| Course title | Algebra 1 |
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| 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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```text Syllabus of the course Algebra 1 1. Relations and Mappings - Ordered pairs and Cartesian products of sets - Binary and n-ary relations - Properties of relations - Equivalence relations and partitions of sets - Order relations - Linear orderings - Mappings and their properties - Injective, surjective, and bijective mappings 2. Operations - The concept of an operation - Binary operations - Cayley tables - Closure of a set under an operation - Operations on sets, numbers, and mappings - Composition of mappings 3. Algebraic Structures - The concept of an algebraic structure - Underlying set of an algebraic structure - Operations in algebraic structures - Groupoids and examples of algebraic structures 4. Basic Types of Groupoids - Associative groupoids (semigroups) - Commutative groupoids - Identity element - Inverse element - Special types of groupoids 5. Groups - Definition of a group - Abelian groups - Order of an element and order of a group - Subgroups - Cyclic groups - Generators of groups 6. Divisibility Theory - Divisibility in the ring of integers - Greatest common divisor - Euclidean algorithm - Congruences - Fundamentals of modular arithmetic Základní studijní literatura: Blažek, J. a kol.: Algebra a teoretická aritmetika I. SPN, Praha, 1983. Blažek, J. a kol.: Algebra a teoretická aritmetika II. SPN, Praha, 1985. Miroslav Haviar, Pavel Klenovčan: Basic Algebra for Future Teachers. ```
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| Learning activities and teaching methods |
Lecture
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| Learning outcomes |
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Course objectives Upon successful completion of the course, the graduate will be able to: - Define and analyze relations and mappings: Understand the concepts of binary and n-ary relations, determine and verify the properties of relations (reflexivity, symmetry, and transitivity), and work with equivalence and order relations. Furthermore, define and analyze mappings, including injective, surjective, and bijective mappings. - Work with binary operations: Define a binary operation on a set, construct and interpret Cayley tables, and verify whether a set is closed under a given operation. - Understand basic algebraic structures: Define the concept of an algebraic structure, identify its underlying set and operations, and work with elementary examples of algebraic structures. - Analyze the properties of groupoids and related structures: Understand the concepts of groupoid, semigroup, and monoid, and determine their basic properties, in particular associativity, commutativity, the existence of an identity element, and the existence of inverse elements. - Work with groups: Define a group and an abelian group, determine the order of an element and the order of a group, and verify whether a given algebraic structure satisfies the group axioms. - Work with subgroups and cyclic groups: Understand the concept of a subgroup, determine whether a given subset is a subgroup, define a cyclic group, identify its generators, and describe its properties. - Apply fundamental concepts of divisibility theory: Work with divisibility in the ring of integers, determine the greatest common divisor using the Euclidean algorithm, and apply basic properties of divisibility in problem solving. - Work with congruences and modular arithmetic: Define congruence modulo n, perform computations in residue class systems, and apply the basic principles of modular arithmetic to mathematical problems.
Competence in the field of basic algebra and divisibility theory; upon successful completion of the course, the graduate will be able to independently solve problems involving relations, mappings, algebraic structures, and groups, work with congruences and modular arithmetic, and understand the connections between these concepts and other areas of mathematics. |
| Prerequisites |
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None.
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| Assessment methods and criteria |
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Mark, Oral exam, Student performance
During the oral examination, students are required to demonstrate an active knowledge of the definitions and theorems contained in the relevant chapters of the primary course literature: - Blažek, J. et al.: Algebra and Theoretical Arithmetic I. SPN, Prague, 1983. - Blažek, J. et al.: Algebra and Theoretical Arithmetic II. SPN, Prague, 1985. The examination also includes solving a randomly selected exercise chosen from those provided at the end of the individual chapters covered during the course in the primary literature. To successfully pass the examination, it is therefore highly advisable to have solved all of these exercises in advance, either through independent study or within the course seminars and exercise sessions. |
| 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 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 (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 focused on education (BB26) | 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 (BB26) | 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: 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 (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 (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 (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 (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 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 (BB23) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Winter |