Ateneo - UPO

Part of Algebra: integers, Euclid's algorithm, Diophantine equations, remainder classes, elements of cryptography.
Part of Geometry: vectors in the plane, in three-dimensional space and extension to the case of R^N, matrices, linear systems, analytic geometry in three-dimensional space.
Part of Logic: propositional logic and relative calculation of natural deduction. Introduction to the logic of predicates.

Recommended texts for the Algebra and Geometry part:
+ Facchini, "Sussidiario di algebra e matematica discreta", ed. Zanichelli.
+ Facchini, "Algebra per informatica. Introduzione ai metodi della matematica discreta e all'algebra astratta", ed. Zanichelli.
+ Bramanti, Pagani, Salsa, "Analisi Matematica 1 con elementi di geometria e algebra lineare", Ed. Zanichelli.

Recommended text for the Logic part:
+ Asperti, Ciabattoni, "Logica a Informatica". Mc Grow Hill Education, 1997 (disponibile solo in formato PDF).

The course aims to provide students with the knowledge of the main elements of algebra, vector and matrix calculus and their applications and to introduce the elementary notions of logic of propositions and of the first order with particular attention to the relationship between truth and derivability. and to the representability in the calculation of predicates. The purpose of the teaching is that students acquire and know how to use an appropriate mathematical language in relation to the topics covered in the course. The educational objective of the teaching is to develop the ability to model simple situations with the help of basic ideas and methods of algebra (elementary number theory) and linear algebra.

Reading and production of short texts in italian. Basic properties of integers. Basic properties of the cartesian plane.

Teaching will take place through lectures on the blackboard. In addition to the theoretical lessons, classroom exercises will be carried out by the teacher with the active involvement of the students to deepen the topics covered during the theoretical lessons. The concepts covered by the course will come stimulate collegially in the classroom and applied directly during classroom exercises for students' critical sense and autonomy of judgment.

The study plan includes a single exam for the courses of "Algebra and Geometry" and "Logic". However, the student is given the opportunity to take the tests separately. In any case, a single grade is recorded for the two courses of "Algebra and Geometry" and "Logic", which will be assigned starting from the average rounded upwards of the two grades obtained following a collegial evaluation of the student.

The exam consists of a written test and an optional oral test. The written test consists in carrying out exercises related to the program carried out during the lessons and has a duration of one hour and thirty minutes. The oral test is a lecturer of the teacher and consists of a discussion on the written test. The student is assured the possibility of taking the two exam modules separately (i.e. "Algebra and Geometry" and "Logic"): in this case the written test will last one hour and half an hour, respectively. However, only one vote is recorded for the two courses in "Algebra and Geometry" and "Logic", which will be assigned starting from the average rounded up by the excess of the marks obtained following a collegial evaluation of the student. During the written test it is allowed to use the calculator. At the end of the written test, the teacher corrects it and communicates the results on the DIR platform and/or on the portal ESSE3. This is usually followed by the oral exam. The results of the exam highlight the degree of understanding of theoretical concepts and the ability to use them to solve new problems.

Integers numbers. Divisibility and prime numbers. Greatest common divisor. Euclid's algorithm. Least common multiple. Diophantine equations. Congruences and remainder classes. Severability criteria. Representation of numbers with a base other than ten. Chinese remainder theorem. Sharing of secrets by means of systems of congruences. Euler's theorem. The discrete logarithm problem. The Diffie-Hellman protocol. The RSA protocol.
Linear systems. Method of elimination of Gauss. Matrices, determinants, rank, inverse matrix. Vector spaces, bases, linear dependence. Linear applications, scalar product. Applications to linear systems.
Logical proposition:
• Syntax.
• Semantics: interpretation, definitions of satisfiable, contradictory, tautology, semantic consequence and related results, semantic equivalence, functional completeness and normal form.
• Natural deduction: rules and proofs, correctness, completeness.
Logic of predicates:
• Syntax: free and bound variables, substitution.
• Semantics: Interpretation, definitions of satisfiable, contradictory, tautology, semantic consequence and related results.

At the end of the course the student must know the basic notions of algebra, linear algebra and propositional logic (both as regards semantics and as regards test methods) and predicate logic and have developed a mastery of the notions learned that allows him to connect them independently and to use them jointly in solving simple theoretical problems. Furthermore, the student must be able to clearly explain both the concepts themselves and their use in general terms or in the specific case of a problem.