1. Naïve theory of set.
2. Cardinality of sets, countable and uncountable sets
3. Relational structures; relations of equivalence and partial order, factor set induced by equivalence
4. Types and properties of functions; surjection (mapping on), injection (one-to-one mapping into), bijection (one-to-one mapping on), inverse functions, composition of functions
5. Propositional Logic; syntax, semantics, formalization of sentences in the language of propositional logic
6. Propositional Logic; equivalent transformations, normal forms of formulae (conjunctive and disjunctive)
7. Propositional Logic; satisfiability, logical validity, contradiction, inference rules (modus ponens, modus tollens, ...)
8. Naïve theory of sets; operations on sets such as union, intersection, complement, power set, Cartesian product, and relations between sets such as being a subset; definition of functions and relations, demonstration by examples.
9. 1st-order predicate logic; syntax, semantics, formalization of sentences in the language of 1st-order predicate logic
10. 1st-order predicate logic; interpretation of non-logical symbols, models, Venn diagrams, demonstration of logical validity, satisfiability, and contradiction by models
11. 1st-order predicate logic; quantified formulae, equivalent transformations of formulae, de Morgan laws and normal forms
12. The notion of proof, proof techniques, structure of proofs, direct and indirect proof
13. Inductive proofs and recursion; proof by mathematical induction on natural numbers, structural induction. Recursive mathematical functions, recursive definitions
Topics dealt with in Seminars:
Topics dealt with in seminars follow the topics of particular lectures. At seminars the students will receive an individual home-work within the scope of two lessons.
1. Naïve theory of set.
2. Cardinality of sets, countable and uncountable sets.
3. Relations of equivalence and partial order, factor set induced by equivalence
4. Types and properties of functions. Algebraic structures.
5. Propositional Logic, syntax, semantics.
6. Formalization of sentences in the language of propositional logic.
7. Propositional Logic; equivalent transformations, using the deduction rulles.
8. 1st-order predicate logic; syntax, semantics.
9. Formalization of sentences in the language of 1st-order predicate logic.
10. Venn diagrams, equivalent transformations of formulae.
11. Proof techniques, structure of proofs, direct and indirect proof.
12. Inductive proofs and recursion. Recursive mathematical functions.
2. Cardinality of sets, countable and uncountable sets
3. Relational structures; relations of equivalence and partial order, factor set induced by equivalence
4. Types and properties of functions; surjection (mapping on), injection (one-to-one mapping into), bijection (one-to-one mapping on), inverse functions, composition of functions
5. Propositional Logic; syntax, semantics, formalization of sentences in the language of propositional logic
6. Propositional Logic; equivalent transformations, normal forms of formulae (conjunctive and disjunctive)
7. Propositional Logic; satisfiability, logical validity, contradiction, inference rules (modus ponens, modus tollens, ...)
8. Naïve theory of sets; operations on sets such as union, intersection, complement, power set, Cartesian product, and relations between sets such as being a subset; definition of functions and relations, demonstration by examples.
9. 1st-order predicate logic; syntax, semantics, formalization of sentences in the language of 1st-order predicate logic
10. 1st-order predicate logic; interpretation of non-logical symbols, models, Venn diagrams, demonstration of logical validity, satisfiability, and contradiction by models
11. 1st-order predicate logic; quantified formulae, equivalent transformations of formulae, de Morgan laws and normal forms
12. The notion of proof, proof techniques, structure of proofs, direct and indirect proof
13. Inductive proofs and recursion; proof by mathematical induction on natural numbers, structural induction. Recursive mathematical functions, recursive definitions
Topics dealt with in Seminars:
Topics dealt with in seminars follow the topics of particular lectures. At seminars the students will receive an individual home-work within the scope of two lessons.
1. Naïve theory of set.
2. Cardinality of sets, countable and uncountable sets.
3. Relations of equivalence and partial order, factor set induced by equivalence
4. Types and properties of functions. Algebraic structures.
5. Propositional Logic, syntax, semantics.
6. Formalization of sentences in the language of propositional logic.
7. Propositional Logic; equivalent transformations, using the deduction rulles.
8. 1st-order predicate logic; syntax, semantics.
9. Formalization of sentences in the language of 1st-order predicate logic.
10. Venn diagrams, equivalent transformations of formulae.
11. Proof techniques, structure of proofs, direct and indirect proof.
12. Inductive proofs and recursion. Recursive mathematical functions.