## LOGIC, ALGEBRA AND GEOMETRY LOGICA, ALGEBRA E GEOMETRIA

#### Logic, Algebra and Geometry Logica, Algebra e Geometria

A.Y. Credits
2020/2021 6
Lecturer Email Office hours for students
Giovanni Molica Bisci Monday from 16:00 to 17:00 p.m.
Teaching in foreign languages
Course with optional materials in a foreign language English
This course is entirely taught in Italian. Study materials can be provided in the foreign language and the final exam can be taken in the foreign language.

### Assigned to the Degree Course

Applied Informatics (L-31)
Curriculum: PERCORSO COMUNE
Date Time Classroom / Location
Date Time Classroom / Location

### Learning Objectives

Aim of the course is to give students some basic tools and topics in Logic, Algebra and Geometry.

### Program

01. Propositional logic:

01.01  Formal languages, alphabet, sintax, semantic.

01.02  The language of propositional calculus.

01.03  Logical operators, truth tables.

01.04  Interpretations, satisfiability.

01.05  Algebraic properties of logical operators and quantifiers. Semantic equivalence.

01.06  Functional completeness.

01.07  Normal forms: conjunctive normal form and disjunctive normal form. Construction of a formula in conjunctive normal form and disjunctive normal form starting from truth tables.

01.08  Sets of functionally complete logical operators.

02. Predicate logic:

02.01  The language of predicate calculus. Quantifiers.

02.02  Symbols, atomic formulas and well-formed formulas.

02.03  Free and bound variables.

02.04  Closed formulas. Substitution.

02.05  Semantics of predicate calculus.

02.06  Interpretations. Satisfiability, validity and models.

02.07  Universally closed and existentially closed formulas.

02.08  Semantic equivalence.

02.09  Prenex normal form.

02.10  Skolem normal form.

03. Sets:

03.01  Sets and their representation.

03.02  Subsets of a set.

03.03  Cardinality of a set.

03.04  Operations on sets: union, intersection, complement, difference, symmetric difference and Cartesian product. Properties of the operations on sets. De Morgan’s laws.

03.05  Relations. Equivalence relations. Equivalence classes and their properties. Quotient set. Partition of a set. Congruence modulo n: algorithm of the Euclidean division and integers modulo n.

03.06  Order relations: partially ordered set, totally ordered set and well-ordered set. Maximum and minimum of a partially ordered set.

04. Algebraic structures:

04.01  Binary operations and their properties. Identity element and inverse element with respect to a binary operation. Algebraic structures: preliminaries.

04.02  Semigroups: definition and preliminaries. Subsemigroups.

04.03  Monoids: definition and preliminaries. Identity element and its unicity. Submonoids.

04.04  Groups: definition and preliminaries. Inverse element and its properties*. Abelian groups. Subgroups. Cancellation rule. Necessary and sufficient condition for a subset to be a subgroup of a group.

04.05  Rings: definition and preliminaries. Properties of rings*. Subrings. Commutative rings and rings with identity. Zero divisors. Integral domains. Necessary and sufficient condition for a commutative ring to be an integral domain*.

04.06  Fields: definition and preliminaries. Completenees of the real field.

04.07  The polynomial ring. Division algorithm.

04.08  Ring of integers modulo p.

04.09  Boolean rings and Boolean algebras.

05. Vector spaces:

05.01  Vectors and operations on them: sum, difference, scalar multiplication, scalar product and vector product.

05.02  Vector spaces: definition and properties.

05.03  Vector subspaces. Necessary and sufficient condition for a subset to a subspace.

05.04  Linear combination of vectors. Subspaces spanned by a set of vectors.

05.05  Linear dependence and linear independence of vectors.

05.06  Basis and dimension of a vector space. Uniqueness of the representation of a vector as linear combination of the elements of a basis.

05.07  Operations on subspaces: sum and intersection. Grassmann Theorem.

05.08  Direct sum of vector subspaces. Necessary and sufficient condition for a vector space to be direct sum of its subspaces.

06. Matrices:

06.01  Preliminaries and operations with matrices: sum, scalar multiplication, product and their properties. Vector spaces of matrices (m,n).

06.02  Square matrices and diagonal matrices.

06.03  Inverse matrix and its unicity. Inverse matrix of the product.

06.04  The transpose of a matrix  and its properties. The transpose of the inverse matrix.

06.05  Symmetric and antisymmetric matrices.

06.06  Matrix associated with a set of vectors with respect to a basis.

06.07  The determinant of a square matrix  and its properties.

06.08  Sarrus rule and Laplace Theorem for the determinant.

06.09  Construction of the inverse matrix.

06.10  Binet Theorem and its consequences.

06.11  Orthogonal matrices and their properties.

06.12  Rank of a matrix. Kronecker Theorem.

06.13  Elementary transformations of a matrix and diagonal canonical form.

06.14  Eigenvalues and eigenfunctions of a matrix. Characteristic polynomial and characteristic equation.

07. Linear systems:

07.01  Preliminaries and methods of solution.

07.02  Matrix equation.

07.03  Homogeneous and non-homogeneous linear systems.

07.04  Cramer Theorem.

07.05  Rouché-Capelli Theorem.

07.06  Linear systems depending on parameters.

08. Linear maps:

08.01  What is a linear map. Necessary and sufficient condition for a map to be linear. Image of the zero vector by a linear map. Operations with linear maps: sum, scalar multiplication and composition.

08.02  Invertible linear maps.

08.03  Kernel and image of a linear map and their properties.

08.04  Rank-nullity theorem.

08.05  Matrix associated with a linear map.

08.06  First and second theorem of equivalence for linear maps.

08.07  Isomorphic vector spaces. Vector spaces with finite dimension n are isomorphic.

08.08  Eigenvalues and eigenfunctions of an endomorphism. Necessary and sufficient condition for a scalar to be an eigenvalue of an endomorphism. Eigenspace related to an eigenvalue and its dimension.

08.09  Diagonalizable matrices and endomorphisms. Diagonalizability criteria. Sufficient condition for an endomorphism to be diagonalizable.

09. Geometrical structures:

09.01 Euclidean spaces.

09.02 Unitary spaces.

09.03 Affine spaces.

09.04 Multilinear Algebra.

09.04 Projective spaces.

09.05 Algebraic curves: conics.

09.06 Multilinear Algebra.

### Bridging Courses

There are no mandatory prerequisites. It is recommended to take the exam of Logic, Algebra and Geometry during the first year of the Laurea Degree Program in Applied Computer Science.

### Learning Achievements (Dublin Descriptors)

Knowledge and understanding:

At the end of the course the student will learn the basic notions of Logic, Algebra and Geometry.

Applying knowledge and understanding:

At the end of the course the student will learn the methodologies of Logic, Algebra and Geometry and will be able to apply them to the study of various problems.

Making judgements:

At the end of the course the student will be able to apply the techniques of Logic, Algebra and Geometry in order to solve new problems, also coming from real-world applications.

Communications skills:

At the end of the course the student will have the ability to express the fundamental notions of Logic, Algebra and Geometry using a rigorous terminology.

Learning skills:

During the course the student will learn the ability to study the notions of Logic, Algebra and Geometry, also in order to use it in solving different kind of problems.

### Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

### Didactics, Attendance, Course Books and Assessment

Didactics

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

M. Curzio, P. Longobardi, and M. Maj, Lezioni di Algebra. Napoli: Liguori Editore, 2014.

E. Mendelson, Introduction to mathematical logic. Sixth edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015. xxiv+489 pp.

T. Jech, Set Theory. The Third Millennium Edition, revised and expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xiv+769 pp.

Assessment

The exam of Logic, Algebra and Geometry consists of a written exam and an oral one, both of them mandatory.

The written exam, to carry out in two hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30. During the written exam it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.

The oral exam consists of a discussion related to the topics of the course.  The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.

The final mark of Mathematical Analysis 2 is the average of the marks of the written exam and the oral one.

### Additional Information for Non-Attending Students

Didactics

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

M. Curzio, P. Longobardi, and M. Maj, Lezioni di Algebra. Napoli: Liguori Editore, 2014.

E. Mendelson, Introduction to mathematical logic. Sixth edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015. xxiv+489 pp.

T. Jech, Set Theory. The Third Millennium Edition, revised and expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xiv+769 pp.

Assessment

The exam of Logic, Algebra and Geometry consists of a written exam and an oral one, both of them mandatory.

The written exam, to carry out in two hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30. During the written exam it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.

The oral exam consists of a discussion related to the topics of the course.  The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.

The final mark of Logic, Algebra and Geometry is the average of the marks of the written exam and the oral one.

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