Discrete Mathematics and Logic

Vztahy:

It is not possible to register for the course BIK-DML.21 if the student is concurrently registered for or has already completed the course BIK-ZDM (mutually exclusive courses).

It is not possible to register for the course BIK-DML.21 if the student is concurrently registered for or has previously completed the course BIK-ZDM (mutually exclusive courses).

Garant předmětu:

Eva Pernecká

Lecturer:

Eva Pernecká

Tutor:

Eva Pernecká

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. Types of mathematical proofs.

4. Mathematical induction.

5. Sets, relations, functions. Basic number sets. Cardinalities of sets.

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

7. Equivalence and ordering.

8. Enumerative combinatorics and its basic principles.

9. Classical definition of probability.

10. k-combinations with repetition, permutations with repetition, Stirling numbers, properties of binomial coefficients.

11. Fundamentals of number theory, modular arithmetic.

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

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

Syllabus of tutorials:

1. Introduction to mathematical logics.

2. Formulas, truth tables. Tautology, contradiction, satisfiability; consequence and equivalence.

3. Universal systems of connectives. Disjunctive and conjunctive normal forms, minimalization, Karnaugh maps.

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

5. Formalization of language. Types of mathematical proofs.

6. Mathematical induction.

7. Sets and maps.

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

9. Equivalence and order.

10. Application of combinatorial principles.

11. Advanced combinatorial problems, probability,

12. Divisibility. Diophantine equations solution.

13. 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; Karolinum2007; 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:

• Bachelor specialization Information Security, part-time, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Software Engineering, part-time, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Computer Networks and Internet, part-time, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Computer Systems and Virtualization, part-time, in Czech, 2021 (compulsory course in the program)
• Bachelor program, unspecified specialization, part-time, in Czech, 2021 (compulsory course in the program)