CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2024/2025

# Discrete Mathematics and Logic

Code Completion Credits Range Language
BI-DML.21 Z,ZK 5 2P+1R+1C Czech
Vztahy:
It is not possible to register for the course BI-DML.21 if the student is concurrently registered for or has already completed the course BI-MLO (mutually exclusive courses).
It is not possible to register for the course BI-DML.21 if the student is concurrently registered for or has already completed the course BI-ZDM (mutually exclusive courses).
It is not possible to register for the course BI-DML.21 if the student is concurrently registered for or has previously completed the course BI-MLO (mutually exclusive courses).
It is not possible to register for the course BI-DML.21 if the student is concurrently registered for or has previously completed the course BI-ZDM (mutually exclusive courses).
Garant předmětu:
Jan Spěvák
Lecturer:
Daniel Dombek, Jiřina Scholtzová, Jan Spěvák
Tutor:
Jakub Čermák, Daniel Dombek, Jan Legerský, Jitka Rybníčková, Jiřina Scholtzová, Jan Spěvák, Irena Šindelářová
Supervisor:
Department of Applied Mathematics
Synopsis:

Students will get acquainted with the basic concepts of propositional logic and predicate logic and learn to work with their laws. Necessary concepts from set theory will be explained. Special attention is paid to relations, their general properties, and their types, especially functional relations, equivalences, and partial orders. The course also lays down the basics of combinatorics and number theory, with emphasis on modular arithmetics.

Requirements:

None.

Syllabus of lectures:

1. Propositional logic. Formulas. Truth tables. Logical equivalence. Basic laws.

2. Disjunctive and conjunctive normal forms. Full forms. Logical consequence.

3. Predicate logic. Formalization of language.

4. Sets and functions. Basic number sets. Cardinalities of sets.

5. Mathematical induction. Types of mathematical proofs.

6. Binary relations (properties, representations). Composition of relations.

7. Equivalence and ordering.

8. Enumerative combinatorics and its basic principles.

9. k-combinations with repetition, permutations with repetition, Stirling numbers, properties of binomial coefficients. Classical definition of probability.

10. Fundamentals of number theory, modular arithmetic.

11. Properties of prime numbers, Fundamental theorem of arithmetic.

12. Diophantine equations, linear congruences, Chinese remainder theorem.

Syllabus of tutorials:

1. Introduction to mathematical logics. Formulas, truth tables. Tautology, contradiction.

2. Consequence and equivalence. Universal systems of connectives.

3. Disjunctive and conjunctive normal forms.

4. Syntax of predicate logic. Language, terms, formulas. Formalization of language.

5. Mathematical induction.

6. Sets and maps.

7. Binary relation (properties, representation), composition of relations.

8. Equivalence and order.

9. Application of combinatorial principles, probability,

11. Divisibility. Diophantine equations solution.

12. Solution of linear congruences and their systems.

Study Objective:
Study materials:

1. Mendelson E.: Introduction to Mathematical Logic (6th Edition); Chapman and Hall 2015; ISBN 978-1482237726

2. Chartrand G., Zhang P.: Discrete Mathematics; Waveland;2011; ISBN 978-1577667308

3. Graham R. L., Knuth D. E., Patashnik O.: Concrete Mathematics: A Foundation for Computer Science (2nd Edition); Addison-Wesley Professional; 1994; ISBN 978-0201558029

4. Trlifajová K., Vašata D.: Matematická logika; ČVUT2017; ISBN 978-80-01-05342-3

5. Nešetřil J., Matoušek J.: Kapitoly z diskrétní matematiky; Karolinum 2007; ISBN 978-80-246-1411-3

Note:
Further information:
https://courses.fit.cvut.cz/BI-DML
Time-table for winter semester 2024/2025:
Time-table is not available yet
Time-table for summer semester 2024/2025:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2024-06-16
Aktualizace výše uvedených informací naleznete na adrese https://bilakniha.cvut.cz/en/predmet6533606.html