| 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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Upon successful completion of this course, the student will be able to: - Define and work with basic algebraic structures: The student will understand concepts such as groupoid, semigroup, and group, and will be able to verify whether a given set with an operation forms the respective structure. - Apply knowledge of residue classes: The student will be able to define residue classes of integers and perform addition and multiplication operations with them. They will be able to use this knowledge to solve problems in number theory. - Analyze the properties of groups: The student will be able to identify and describe key properties of groups, such as commutativity, associativity, and the existence of an identity and inverse element. - Work with subgroups and cyclic groups: The student will understand the concept of a subgroup and be able to determine if a given subset is a subgroup. Furthermore, they will be able to define and analyze cyclic groups. - Understand group homomorphisms: The student will be able to define and explain the concepts of homomorphism, isomorphism, endomorphism, and automorphism of groups and will understand their significance for the study of algebraic structures. - Apply knowledge of normal subgroups and quotient groups: The student will be able to define a normal subgroup, explain its significance, and construct quotient groups. - Classify finite groups: The student will gain an overview of the classification of finite groups up to order 15.
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 (BB25) | 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: 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: 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 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 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 (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 (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 (BB25) | 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 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: 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: teaching focus (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 focused on education (BB21) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |