Fundamentals of Mathematics (MAT115) | GSÜ

Content

Objective

The course introduces students to the basic language, methods, and foundational structures of modern mathematics. It includes on propositional and predicate logic, quantifiers, methods of proof, sets, relations, functions, and cardinality. A further objective is to develop familiarity with axiomatic set theory, particularly the Zermelo--Fraenkel axioms with the Axiom of Choice (ZFC), and to explain the role of Cantor's work in the modern understanding of infinite sets.

Course Content

This course provides an introduction to mathematical reasoning and the foundations of mathematics. It begins with propositional logic, predicates, and quantifiers, followed by standard methods of proof. The course then develops the elementary theory of sets, set operations, relations, and functions. The final part introduces Cantor's theory of infinite sets, the axiomatic framework of ZFC, and the comparison of finite and infinite cardinalities. Throughout the course, formal definitions and elementary proofs are used to develop the language and habits of rigorous mathematical argument.

Course Learning Outcomes

By the end of the course, students will formulate and analyze mathematical statements using propositional and predicate logic; correctly use quantifiers and logical connectives; construct direct, contrapositive, contradiction, and induction arguments; perform and justify standard operations on sets; work with equivalence relations and other basic types of relations; analyze functions in terms of injectivity, surjectivity, bijectivity, composition, and inverses; explain the basic principles underlying the ZFC axioms; and compare the cardinalities of finite and infinite sets using injections, surjections, bijections, and Cantor's diagonal argument.

Teaching and Learning Methods

The course combines lectures with regular problem-solving and proof-writing exercises. New concepts will be introduced through precise definitions and elementary examples, followed by progressively more demanding arguments designed to develop mathematical reasoning. Particular attention will be given to the correct use of logical notation, quantifiers, definitions, and proof techniques. Later topics on Cantor, ZFC, and cardinality will place the elementary material in its broader foundational context. Quizzes and examinations will be used to consolidate these skills.

References

-Mathematical Proofs: A Transition to Advanced Mathematics
Gary Chartrand, Albert D. Polimeni, Ping Zhang

-Mathématiques 1ère année, Cours et exercices,
Deschamps et Warusfel

- Matematiğe Giris¸ I-II, Ali Nesin, NMKY

- Math en Ligne de Bernard Ycart: https://ljk.imag.fr/membres/Bernard.Yc art/mel/

-Sezgisel Kümeler Kuramı, Ali Nesin, NMKY

Weekly Contents

Theory Topics
Week	Weekly Contents
1	Introduction to logic
2	Proof methods
3	Quantifiers
4	Set theory
5	Operations of sets
6	Relations
7	Functions
8	Mid-term examination
9	Induction
10	Cantor, Russell et ZFC
11	L'Axiom du Choix
12	Cardinalities
13	Cardinalities
14	Revision

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	5	60
Contribution of final exam to overall grade	1	40
Total	6	100

In-Term Studies
Activities	Number	Contribution
Assignments	0	0
Presentation	0	0
Midterm Examinations (including preparation)	1	30
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	4	30
Term Paper/ Project	0	0
Portfolio Study	0	0
Reports	0	0
Learning Diary	0	0
Thesis/ Project	0	0
Seminar	0	0
Other	0	0
Make-up	0	0
Total	5	60

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways;				X
2	can properly state and use concepts and results of major mathematical interest;					X
3	masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools;			X
4	is able to express one’s mathematical ideas in an organised way both in written and oral forms;				X
5	understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...);				X
6	has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems;
7	has a theoretical and practical knowledge in computer science well adapted for learning a programming language;
8	has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions;
9	has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement;		X
10	masters French language as well as other foreign languages, to a level sufficient to study or work abroad.			X

ECTS

Activities	Number	Period	Total Workload
Class Hours	14	5	70
Working Hours out of Class	14	2	28
Assignments	0	0	0
Presentation	0	0	0
Midterm Examinations (including preparation)	1	10	10
Project	0	0	0
Laboratory	0	0	0
Other Applications	0	0	0
Final Examinations (including preparation)	1	15	15
Quiz	5	15	75
Term Paper/ Project	0	0	0
Portfolio Study	0	0	0
Reports	0	0	0
Learning Diary	0	0	0
Thesis/ Project	0	0	0
Seminar	0	0	0
Other	0	0	0
Make-up	0	0	0
Yıl Sonu	0	0	0
Hazırlık Yıl Sonu	0	0	0
Hazırlık Bütünleme	0	0	0
Total Workload			198
Total Workload / 25			7.92
Credits ECTS			8