| MATHEMATICS | |||||
| Bachelor | TR-NQF-HE: Level 6 | QF-EHEA: First Cycle | EQF-LLL: Level 6 | ||
| Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
| MAT1004 | Abstract Mathematics II | Spring | 3 | 2 | 4 | 5 |
| The course opens with the approval of the Department at the beginning of each semester |
| Language of instruction: | En |
| Type of course: | Must Course |
| Course Level: | Bachelor |
| Mode of Delivery: | Face to face |
| Course Coordinator : | Instructor MOHAMED KHALIFA |
| Course Lecturer(s): |
Instructor TUĞBA KIRAL ÖZKAN RA AYSUN SOYSAL |
| Course Objectives: | To enable the student to obtain skills and logical perspectives that prepare them for subsequent courses inside the department,to understand and use the language and notation of mathematics, to communicate using mathematical language, to comprehend and construct mathematical arguments,to introduce fundamental concepts of mathematics which are essential for mathematical thinking and.to provide most common methods of mathematical proofs , to develop the mathematical maturity of the student,for further studies in mathematics. |
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The students who have succeeded in this course; Learning Outcomes The students who have succeeded in this course,will be able to o describe proof methods. o conclude validity of propositions. o apply concepts of logic to proof methods. o formulate and develop mathematical statements. o distinguish mathematical implications. o adopt proof techniques to fundamental topics:set theory, graphs, correspondences, functions, relations, construct sets and relations of given property. o compare sets (with respect to cardinality etc. ),compare functions (with respect to injectivity, surjectivity, invertibility, the properties of image and preimage under them etc.),demonstrate basic abstract structures. o describe binary operations. o conclude validity of properties of natural numbers, cardinal numbers and basic properties of binary operations, permutation groups. o apply concepts of group theory to number systems. o formulate and develop cardinal arithmetic. o distinguish the properties of finite and infinite numbers. o use group tables, lattice diagrams, qoutient sets effectively. o adapt proof techniques to fundamental topics: construction of numbers, finiteness, countability, mathematical structures leading to number systems. o compare group structures with respect to cardinality, finiteness, the number of subgroups, commutativity and cyclicity, and also be able to demonstrate basic structures. |
| The language of mathematics, Theorems, Theory of logics, Statements and Proofs, Quantifiers, Sets, Graphs and Correspondances, Functions, Union and Intersection of a family of sets, Images of unions and intersections of a family of sets under functions, Product of a family of sets, Relations, Equivalence Relations, qoutient sets , Combinatorial analysis, order relations.Equipotency between the sets. Finite and infinite sets, Cardinal numbers, Cardinal arithmetic, Natural numbers. Properties of Natural Numbers,Mathematical induction,Construction of integers , Construction of rational numbers, Mathematical structures: Binary Operations, Groups, Permutation groups , Rings and Fields, irrational numbers,real numbers and Complex numbers, Countability. |
| Week | Subject | Related Preparation | |
| 1) | The language of mathematics. Theorems, Theory of logics. Statements and Proofs. Quantifiers.Sets. | ||
| 2) | Union and Intersection of a family of sets, coverings and partitions. | ||
| 3) | Product of sets. Graphs.Inverses of Graphs. Correspondances and functions. Images of the family of sets under graphs and functions. | ||
| 4) | Composite functions, graphs and correspondences. Sections, retractions, injections and surjections. | ||
| 5) | Product of a family of sets.Relations.Equivalence relations. | ||
| 6) | Ordering. Partially ordering. Total ordering. | ||
| 7) | Well ordering. Directed sets. Intervals. Axiom of choice. | ||
| 8) | Equipotency between the sets.Cardinal numbers, Cardinal arithmetic. Finite and infinite sets. | ||
| 9) | Natural numbers. Properties of Natural Numbers. Mathematical induction. | ||
| 10) | Combinatorial analysis. | ||
| 11) | Construction of integers, Construction of rational numbers. | ||
| 12) | Countable sets. Examples of countable, non-countable sets. | ||
| 13) | Mathematical structures, Binary Operations. Isomorphism. Groups. Permutation groups. | ||
| 14) | Rings and Fields.Real numbers, Complex numbers. Summary of the course topics, directions and notices for the final exams. | ||
| Course Notes: | Theory of sets, Bourbaki,N A First course in Algebra, Fraileigh,J. Undergraduate Analysis, Serge Lang |
| References: | Naive Set Theory ,Halmos Paul R. Sets, Functions and Logic, An introduction to abstract mathematics, Keith Devlin |
| Semester Requirements | Number of Activities | Level of Contribution |
| Attendance | 0 | % 0 |
| Laboratory | 0 | % 0 |
| Application | 0 | % 0 |
| Field Work | 0 | % 0 |
| Special Course Internship (Work Placement) | 0 | % 0 |
| Quizzes | 0 | % 0 |
| Homework Assignments | 1 | % 10 |
| Presentation | 0 | % 0 |
| Project | 0 | % 0 |
| Seminar | 0 | % 0 |
| Midterms | 2 | % 45 |
| Preliminary Jury | 0 | % 0 |
| Final | 1 | % 45 |
| Paper Submission | 0 | % 0 |
| Jury | % 0 | |
| Bütünleme | % 0 | |
| Total | % 100 | |
| PERCENTAGE OF SEMESTER WORK | % 55 | |
| PERCENTAGE OF FINAL WORK | % 45 | |
| Total | % 100 | |
| Activities | Number of Activities | Duration (Hours) | Workload |
| Course Hours | 14 | 3 | 42 |
| Laboratory | 0 | 0 | 0 |
| Application | 14 | 2 | 28 |
| Special Course Internship (Work Placement) | 0 | 0 | 0 |
| Field Work | 0 | 0 | 0 |
| Study Hours Out of Class | 14 | 1 | 14 |
| Presentations / Seminar | 0 | 0 | 0 |
| Project | 0 | 0 | 0 |
| Homework Assignments | 1 | 5 | 5 |
| Quizzes | 0 | 0 | 0 |
| Preliminary Jury | 0 | 0 | 0 |
| Midterms | 2 | 15 | 30 |
| Paper Submission | 0 | 0 | 0 |
| Jury | 0 | 0 | 0 |
| Final | 1 | 15 | 15 |
| Total Workload | 134 | ||
| No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
| Program Outcomes | Level of Contribution |