# Fundamental Concepts of Mathematics (MAY112)

### General

School School of Science Department of Mathematics Undergraduate MAY112 1 Fundamental Concepts of Mathematics Lectures (Weekly Teaching Hours: 5, Credits: 7.5) General Background - Greek Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes As a first step, the students get familiar with basic tools of logic, set theory (set operations and properties), relations and functions. Emphasis is given to notions such as collections and families (coverings) bounds (max, min, sup, inf) as well as to images and pre-images of sets under functions. Part of the kernel of the course is a detailed axiomatic construction of the real numbers aiming that the students acknowledge this set as result of an axiomatic construction rather than of an empiric approach, yet the value and the significancy of the axiomatic foundation of mathematical structures be apparent. In the section concerning cardinality of sets, besides arithmetic of finite sets, students classify types of infinite sets (finite, numerable, denumerable) and approach in an abstract way the notion of infinity in relation with sets in common use as the sets of naturals, integers, rationals, and reals. A major course learning outcome is that assimilation of the offered knowledge will create a good qualitative background so that students be able to proceed with adequacy to studying other branches of mathematics. Analysis and synthesis of data and information Individual work Team work Production of creative and inductive thinking Production of analytical and synthetic thinking

### Syllabus

Definition of trigonometric numbers, trigonometric cycle. Trigonometric numbers of the sum of two angles and trigonometric numbers of the double of an arc. Trigonometrical functions. Trigonometrical equations. Transformations of products to sum and of sums to products.
Elements of Logic. Basic set theory, operations and properties, power set, Cartesian products, collections. Relations, properties, equivalence relations, order relations, bounded sets, well ordered sets, principle of infinite reduction, functions, one to one functions, onto functions.
Image and preimage of a set, functions and ordered sets. Families. The set of real numbers: axiomatic approach. The sets of natural numbers, integers. The field of rational numbers. Roots of nonnegative real numbers. The set if irrational numbers.
The axiom of completeness and equivalent statements. Equivalent sets. Finite sets. Infinite sets. Schroder-Bernstein theorem. Numerable sets. At most numerable sets. Denumerable sets. Cantor’ theorem. Axiom of Choice and equivalent statements. A first approach to the necessity of an axiomatic foundation of sets.

### Teaching and Learning Methods - Evaluation

Delivery Face-to-face
Use of Information and Communications Technology Use of ICT (Tex, Mathematica etc.) for presentation of essays and assignments.
Teaching Methods
Lectures 65
Study and analysis of bibliography 22.5
Preparation of assignments and interactive teaching 100
Course total 187.5
Student Performance Evaluation Written examination at the end of the semester including theory and problems-exercises.

### Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• K. G. Binmore, Logic, Sets and Numbers, Cambridge University Press, 1980.
• W. W. Fairchild and C. I. Tulcea, Sets, W. B. Shaunders Co. Philadelphia, 1970.
• S. Lipschutz, Set Theory and Related Topics, Schaum’s Outline Series, New York, 1965.
• D. Van Dalen, H. C. Doets and H. Deswart, Sets: Naïve, Axiomatic and Applied, Pergamon Press, Oxford, 1987.