRODUCTION TO THE PROBLEM P = NP.
(7/6)

FIRST-ORDER LOGIC
ALPHABET, FORMULAS, STRUCTURES, TRUTH. SEMANTICS AND SYNTAX: LOGICAL CONSEQUENCES AND PROOFS. NOTES ON THE THEOREMS OF CORRECTNESS AND COMPLETENESS. FORMAL METHODS OF DEMONSTRATION. EXPRESSIVE LIMITATIONS OF FIRST-ORDER LOGIC. PROBLEM OF SATISFIABILITY IN FIRST-ORDER LOGIC: REDUCTION OF A FORMULA INTO PRENEX NORMAL FORM, SKOLEM PROCEDURE, HERBRAND'S THEOREM, RECOURSE TO THE ALGORITHMS OF PROPOSITIONAL LOGIC. NOTES ON THE METHOD OF RESOLUTION AND UNIFICATION.
(7/6)

MATHEMATICAL MODELS
GENERAL INFORMATION ON MODELLING TECHNIQUES. NOTES ON THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS. SIMULATION MODELS. MICROSCOPIC AND MACROSCOPIC MODELS. SECURITY APPLICATIONS.
(4/4)

TOTAL HOURS: (20/22)
Teaching Methods
THE TEACHING CONSISTS OF FRONTAL LECTURES FOR A TOTAL OF 22 HOURS AND CLASSROOM EXERCISE SESSIONS FOR A TOTAL OF 20 HOURS.
ATTENDANCE TO THE COURSE IS MANDATORY. IN ORDER TO PARTICIPATE TO THE FINAL ASSESSMENT AND TO GAIN THE CREDITS CORRESPONDING TO THE COURSE, THE STUDENT MUST HAVE ATTENDED AT LEAST 70% OF THE HOURS OF ASSISTED TEACHING ACTIVITIES.
Verification of learning
WITH REGARD TO THE LEARNING OUTCOMES OF THE TEACHING, THE FINAL EXAM AIMS TO EVALUATE: THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE THEORETICAL LECTURES AND THE CLASSROOM EXERCISE SESSIONS; THE MASTERY OF THE MATHEMATICAL LANGUAGE IN WRITTEN AND ORAL TESTS; THE SKILL OF PROVING THEOREMS; THE SKILL OF SOLVING PROBLEMS; THE ABILITY TO IDENTIFY AND APPLY THE BEST AND EFFICIENT METHODS IN PROBLEMS SOLVING; THE ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO DIFFERENT CONTEXTS FROM THOSE PRESENTED DURING THE LESSONS.
THE EXAM NECESSARY TO ASSESS THE ACHIEVEMENT OF THE LEARNING OBJECTIVES CONSISTS IN A WRITTEN TEST, PRELIMINARY WITH RESPECT TO THE ORAL EXAMINATION, AND IN AN ORAL INTERVIEW.
THE WRITTEN TEST CONSISTS IN SOLVING PROBLEMS IMPLEMENTED ON THE BASIS OF WHAT HAS BEEN PROPOSED IN THE FRAMEWORK OF THE THEORETICAL LECTURES AND EXERCISE SESSIONS. SUCH A WRITTEN TEST, THAT THE STUDENT WILL HAVE TO FACE IN TOTAL AUTONOMY, WILL LAST 2 AND HALF HOURS
THERE WILL BE TWO MID-TERM TESTS CONCERNING THE TOPICS ALREADY PRESENTED IN THE COURSE, WHICH IN CASE OF A SUFFICIENT MARK, WILL EXEMPT THE STUDENT ON THESE TOPICS AT THE FINAL WRITTEN TEST.
IN THE CASE OF PRODUCTION OF A SUFFICIENT PROOF, IT WILL BE EVALUATED THROUGH QUALITATIVE SCALES (RANGES OF MARKS).
THE INTERVIEW IS DEVOTED TO EVALUATE THE DEGREE OF KNOWLEDGE OF ALL THE TOPICS OF THE TEACHING, AND WILL COVER DEFINITIONS, THEOREMS PROOFS, EXERCISES SOLVING.
THE FINAL MARK, EXPRESSED IN THIRTIETHS (EVENTUALLY CUM LAUDE), WILL DEPEND ON THE RANGE OF MARKS OF THE WRITTEN TEST, WITH CORRECTIONS IN EXCESS OR DEFECT ON THE BASIS OF THE ORAL INTERVIEW.

Texts
WRITTEN NOTES GIVEN BY THE TEACHER.
More Information
TEACHING PROVIDED IN ITALIAN.
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