Ateneo - UPO

Part of Algebra: sets, basic arithmetics, complex numbers, matrices, relations, cardinality, graphs. Sets with operations: semigroups, monoids, groups, rings, fields. Applications to cryptography. Linear algebra: vectors, bases, scalar product, linear applications. Systems of linear equations. Determinants.
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Part of Logic: Propositional logic and relative calculation of the natural deduction. Introduction to the Logic of Predicates.

Suggested book for the part of Algebra:
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.

Suggested book for the part of Geometry:
Bramanti, Pagani, Salsa, "Analisi Matematica 1 con elementi di algebra lineare", Ed. Zanichelli.

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Suggested book for the part of Logic:
"Logica a Informatica" di A. Asperti e A. Ciabattoni, Mc Grow Hill Education (1997)

Only available in PDF format

Acquire the basic notions of algebra, vector and matrix calculus and their applications.
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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 representability in the calculation of predicates.

Basic notions of elementary mathematics.
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Knowledge acquired in the course of algebra and geometry.

The course is organized with frontal lessons with theoretical sessions and exercises. Each subject of the course is introduced by a general
discussion. In a second time the basic notions of each subject are introduced; and are followed by examples with the purpose of clarifying their meaning. The third step is devoted to the statements of the main theorems and their proofs. The last part is devoted to the exercises.
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Part of Logic: Frontal lessons that include exercises.

In addition to the suggested books for the theoretical part and the exercises, further material for the preparation of the exam will be provided during the development of the course.
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The grade of the logic module is a weighted average with the grade of the algebra and geometry module to obtain the final grade of discrete mathematics.

The exam consists of a written online test (Moodle) made of 33 multiple choice questions, and possibly an additional oral examination.
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Part of Logic: The final exam consists of a written test lasting half an hour and an optional oral test.

Sets and basic arithmetics, GCD and MCM, Euclid's algorithm. Induction principle. Positional systems ina arbitrary basis. Complex numbers. Matrices, their addition and multiplication. Relations, equivalence classes, partitions. Modular arithmetics. Application to cryptography, RSA algorithm. Cardinality, binomial coefficients, infinite sets, orderings. Graphs, Eulerian and Hamiltonian circuits, trees and planar graphs. Sets with operations: semigroups, monoids, groups, rings, fields. Permutations. Symmetry groups. Lateral classes, normal subgroups, Lagrange's theorem. Linear algebra: vector spaces, linear independence, bases. Analytic geometry in 3D space: equation of the line and of the plane. Linear applications and their matrix representation.
Kernel and image. Linear operators. Scalar product. Vector product in 3D and its geometrical interpretation. Systems of linear equations, homogeneous and non-homogeneous, rank of a matrix, Rouché-Capelli's theorem. Determinants, Cramer's rule. Inverse matrix. Hermitian, unitary, normal operators.
Eigenvectors and eigenvalues.
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Propositional Logic:
• 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 statement.
Logic of Predicates:
• Syntax: free and bound variables, substitution.
• Semantics: Interpretation, definitions of satisfiable, contradictory, tautology, semantic consequence and related results.

The first purpose is to provide the students with a method of study based on rigorous proofs which allows them to exploit such an acquired skill
also in subjects different from Mathematics. Coming now to the more technical part, one of the main purposes of the course is to provide the students with the basic notions of algebra, in particular vector and matrix calculus, of fundamental importance in the applications. These are necessary notions to be applied in many other courses, as for example Physics.
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The skills to be tested, and necessary for passing the course, consist in the understanding of the basic concepts of propositional logic (both as regards semantics and as regards test methods) and predicate logic.