Mathematical language, sets and numbers

Description

Subject objectives: This is a course in the fundamentals of mathematics and provides preparation for the other subjects in the mathematics major. Students will develop good habits of understanding, communicating, and writing mathematics. Methods and techniques of reasoning will be worked on. The methods will be applied to solve various interesting problems. It could be said that this is a course about understanding and thinking, not about calculating and memorizing rules. The program explores topics involving numbers, sets, and functions. With elementary properties of these, it moves on to induction and cardinality. The study of natural numbers includes the properties of divisibility and modular arithmetic. Contents: 1. Introduction to mathematical logic (2 hours of lectures). 2. Sets. (4 expository hours). 2.1. Sets and elements. Subsets: Parts of a set. 2.2. Operations with sets: Properties. Boolean algebra of the parts of a set. 2.3. Covering and partition. Disjoint union and Cartesian product. 3. Applications (5 expository hours) 3.1. Concept of map. Graph of a map: Examples. 3.2. Injective, surjective and bijective applications. 3.3. Composition of applications; properties; inverse application. 3.4. Extensions of an application to the power set. 4. Relationships (6 expository hours) 4.1. Notion of relation. Composition of relations. Inverse relation. 4.2. Graphical representations. Binary relations in a set; properties. 4.3. Induced relation. 4.4. Equivalence relations: Equivalence classes: Properties. Example: rational and real numbers. 4.5. Canonical factorization of an application. 4.6. Order relations: Graphical representations: Hasse diagrams (trees). Total and partial order. Salient elements of an ordered set. Chains, lattices and well-ordered sets. 5. Infinite sets (6 expository hours). 5.1. Finite and infinite sets. 5.2. Principle of induction. Operations and order in N. 5.3. Cardinality. Cantor-Bernstein's theorem. Order relation. 5.4. Numerable and non-numerable sets. Numerability of Q and non-numerability of R. 5.5. Cardinality of unions, products, the set of parts, etc. 5.6. The axiom of choice and Zorn's lemma. Application 1: the order relation between cardinals is of total order. Application 2: cardinality of AxA when A is an infinite set. 6. Combinatorics (4 expository hours). 6.1. Number of applications and injective applications between finite sets. 6.2. Permutations. Permutations with repetition. 6.3. Combinatorial numbers. Combinations. Newton's binomial. Combinations with repetition. 6.4. Principle of inclusion-exclusion. Number of surjective applications between finite sets. 7. Algebraic structures (4 expository hours). 7.1. Groups, homomorphisms and isomorphisms of groups. 7.2. Cyclic groups. 7.3. Symmetric groups. Sign of a permutation. 8. The ring of integers (5 expository hours). 8.1. Rings and ideals; homomorphisms and isomorphisms of rings. 8.2. Integers and st