| Course title | Algebra Seminar |
|---|---|
| Course code | KMT/SAL@ |
| Organizational form of instruction | Seminar |
| Level of course | Bachelor |
| Year of study | not specified |
| Semester | Summer |
| Number of ECTS credits | 3 |
| Language of instruction | Czech |
| Status of course | Compulsory-optional |
| 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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- Ordered sets (posets): Quasi-orders, symmetric and antisymmetric relations, special elements in poses, chains, subposets, isomorphism of poses and construction of Hasse diagrams. - Bundles as poses and algebraic structures: Semibundles, operations of intersection (in the sense of infima) and union (in the sense of suprema), subbundles, ideals, filters, direct product, homomorphisms and bundle congruences. - Complete, modular, distributive and complementary bundles: Properties of complete bundles, completion of poses, modularity, distributivity, set representation and Stone's theorem. - Boolean algebras and Boolean circuits: Definition, basic properties, Boolean subalgebras, direct product of Boolean algebras, algebraic relation with Boolean circuits. - Boolean homomorphisms, filters and congruences: Ideals and filters in Boolean algebras, properties of homomorphisms, bundle congruences and set representation. - Boolean functions, polynomials and logic circuits: Analytical expression of Boolean functions, algebraic and graphical methods of minimization (Karnaugh maps, Quine-McCluskey algorithm), synthesis of logic systems (AND, OR, NOT gates) and their didactic transformation for secondary schools.
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| Learning activities and teaching methods |
Monologic Lecture(Interpretation, Training)
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| Learning outcomes |
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The course focuses on ordered structures, lattice theory, and Boolean algebra, including their applications in mathematical logic and logic circuit design. Students learn to formulate precise prompts for generative AI, which helps them design, optimize, and prepare functional program codes for minimizing Boolean functions and analyzing properties of lattice structures. These codes and visualizations are then run and verified in the CAS (Wolfram Mathematica) system.
After completing the course, the student will: - Apply theoretical knowledge of algebra and order theory (sets, networks, modularity, distributivity, Boolean algebras and homomorphisms) in solving specific algebraic problems and didactic justification of structural relations. - Perform independent algorithmization and software calculations (including Hasse diagram generation, Boolean function minimization and logic circuit simulation) using CAS (Wolfram Mathematica) and with the assistance of AI. - Design didactic materials and model problems for teaching elements of mathematical logic and graph theory in elementary and secondary schools with the aim of demonstrating the real applicability of Boolean algebra. - Critically analyze and compare the effectiveness of minimization algorithms (Karnaugh maps vs. Quine-McCluskey) and assess the complexity of logical systems in terms of the number of logical terms. - Reflects and evaluates the didactic projects of his colleagues, suggests adjustments and corrective measures to increase the comprehensibility of abstract explanations using interactive elements. |
| Prerequisites |
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Successfully completed the course Algebra 1 and Algebra 2.).
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| Assessment methods and criteria |
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Student performance
The student must complete the following in different parts: - written: colloquial test - practical implementation of algorithms, poset analysis, Boolean function minimization and logic circuit design using the Wolfram Mathematica system (40%), - creation of the assigned task: independent didactic or computational project - creation of interactive visualization material for poset theory (Hasse diagrams) or simulation scheme of logic circuits with AI support, run in Wolfram Mathematica (40%), - active participation in seminars: continuous analysis of algebraic and structural principles, active solution of assigned examples at the blackboard and presentation of the theoretical basis (20%), - analysis and evaluation of the completed task based on predetermined criteria (ungraded, but mandatory self-assessment requirement), - self-study (preparation for continuous tasks, colloquial test and project creation). |
| 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 (BB19) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB21) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB24) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB21) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB20) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB24) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB23) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB20) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB26) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB25) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB19) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB22) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB22) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB26) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB23) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB25) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |