Fundamental Concepts of Mathematics (MAY112)

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	MAY112
Semester	1
Course Title	Fundamental Concepts of Mathematics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 5, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	http://www.math.uoi.gr/GR/studies/undergraduate/courses/may112.html

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.
General Competences	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

Activity	Semester Workload
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

Παναγιώτης Χρ. Τσαμάτος, Θεμελιώδεις Έννοιες Μαθηματικής Ανάλυσης, Εκδόσεις Τζιόλα, 2009.
Α. Τσολομύτης, Σύνολα και Αριθμοί, Leader Books, 2004.
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.