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Discrete Mathemathics

Code:

INF32155

Sigla:

MD

Áreas Científicas
Classificação	Área Científica
OFICIAL	Matemática

Ocorrência: 2021/2022 - 1S

Ativa?	Yes
Página Web:	https://moodle.ips.pt/2122/course/view.php?id=95
Unidade Responsável:	Departamento de Matemática
Curso/CE Responsável:	Informatics Engineering

Ciclos de Estudo/Cursos

Sigla	Nº de Estudantes	Plano de Estudos	Anos Curriculares	Créditos UCN	Créditos ECTS	Horas de Contacto	Horas Totais
INF	163	Plano de Estudos	2	-	6	60	162

Língua de trabalho

Portuguese

Objetivos

The objectives of this course include the solidification of some knowledge about whole numbers and combinatorial analysis acquired up to the 12th grade, as well as learning a new set of mathematical tools used in other subjects of the course. This knowledge is aimed, in particular, at the understanding of discrete mathematical models very common in the study of various technological systems. It will insist on the development of the students' logical-deductive reasoning, and, in this sense, the content should also be seen as a means to achieve this end.

Resultados de aprendizagem e competências

At the end of the course the student should be able to:
1. Count elements of a finite set;
2. Use the binomial theorem and the distribution principle in solving problems;
3. Use the principle of induction to prove properties of integers;
4. Identify and apply the Euclid algorithm for solving integer linear equations and decomposing a number into prime factors;
5. Operate with congruences and solve linear congruences;
6. Determine whole remainders by applying the Chinese remainder theorem, Euler's theorem and Fermat's theorem;
7. Identify and obtain matrix representations of oriented and non-oriented graphs;
8. Apply different graph techniques to solve the following problems: shortest path problem, least cost tree determination problem and graph coloring problem.

Modo de trabalho

Presencial

Pré-requisitos (conhecimentos prévios) e co-requisitos (conhecimentos simultâneos)

Knowledge acquired up to the 12th grade of schooling.

Programa

1. Combinatorics
Cardinal of a finite set, basic counting principles (addition and product). Arrangements, permutations and combinations. Binomial theorem. Inclusion-exclusion and distribution principles.
2. Rational Number Arithmetic
a.The. Integer Arithmetic: integer axiomatics, principle of induction, integer division, divisibility, greatest common divisor, Euclid's algorithm, prime numbers among themselves, least common multiple, Diophantine linear equation, prime numbers, fundamental theorem of arithmetic, computational considerations.
b. Modular Arithmetic: congruences and their properties, solving linear congruences; Chinese theorem of remains; Euler's theorem and (small) Fermat's theorem.
3. Graphs
Basic concepts, paths and cycles, connectivity; count the number of rides, the shortest path problem. Notion of tree, the problem of the minimum cost support tree. Coloring of graphs.

Bibliografia Obrigatória

Carlos Luz; Sebenta de Matemática Discreta
Artur Brito da Cruz; Slides Teóricos
Artur Brito da Cruz; Slides dos Exercícios

Bibliografia Complementar

N. L. Biggs; Discrete Mathematics, 2nd edition, Oxford University Press, 2008
D. M. Cardoso, J. Szymański e M. Rostami; Matemática Discreta, Escolar editora, 2009
R. L. Graham, D. E. Knuth e O. Parashnik; Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley, Reading, MA, 1989
L. Lovász, J. Pelikán e K. Vesztergombi; Discrete Mathematics, Springer, 2009
S. Lipschutz, M. Lipson; Matemática Discreta, Colecção Schaum, 1987
K. H. Rosen; , Discrete Mathematics and Its Applications, 6ª edição, McGraw-Hill, 2007

Métodos de ensino e atividades de aprendizagem

Discrete Mathematics UC has a load of 4h/week of classes are theoretical and practical
face-to-face teaching
Classroom classes will have two components:
• An expositive part, where the fundamental concepts of the different items of the program are presented together with the demonstration of the main results, intending in this way that the students acquire a global vision of the approached themes and their interconnections;
• A practical part, where students will apply the knowledge acquired improving their understanding of the subjects taught.
Distance learning
The UC will have all the necessary information and materials centralized in the MOODLE platform. The minimum goals will be presented weekly, accompanied by documents that allow the student to achieve these goals. These documents will consist of textbooks (they are the UC textbooks, which contain all the syllabus), PDF presentations where the week's contents are exposed, with a wide variety of examples and solved exercises, sheets with additional solved exercises and exercises to solve . Whenever necessary, online tools will also be included to complement the exposure of the matter, namely Youtube videos, use of websites, such as Wolframalpha, etc. So that students can better distribute their work throughout the week, presentations will be made class by class. Students will be informed that, for each class, they must read the respective presentation in advance.
Classes will take place on the TEAMS platform. The classes will have an expository component with a slide presentation, which will always have a more practical part, where the exercises previously made available will be solved, with additional explanations. If necessary, new examples and exercises will be presented.
The consolidation of knowledge by students will be based on a new reading of the materials made available at the beginning of the week. In addition, they will always have the possibility to connect via TEAMS with the teacher, both in scheduled classes and in times of doubts clearing classes.

Tipo de avaliação

Distributed evaluation with final exam

Componentes de Avaliação

Designation	Peso (%)
Teste	100,00
Total:	100,00

Componentes de Ocupação

Designation	Tempo (Horas)
Estudo autónomo	102,00
Frequência das aulas	60,00
Total:	162,00

Obtenção de frequência

The student can choose either the assessment by tests or the assessment by exam:

Test evaluation
This assessment consists of performing 2 (two) tests.
The final CF grade for the subject will be calculated using the arithmetic average of the two tests rounded to units and the conditions for approval are:
If CF is greater than or equal to 17, the student must take an oral exam. The final grade is given by the arithmetic average (rounded to the units) of CF and the classification obtained in this test. If the student deso not attend the oral exam, the final classification will be 16 points;
If CF is greater than or equal to 10 and less than 17, the student is approved with a final grade of CF, provided that the classification in both tests has been equal to or greater than 7.0.

Exam evaluation
The assessment based on the performance of exams follows the usual rules, that is, students who choose not to perform test evaluation, or who, having opted for it, have not been approved, may attend the regular exam periods.
If E is the classification obtained in the exam (rounded to the units), if E is greater than or equal to 17, the student will have to take an oral exam, obtaining as a final grade the average of the classifications of the referred oral exam and the written exam . If the student does not attend the oral exam, the final classification will be 16 points;
If E (rounded to the units) is greater than or equal to 10 and less than 17, the student is approved with a final grade of E.

Fórmula de cálculo da classificação final

Test evaluation
Se T1 e T2>=7 e T1+T2<=32, então CF=(T1+T2)/2
Se TE e T2>=7 e T1+T2>32, CF=(((T1+T2)/2)+OE)/2

Exam evaluation
Se E<17, CF=E
Se E>=17, CF=(E+OE)/2

Avaliação especial (TE, DA, ...)

Students covered by article 217 of the Regulation of Academic Activities and Guidelines for Assessment and School Performance of Students of the Polytechnic Institute of Setúbal (September 2018) must, until the second week of the beginning of the semester, contact the person responsible for curricular unit, in person or by e-mail artur.cruz@estsetubal.ips.pt, to present their relevant specificities, under the terms foreseen in the respective diplomas, otherwise they cannot be executed due to lack of objective conditions.

Observações

Summative evaluations will preferably be in person.
If the assessments are remote, for any time of assessment (mini-test, test or exam), a student may be asked to take an oral test to confirm the knowledge revealed in these summative assessments. The classification of the oral exam replaces the classification of the moment of assessment in question. If the student does not attend the oral exam, without due justification, the assessment in question will be canceled.
The tests last for 1 hour and 30 minutes, the exams will last for 2 hours and 30 minutes and they are both developmental.

Moodle:

https://moodle.ips.pt/2122/course/view.php?id=95

Password: MD2121