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:
+ Bramanti, Pagani, Salsa, "Analisi Matematica 1 con elementi di geometria e algebra lineare", Ed. Zanichelli.
+ Notes of the teacher's upload on the IT platform DIR.

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.

Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the __Staff
Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/servicesstudents-physical-or-learning-disabilities.

Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the
tutor in charge of the course to define the examination modalities, concerning academic aspects.

The exam consists of a written test and an oral test. The written test involves solving exercises based on the material covered during the lectures and lasts one hour and thirty minutes. The oral test is at the discretion of the teacher and consists of a discussion of the written test. Students may take the two exam modules—"Algebra and Geometry" and "Logic"—separately. The written test for the "Algebra and Geometry" module is divided into two parts, which can also be taken separately, each lasting 30 minutes: one with exercises on Modular Arithmetic and the other with exercises on Linear Algebra. Each part is considered passed if a sufficient grade is obtained and is valid until December 2026. If all parts are passed, the Discrete Mathematics exam is considered passed, and a single grade is recorded for the two modules, "Algebra and Geometry" and "Logic." This grade is calculated as the rounded average of the grades obtained in each part, following a collegial evaluation of the student. At the end of each written test, the teacher corrects it and communicates the results via the DIR platform and/or the ESSE3 portal. This is generally followed by the oral test. The exam results reflect the student’s understanding of theoretical concepts and their ability to apply them to solve new problems. Only the use of a calculator is allowed during the written tests.

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.