| Course title | Algebra 1 |
|---|---|
| Course code | KMT/AL1@ |
| Organizational form of instruction | Lecture + Exercise |
| Level of course | Bachelor |
| Year of study | not specified |
| Semester | Winter |
| Number of ECTS credits | 3 |
| Language of instruction | Czech |
| Status of course | Compulsory |
| Form of instruction | Face-to-face |
| Work placements | This is not an internship |
| Recommended optional programme components | None |
| Lecturer(s) |
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| Course content |
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1. Introduction Basic mathematical concepts: ordered pairs, Cartesian product Relations: binary and n-ary, properties of relations (reflexive, symmetric, transitive) Equivalence and order: partitioning of sets, linear ordering Mappings: definition, injective mappings, functions Divisibility theory: division, greatest common divisor, modular arithmetic 2. Operations Binary operations: definition, Cayley tables Examples of operations: addition, multiplication, function composition, logical operations, set operations Closure of a set with respect to an operation 3. Algebraic Structures Definition of algebraic structure: carrier set, operations Examples: groupoids, mathematical structures 4. Basic Types of Groupoids Associative and commutative groupoids The concept of identity and inverse elements Examples of special groupoids 5. Groups Definition of a group: associativity, existence of a neutral element, and inverse element Types of groups: Abelian (commutative) groups Order of elements and groups Subgroups: definition, examples Cyclic groups and group generators
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| Learning activities and teaching methods |
Lecture
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| Learning outcomes |
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The objective of the course is to introduce students to the fundamentals of algebraic structure theory, with a focus on understanding groups and their properties. Students should acquire both theoretical and practical knowledge of basic algebraic concepts and operations, which are crucial for further study in mathematics, particularly in the areas of abstract algebra and number theory. Upon completing the course, students should be able to: Understand basic algebraic concepts: Define and work with ordered pairs, Cartesian products, relations, and mappings. Recognize and apply the properties of various types of relations, such as reflexive, symmetric, and transitive relations. Apply theoretical knowledge to solve practical problems: Perform operations in various mathematical structures, such as numerical sets and sets of functions. Use Cayley tables to represent and analyze binary operations. Identify and work with algebraic structures: Define and analyze different types of algebraic structures, such as groupoids and groups. Recognize and describe the properties of special groupoids, such as associativity and commutativity. Analyze and work with groups: Define groups and distinguish between different types of groups, including Abelian groups. Identify subgroups and cyclic groups, and understand their significance in algebraic structures. Prepare for advanced studies: Acquire a foundation for more advanced study in abstract algebra, including the theory of rings and modules.
Competence in the basics of algebra; the student will be able to independently solve the problems and problems of some algebraic structures and, last but not least, he will become aware of the connections with the other mathematical substance. |
| Prerequisites |
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None.
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| Assessment methods and criteria |
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Mark, Oral exam, Student performance
Active knowledge of the course material. Specific requirements for the exam (including individual assignments for semester projects) are provided in the relevant MS Teams group. Regarding seminars, the requirements are as follows: Credit: Active participation in exercises, completion of the credit test. Students with an Individual Study Plan (ISP): A minimum of 30% attendance in exercises is required. Absences beyond the allowed limit will be addressed through homework assignments.Active knowledge of subject matter. |
| Recommended literature |
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| Study plans that include the course |
| Faculty | Study plan (Version) | Category of Branch/Specialization | Recommended semester | |
|---|---|---|---|---|
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB21) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB23) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB20) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB21) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB19) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB23) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB19) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB22) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB20) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB22) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB24) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB24) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |