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Information
| Code | MATZ116 |
| Name | Abstract Mathematics II |
| Term | 2024-2025 Academic Year |
| Semester | 2. Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 5 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Lisans Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. KAMURAN TARIM |
| Course Instructor |
1 |
Course Goal / Objective
Introduce order preserving functions, lattices, well ordered sets, axiom of choice, Zorn?s lemma, Hausdorff maximal principal, well ordering principle, finite and infinite sets, countable and uncountable sets. Construction of IR real numbers set
Course Content
In this course, the concepts of order preserving functions, lattices and well ordered sets will be elaborated. Basic mathematical principles such as the Axiom of Choice, Hausdorff's Size Principle, Zorn's Theorem and the Good Order Principle will be discussed. In addition, basic properties of set theory such as finite, infinite, countable and uncountable sets and the IR set of real numbers will be emphasized.
Course Precondition
none
Resources
Kenneth H. Rosen, Discrete Mathematics and Its Applications, 7th Edition, McGraw-Hill Companies, Inc. 2012. L.Gilbert, J. Gilbert, Elements of Modern Algebra, 7th Edition, Brooks/Cole, Cengage Learning, 2009. Fethi Çallıalp, Soyut Matematik, 1.Baskı, Birsen Yayınevi, 2012. Hüseyin Bilgiç, Soyut Matematik Ders Notları, Kahramanmaraş Sütçü İmam Üniversitesi, 2014. https://acikders.ankara.edu.tr/course/view.php?id=25 Timur Karaçay, Soyut Matematik, 1. Baskı, Seçkin Yayıncılık, 2014 Okay Çelebi, Öner Çakar, Soyut Matematik, 6. Baskı, Ankara Üniversitesi Fen Fakültesi Yayınları, Ankara, 2009. Sait AKKAŞ, H. Hilmi HACISALİHOĞLU, Zühtü ÖZEL, Arif SABUNCUOĞLU, Soyut Matematik, 3. Baskı, Gazi Üniversitesi, Fen-Ede. Fak. yayıları, ANKARA, 1998. Orhan Özer, Doğan Çoker, Kenan Taş, Soyut Matematik, İzgiYayınevi, Ankara, 1996
Notes
TİMUR KARAÇAY, SOYUT MATEMATİK - AKIL YÜRÜTMENİN BAŞLANGICI, SEÇKİN Yayınevi, 2016
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Defines order preserving function and order isomorphism |
| LO02 | Proves basic propositions about maps which preserve ordering |
| LO03 | Defines lattice, sublattice and complete lattice |
| LO04 | Defines well-ordered set, initial segment and |
| LO05 | Explains statements of axiom of choice, Hausdorff Maximal Principle, Zorn's Lemma and well ordering principle |
| LO06 | Decides whether the given set countable or not, |
| LO07 | Proves fundamental theorems about partially ordered sets and totally ordered sets. |
| LO08 | Gives examples on notions used in the construction of real number system. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Has enough knowledge about mathematics. | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Has pedagogical knowledge about teaching profession and field. | 2 |
| PLO03 | Bilgi - Kuramsal, Olgusal | Implements classroom management approaches to be used in educational environments effectively. | |
| PLO04 | Bilgi - Kuramsal, Olgusal | Prepares the learning environments in which appropriate teaching methods are used for effective mathematics education in accordance with development and age levels. | |
| PLO05 | Bilgi - Kuramsal, Olgusal | Knows the relationship between Mathematics-Society-Environment-History and uses it in professional and daily life. | 2 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Uses Turkish properly and effectively according to the rules. | |
| PLO07 | Bilgi - Kuramsal, Olgusal | Selects and designs appropriate materials, in mathematics teaching. | |
| PLO08 | Bilgi - Kuramsal, Olgusal | Monitors students' progress using different assessment and evaluation methods and techniques. | |
| PLO09 | Bilgi - Kuramsal, Olgusal | Takes responsibility as an individual and as a team member to solve problems related to the field. | 2 |
| PLO10 | Beceriler - Bilişsel, Uygulamalı | Has life-long learning awareness. | |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | Shares his/her knowledge and skills, problems and solutions that he/she identified by means of oral and written communication with the expert and non-expert people. | |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | Uses information and communication technologies and other related materials for an effective mathematics teaching. | |
| PLO13 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | Has enough foreign language knowledge to follow foreign resources related to the field. | |
| PLO14 | Yetkinlikler - Öğrenme Yetkinliği | Has the knowledge of the purpose, structure and functioning of the Turkish education system. | |
| PLO15 | Yetkinlikler - Öğrenme Yetkinliği | Becomes a teacher who adheres to Atatürk's principles and revolutions. |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Increasing functions , order isomorphisms, lattices | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap, Alıştırma ve Uygulama |
| 2 | Lattice and Borel algebras | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 3 | Initial segments and well-ordered sets | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 4 | Well-ordered sets and transfinite induction | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 5 | Choice function ,axiom of choice and generalized Cartesian product | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 6 | Zorn's lemma, Hausdorff maximal principle, well ordering principle and their equivalence | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 7 | Finite and infinite sets | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 8 | Mid-Term Exam | Source 1,2 | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Countable and uncountable sets | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 10 | Cantor theorem, Schröder-Benstein theorem. Countability of Q, uncountability of IR and similar applications | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 11 | Number systems | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 12 | Dedekind cuts 1 | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 13 | Dedekind cuts 2 | Source 1 | Öğretim Yöntemleri: Soru-Cevap, Anlatım |
| 14 | Construction of IR real number system 1 | Source 1 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 15 | Construction of IR real number system 2 | Sorce 2 | Öğretim Yöntemleri: Anlatım, Soru-Cevap |
| 16 | Term Exams | Prepare to exam | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | Prepare to exam | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 3 | 42 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 3 | 42 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 0 | 0 | 0 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 12 | 12 |
| Final Exam | 1 | 18 | 18 |
| Total Workload (Hour) | 114 | ||
| Total Workload / 25 (h) | 4,56 | ||
| ECTS | 5 ECTS | ||