2025
Algebra and combinatorics
Name: Algebra and combinatorics
Code: MAT14989L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Regime de Frequência: Presencial
Sustainable Development Goals
Learning Goals
1.Ability to use sets and their operations in a flexible, current way, in an abstract or concrete environment.
2. Ability to use the notions of order relationship and equivalence relationship in different contexts.
3. Understand counting principles in the context of calculating cardinals of sets.
4. Ability to define, use and understand the abstract properties of operations defined on sets.
5. Ability to formulate and solve problems within the scope and/or with algebra and combinatorics techniques
6. Ability to relate concepts from other areas of mathematics and other disciplines with concepts from algebra and combinatorics.
2. Ability to use the notions of order relationship and equivalence relationship in different contexts.
3. Understand counting principles in the context of calculating cardinals of sets.
4. Ability to define, use and understand the abstract properties of operations defined on sets.
5. Ability to formulate and solve problems within the scope and/or with algebra and combinatorics techniques
6. Ability to relate concepts from other areas of mathematics and other disciplines with concepts from algebra and combinatorics.
Contents
1 Sets. Operations and order relations. Hasse diagrams. Set of parts. Combinations.
A2 Equivalence relations. Classifications of words in Portuguese. Classification in biology.
A3 Principle of inclusion exclusion.
B1 Functions. Properties: injective, surjective, invertible. Function composition.
B2 Count functions in finite sets. Arrangements, permutations.
C1 Operations defined by tables. Associativity, neutral element, invertible elements. Inverses. Commutativity. Compare with modular arithmetic.
C2 Concrete algebraic structures: natural, integers, rational, complex. Function sets with composition.
C3 Concrete algebraic structures: geometric transformations, translations, reflections, rotations.
D1 Groups of permutations. Invariant sets and orbits.
D2 Symmetries of regular polygons and polyhedra.
E1 Graphs. Paths in graphs. Counts. Example of the streets of a city (Évora).
A2 Equivalence relations. Classifications of words in Portuguese. Classification in biology.
A3 Principle of inclusion exclusion.
B1 Functions. Properties: injective, surjective, invertible. Function composition.
B2 Count functions in finite sets. Arrangements, permutations.
C1 Operations defined by tables. Associativity, neutral element, invertible elements. Inverses. Commutativity. Compare with modular arithmetic.
C2 Concrete algebraic structures: natural, integers, rational, complex. Function sets with composition.
C3 Concrete algebraic structures: geometric transformations, translations, reflections, rotations.
D1 Groups of permutations. Invariant sets and orbits.
D2 Symmetries of regular polygons and polyhedra.
E1 Graphs. Paths in graphs. Counts. Example of the streets of a city (Évora).
Teaching Methods
Classes in theoretical-practical format, expository of content, articulated with the presentation of examples and exercises. The presentation moments alternate with the presentation of applications and discussion on solving associated problems. Applications can be proposed and presented by students and can be related to algebra, combinatorics, or other disciplinary areas. Moments to perform exercises in groups and individually. Calls to the board for students to present concepts and solve problems to their colleagues. It is suggested to use free software to solve certain problems. Python, R, Excel, or something else to be decided by the teacher. In this way, the recommendations and guidelines of the pedagogical model of the university of Évora are taken into account.
Assessment
Continuous assessment regime or a final exam assessment regime.
Continuous assessment will consist of several short moments - questions in class, with these assessment moments being seen simultaneously as assessment and practice/learning. They can be held every week or with a schedule to be agreed by the teacher at the beginning of the academic year. These moments must be more than 5 throughout the semester (1/3 of the total number of weeks).
In addition to the questions in class, a final written test will be carried out.
Each question in will have a fixed price (e.g. 2 points). The final grade for questions in is obtained by adding the partial grades for each question and normalized to 20 points. The weight of questions in will be 30% of the final grade. The final test will weigh 70% of the final grade.
Continuous assessment will consist of several short moments - questions in class, with these assessment moments being seen simultaneously as assessment and practice/learning. They can be held every week or with a schedule to be agreed by the teacher at the beginning of the academic year. These moments must be more than 5 throughout the semester (1/3 of the total number of weeks).
In addition to the questions in class, a final written test will be carried out.
Each question in will have a fixed price (e.g. 2 points). The final grade for questions in is obtained by adding the partial grades for each question and normalized to 20 points. The weight of questions in will be 30% of the final grade. The final test will weigh 70% of the final grade.
Teaching Staff
- Carlos Correia Ramos [responsible]
Certificações
