The aim of the course is to provide foundational concepts in Logic, Algebra, and Geometry.

01. Set Theory and Functions
01.01 Set operations and Boolean algebra
01.02 Relations: equivalence, partial and total orders
01.03 Functions: injective, surjective, bijective
01.04 Image, preimage, composition
01.05 Quotient sets and paradoxes (overview)
01.06 Axiomatic foundations: NBG axioms, axiom of choice

02. Algebraic and Numeric Structures
02.01 Groups, rings, fields (basic definitions and examples)
02.02 The ring ℤ, the field ℚ, and fractions
02.03 Orderings and Archimedean property
02.04 Induction, recursion, and constructive language

03. Cardinality and Infinite Sets
03.01 Finite, countable, and uncountable sets
03.02 Equipotence, Cantor-Bernstein, Cantor, Dedekind
03.03 Overview of the continuum hypothesis and classical cardinals

04. Metric Spaces
04.01 Absolute value and distance
04.02 Intervals, interior points, boundary, isolated points
04.03 Open and closed sets
04.04 Sequences, limits, accumulation points
04.05 Cauchy sequences, completeness, ℚ as an incomplete space
04.06 Power series (basic examples)

05. The Real Field
05.01 Constructions: Dedekind and/or Cantor (choose one)
05.02 Completeness of ℝ
05.03 Extended real line, basic topology
05.04 Brief introduction to the complex field

06. Linear Algebra
06.01 Vector spaces and subspaces
06.02 Linear maps, matrices
06.03 Linear systems, Gaussian elimination
06.04 Brief notes on quadratic forms

07. Propositional Logic
07.01 Syntax and semantics, truth tables
07.02 Equivalences, normal forms (CNF, DNF)
07.03 Functional and constructive completeness
07.04 Building formulas from truth tables

08. Predicate Logic
08.01 Syntax: terms, formulas, quantifiers
08.02 Interpretations, models, validity
08.03 Equivalences, closures, prenex form
08.04 Brief notes on Skolem form

There are no mandatory prerequisites. It is recommended to take the Logic, Algebra, and Geometry exam during the first year of study.

Knowledge and Understanding.
By the end of the course, the student is expected to have acquired a solid grasp of the mathematical topics covered. They should be able to reason clearly and correctly about the subjects in the syllabus, using appropriate terminology. Examples and working methods are demonstrated during lectures and further explored in problem sessions.

Applied Knowledge and Understanding.
By the end of the course, the student should be able to effectively use the main tools of elementary mathematics. They should be capable of correctly applying the studied formulations and of solving general mathematical problems similar to those addressed in class. In particular, they should be able to transfer acquired knowledge to slightly different contexts and demonstrate the ability to independently solve problems that may appear novel. Examples of such applications are shown in class and practiced in the exercises.

Independent Judgment.
By the end of the course, the student is expected to have developed a solid ability to analyze topics and problems in general mathematics, critically assess proposed solutions, and accurately interpret similar topics.

Communication Skills.
By the end of the course, the student should be able to clearly communicate their reasoning and ideas related to general mathematics problems. Working methods are presented during lectures and reinforced through exercises.

Learning Skills.
By the end of the course, the student should have developed good autonomy in studying the subject, interpreting qualitative phenomena, and seeking additional information to deepen their understanding of the topics discussed.

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

Self-assessment exercises for evaluating the level of preparation are available on the Moodle platform for blended learning.

Teaching

Theoretical lectures and exercises. The course is delivered in a hybrid format, with lessons held in person in the classroom and simultaneously broadcast online via the Moodle platform.

Attendance

Though strongly recommended, attendance is not required.

Course books

L. Carlucci Aiello - F. Pirri, Strutture, logica, linguaggi, Ediz. Mylab. Editore: Pearson. Collana: Informatica - Codice EAN: 9788891907844, 2018.

G. Devillanova - G. Molica Bisci, Elements of Set Theory and Recursive Arguments, Atti della Accademia Peloritana dei Pericolanti
Classe di Scienze Fisiche, Matematiche e Naturali, ISSN 1825-1242 - Vol. 99, No. S1, (2021).

G. Devillanova - G. Molica Bisci, The Faboulous Destiny of Richard Dedekind, Atti della Accademia Peloritana dei Pericolanti
Classe di Scienze Fisiche, Matematiche e Naturali, ISSN 1825-1242 - Vol. 98, No. S1,  (2021).

Per approfondimenti:

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

Assessment of Learning

Student learning is assessed primarily through a comprehensive final written exam, supplemented—when provided by the academic calendar—by an optional midterm written test.

Final Exam

The final exam is mandatory, lasts two hours, and consists of six exercises designed to evaluate the student’s understanding of the following core topics covered during the course:

• Set theory

• Relations and functions

• Induction and recursion

• Algebraic structures

• Vector spaces and linear systems

• Propositional and predicate logic

Midterm Exam (if applicable)

If scheduled in the academic calendar, a one-hour written midterm may be held during the semester break. The midterm consists of three exercises on the following topics:

• Set theory

• Relations and functions

• Induction and recursion

Successful completion of the exercise on set theory with a score of at least 5/30 exempts the student from the corresponding question in the final written exam. This exemption is valid only for the exams in the first examination session following the midterm.

Evaluation Criteria

The written exam will be graded on a 30-point scale, based on the following criteria:

• Relevance and correctness of answers with respect to the course content

• Level of articulation and completeness in the solutions

• Appropriateness and precision of mathematical language used

Passing Grade

The exam is considered passed with a minimum score of 18/30.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Teaching

Same as for attending.

Attendance

Same as for attending.

Course books

Same as for attending.

Assessment

Same as for attending.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.