Logic and sets

2022/2023
Programme:
Mathematics, First Cycle
Year:
1 year
Semester:
first
Kind:
mandatory
ECTS:
6
Language:
slovenian
Hours per week – 1. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Mathematical terminology, definitions, constructions, propositions, and proofs. Propositional and predicate calculus, rules of inference.
Basic set theory. Relations and functions. Equivalence relations and quotient sets. Ordered structures. Finite, countable and infinite sets. Cardinality of sets. Sets and classes. Axioms of set theory. Axiom of choice and Zorn's lemma.

Readings

N. Prijatelj: Osnove matematične logike I, DMFA-založništvo, Ljubljana, 1992.
N. Prijatelj: Osnove matematične logike II, DMFA-založništvo, Ljubljana, 1992.
N. Prijatelj: Matematične strukture I : Množice - relacije – funkcije, DMFA-založništvo, Ljubljana, 1996.
M. Juvan, P. Potočnik: Teorija grafov in kombinatorika, DMFA-založništvo, Ljubljana, 2000.

Objectives and competences

Students learn the basics about mathematical proofs and correct logic inference, basic discrete structures, basic terminology from combinatorics and basics about set theory.

Intended learning outcomes

Knowledge and understanding: Capability of forming exact mathematical expressions. Basic understanding of the concept of a mathematical proof. Basic knowledge about discrete structures and sets.
Application: Propositional calculus is an elementary language for expressing mathematical content. A proof is a basic mathematical tool. Hence, the knowledge obtained in this course is used in all subseqent mathematical courses.
Reflection: Mathematical logic is a mathematical reflection on mathematics as an axiomatic method. The course encourages reflection on the nature of mathematics itself.
Transferable skills: Correct proving, as well as the knowledge from set theory, represent a basis for all mathematical courses. Knowledge about discrete structures is the basis for further courses on discrete mathematics and computer science.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Andrej Bauer:
BAUER, Andrej, LEŠNIK, Davorin. Metric spaces in synthetic topology. V: 3rd Workshop on Formal Topology, Padua, 7-12 May 2007. Third workshop on formal topology : special issue, (Annals of pure and applied logic (Print), ISSN 0168-0072, Vol. 163, iss. 2 (February 2012)). Amsterdam: Elsevier, 2012, vol. 163, issue 2, str. 87-100. [COBISS-SI-ID 16073305]
AWODEY, Steve, BAUER, Andrej. Propositions as [Types]. Journal of logic and computation, ISSN 0955-792X, 2004, vol. 14, no. 4, str. 447-471. [COBISS-SI-ID 13374809]
BAUER, Andrej, SIMPSON, Alex. Two constructive embedding-extension theorems with applications to continuity principles and to Banach-Mazur computability. Mathematical logic quarterly, ISSN 0942-5616, 2004, vol. 50, no. 4/5, str. 351-369. [COBISS-SI-ID 13378649]