Description:

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.

Contents:

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. Types of mathematical proofs. Mathematical induction.
6. Binary relations (properties, representations). Composition of relations.
7. Equivalence and ordering.
8. Combinatorics and its basic principles.
9. Classical definition of probability. k-combinations with repetition, permutations with repetition, Stirling numbers, properties of binomial coefficients.
10. Fundamentals of number theory, modular arithmetic.
11. Properties of prime numbers, Fundamental theorem of arithmetic.
12. Diophantine equations, linear congruences, Chinese remainder theorem.

Seminar contents:

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.
4. Syntax of predicate logic. Language, terms, formulas. Formalization of language.
5. Sets and maps
6. Types of mathematical proofs. Mathematical induction.
7. Binary relation (properties, representation), composition of relations.
8. Equivalence and order.
9. Application of combinatorial principles.
10. Advanced combinatorial problems, probability,
11. Divisibility. Diophantine equations solution.
12. Solution of linear congruences and their systems.

Recommended literature:

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; ČVUT 2017; ISBN 978-80-01-05342-3
5. Nešetřil J., Matoušek J.: Kapitoly z diskrétní matematiky; Karolinum 2007; ISBN 978-80-246-1411-3

Keywords:

logic, sets, maps, relations, combinatorics, diophantine equations, linear congruences

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar