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

01. Elements of Set Theory
01.01 The Boolean algebra of subsets of a set.
01.02 Relations: equivalence; pre-order; order; total order. Lattices.
01.03 Bounded sets: generalities.
01.04 Relations. Operations between relations.
01.05 Functions between sets: generalities.
01.06 Image and inverse image of a function.
01.07 Injective, surjective and biunivocal functions.
01.08 Quotient set.
01.09 Paradoxes of naive set theory.
01.10 NGB Theory: Axioms.
01.11 Russell's class. Some important sets.
01.12 The axiom of choice and some of its equivalent formulations.

02. Algebraic structures
02.01 Commutative rings.
02.02 Groups; semigroups; monoids; semirings; fields.
02.03 The quotient ring.
02.04 Compatibility of the order with respect to the operations.
02.05 Totally ordered fields.
02.06 Morphisms between structures.
02.07 Characterization of the upper (lower) bound in totally ordered fields.
02.08 Total orders on a field.

03. Numerical structures
03.01 The set of natural numbers: Peano's axioms.
03.02 The weak recursion theorem.
03.03 The principle of mathematical induction and the well-order principle.
03.04 Semantic versions of the mathematical induction principle.
03.05 The Bernoulli inequality. Newton's binomial. The MA-MG theorem.
03.06 Generalized induction and the recursion principle.
03.07 The generalized recursion theorem.
03.08 Language and the Bottom up contruction.
03.09 The ring of integers.
03.10 The field of rationals.
03.11 The fraction field of an integrity domain.
03.12 Characteristic of a ring.
03.13 Property of Archimedes.
03.14 Integer (fractional) part.

04. Cardinality
04.01 Cardinality in the sense of Frege.
04.02 Injective (surjective) functions between finite sets.
04.03 Equivalence relationship in the class of sets. Cardinality of a set.
04.04 Problem of hyperclasses. Operations between cardinals.
04.05 The restriction to subsets of a set.
04.06 The total order in the power set modulo equipotence.
04.07 The Cantor-Berenstein theorem.
04.08 Hartgos' theorem.
04.09 Cantor's theorem.
04.10 The continuum hypothesis.
04.11 Characterization of infinite sets according to Dedekind.
04.12 Finite sets (countable).
04.13 Cardinality of infinite sets.
04.14 Cardinality of classical sets.

05. Metric structure on totally ordered fields
05.01 Absolute value.
05.02 Intervals.
05.03 Metrics.
05.04 The order topology.
05.05 Internal points; external; boundary; accumulation; isolated.
05.06 Open (closed) sets.
05.07 Archimedean fields.
05.08 Successions in totally ordered fields.
05.09 Power series.
05.10 The ring of sequences on a totally ordered field.
05.11 Cauchy sequences.

06. The real field
06.01 Dedekind completeness.
06.02 Characterization of D-completeness.
06.03 Uniqueness theorem of ordered and D-complete fields (up to increasing isomorphisms).
06.04 The rational field is not D-complete.
06.05 The extended real line. Topology of the extended real line.
06.06 The complex field.
06.07 Characterization theorem of D-completeness.
06.08 Examples: Laurent's series.
06.09 The real field (Cantor).
06.10 The real field (Dedekind).
06.11 The real field (Stevin).

07. Elements of Linear Algebra 
07.01 Vector spaces over a field K.
07.02 Subspaces. Subspaces finitely generated.
07.03 Classical examples.
07.04 Linear applications and matrices.
07.05 Simple endomorphisms.
07.06 Quadratic forms: generality and theorems.
07.07 Linear systems. 

08. Propositional Logic
08.01 Formal languages, alphabet, syntax, semantics.
08.02 The language of Propositional Calculus.
08.03 Connectives, truth tables.
08.04 Interpretations, satisfiability.
08.05 Algebraic properties of connectives and quantifiers. Semantic equivalence.
08.06 Functional completeness.
08.07 Normal forms: conjunctive normal form and disjunctive normal form.
08.08 Construction of a formula in conjunctive or disjunctive normal form starting from the truth table.
08.09 Sets of functionally complete connectives.

09. Logic of predicates
09.01 The language of the Calculus of Predicates. Quantifiers.
09.02 Terms, atomic formulas and well-formed formulas.
09.03 Free variables and bound variables.
09.04 Closed formulas. The replacement.
09.05 The semantics of the Calculus of Predicates.
09.06 Interpretations. Satisfiability, validity and models.
09.07 Universal closure and existential closure of a formula.
09.08 Semantic equivalence.
09.09 Normal preness form.
09.10 Skolem normal form.

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.

Knowledge and understanding. At the end of the course the student must have acquired a good knowledge of the mathematical topics covered in the course. He must be able to argue correctly and with language properties on the topics covered in the program. Examples and working methods are shown in the classroom during the lessons and proposed in the exercises.

Applied knowledge and understanding. At the end of the course the student must have acquired a good ability to use the main tools of basic mathematics. He must be able to correctly apply the formulation studied and must be able to solve general mathematical problems similar to those studied. In particular, he must be able to apply the acquired knowledge even in contexts slightly different from those studied, and have the ability to use the acquired knowledge to independently solve problems that may appear new. Examples of such applications are shown in the classroom during the lessons and proposed in the exercises.

Autonomy of judgment. At the end of the course the student must have acquired a good ability to analyze topics and problems in general mathematics, the ability to critically evaluate any proposed solutions, and a correct interpretation of similar topics.

Communication skills. At the end of the course the student must have acquired a good ability to clearly communicate his / her statements and considerations concerning general mathematics problems. The working method is shown in the classroom during the lessons and proposed in the exercises.

Ability to learn. At the end of the course the student must have acquired a good capacity for autonomy in the study of the discipline, in the reading and interpretation of a qualitative phenomenon, in the search for useful information to deepen the knowledge of the topics covered.

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

Exercises carried out for the self-assessment of the level of preparation are available within the Moodle platform for Blended Learning.

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

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

The expected objectives are verified through the following two tests, both mandatory. Furthermore, according to the academic calendar, an intermediate written test, lasting 1 hour, will be carried out during the period of suspension of teaching activities for the semester in which the course is held. There will be three exercises proposed on the following topics:

Set Theory;

Relations and functions;

Induction principle and ricurrence.

Correct performance of the exercise referred to in point i) corresponds to exemption from carrying out the same in the final written test.  The exercise referred to in point i) is considered passed with a score of 5/30. 

This exemption remains valid only for the exam sessions of the first exam session following the semester in which the intermediate test takes place.

1. A formative assessment test: consisting of a written paper - lasting 2 hours - divided into six exercises on the following topics:

Set Theory;

Relations and functions;

Induction principle and ricurrence;

Algebraic structures; 

Vectorial spaces and linear systems;

Propositional and predicative logic.

2. An oral interview: consisting of the discussion of the written paper and three open questions on the theoretical topics covered in the course.

For both tests, the evaluation criteria are as follows:
- relevance and effectiveness of the responses in relation to the contents of the program;
- the level of articulation of the response;
- adequacy of the disciplinary language used.

The overall evaluation is expressed with a mark out of thirty taking into account both tests.

Precisely: the written exam consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30.  

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.

The final mark of LAG is the average of the marks of the written exam and the oral one.

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

As for attending students.

Attendance

As for attending students.

Course books

As for attending students.

Assessment

As for attending students.

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.

During the course, several exercises will be proposed to be carried out on the theoretical topics covered in class.

These exercises will be functional to the overcoming of the expected written paper.

Students are strongly advised to do them.