Introduction to mathematics 1000-111bWMAa

The concept of a set and the membership relation. Various ways of defining sets, the empty set. The set inclusion. The union and the intersection of two sets. The union and the intersection of a family of sets. The difference of two sets, the complement of a set. The de Morgan laws. Ordered pairs and the Cartesian product. The power set.
Basics of propositional logic: logical connectives, formulas, valuations, propositional tautologies. Basics of first-order logic: quantifiers, de Morgan laws. Connections between logical operations and set operations.
The notion of a function as a set of ordered pairs. The domain and the range of a function. The graph of a function. One-to-one functions, functions "onto", permutations. The composition of functions, the inverse function. Transformation groups. The image and the inverse image of a set under a function. Finite and infinite sequences. Indexed families of sets, their unions and intersections. Double indexed families of sets, double sequences, matrices.
Equipotency of sets. Countable and uncountable sets. Proofs of the existence of uncountable sets, examples of diagonalization arguments. Comparison of cardinalities of sets, the Cantor-Bernstein theorem. Properties of countable sets. Uncountability of the set of real numbers. Sets of cardinality continuum, examples and properties. The Cantor theorem. A remark on the Continuum Hypothesis.
The notion of a relation as a set of ordered pairs, examples of binary relations. The domain, the range and the field of a relation. The inverse of a relation. Functions as relations. General properties of relations.
Partial orders, linear orders, Hasse diagrams, elements with special properties. Order isomorphism, isomorphism invariants. The Kuratowski-Zorn Lemma, the existence of a basis in every vector space.
Equivalence relations. Equivalence classes, the quotient set. Partitions of a set, the correspondence between equivalence relations on a set and its partitions.
Natural numbers, the Peano axioms. Induction and recursion over the set of natural numbers. A remark on the Peano construction of natural numbers.
Sets of integers, rationals and reals -- constructions, definitions of arithmetical operations and orders.