Discrete Mathemathics

Scholar Year: 2019/2020 - 1S

Courses

Head Teacher Responsability Patrícia Santos Ribeiro Head

Weekly workload Hours/week T TP P PL L TC THE EL OT OT/PL TPL S Type of classes 4 Lectures Type Teacher Classes Hours Theorethical and Practical classes Totals 5 20,00 Artur Cruz   4,00 Patrícia Ribeiro   8,00 Acompanhamento Totals 1 1,00

Teaching language

Portuguese

Intended learning outcomes (Knowledges, skills and competencies to be developed by the students)

The main goals of this course are to consolidate some knowledge about whole numbers acquired in school as well as to learn a set of basic tools used in many other courses. This knowledge is intended to understand discrete models that are very common in the approach to various technological systems and also in the operation of many industrial and service organizations. It will be emphasized in the development of the students' logical-deductive reasoning, and in this sense, the contents must be interpreted as a means to achieve this end.

Syllabus

1. Combinatorics
Cardinal of a finite set, basic principles of counting (addition and product). Arrangements, permutations and combinations. The binomial theorem. Principles of inclusion-exclusion and distribution.

2. Rational Arithmetic
The. Arithmetic of Integers: axiomatic of integers, principle of induction, whole division, divisibility, greatest common divisor, Euclid's algorithm, prime numbers among each other, least common multiple, Diophantine linear equation, prime numbers, fundamental theorem of arithmetic.
B. Modular arithmetic: congruences and their properties, resolution of linear congruences; Chinese remainder theorem; Euler's theorem and (small) Fermat's theorem.

3. Graphs
Basic concepts, paths and cycles, connectivity; counting the number of walks, the problem of the shortest way. Notion of tree, the problem of the tree of support of minimum cost. Coloring of graphs.

Demonstration of the syllabus coherence with the UC intended learning outcomes

Understand and apply the principles of addition and product counting
Apply the binomial theorem to the calculation of binomial coefficients
Apply the Principle of Inclusion-Exclusion
Apply the Principle of Distribution
Represent an integer in different bases
Using the Euclid algorithm for gcd determination
Solve problems through linear diophantine equations
Apply the sieve of Eratosthenes
To enunciate the fundamental theorem of arithmetic
Solve, where possible, linear congruences
Solve linear congruence systems - Chinese remainder theorem
Apply the theorems of Euler and Fermat
Represent a graph through the adjacency matrix, the adjacency lists or the incidence matrix
Count the number of walks of a graph through the powers of the adjacency matrix
Determine the shortest path between two vertices of a graph
Determine the minimum cost support tree of a graph
Determine coloring of a graph using a greedy algorithm

Teaching methodologies

The classes are theoretical-practical, constituted by an expositive part, where the concepts are presented
of the different headings of the program together with the demonstration of the main results. And a practical part, where the students will apply, under the guidance of the teacher, the acquired by improving their understanding of the subjects taught.

Assessment methodologies and evidences

The student can choose either the evaluation by tests or the evaluation simply by examination:

Evaluation by Tests:

The evaluation by Tests consists in the accomplishment of 2 (two) tests.
The final grade CF to the discipline will be calculated by the arithmetic mean of the two tests rounded to the units and the conditions of approval are:
- If CF is greater than or equal to 17, the student has to take an oral test. The final grade is given by the arithmetic mean (rounded to the units) of FC and the grade obtained in that test. If you do not attend the oral test, the final classification will be 16 values;
- If CF is greater than or equal to 10 and less than 17, the student is approved with final grade CF, provided that the classification in both tests was equal to or greater than 7.0 values.


Grade improvement:
According to Article 〖11〗〗 of the IPS Student Performance Assessment Guidelines, grade improvement may occur in the school year of enrollment, at the time of appeal, or in the school year following the year of approval, in any of the evaluation periods by examination, except for the special period.

Assessment by Exam:

The evaluation based on the examinations follows the usual rules, that is, students who choose not to carry out the continuous evaluation, or who, having opted for the same, have not obtained approval, may attend at regular times of examination.

 - If E is the grade obtained in the exam (rounded to the units), if E is greater than or equal to 17, the student will have to present himself to an oral test, obtaining as a final mark the average of the classifications of said oral test and the exam written. If the student does not attend the oral test, the final classification will be 16 values;
 - If E (rounded to units) is greater than or equal to 10 and less than 17, the student is approved with final grade E.

Note: The tests last 2 hours, the tests will last 2 hours and 30 minutes and are both developmental.

Special situations: Students covered by article 〖217〗 do of the Regulation of Academic Activities and Guidelines for the Evaluation and School Performance of Students of the Polytechnic Institute of Setúbal (September 2018) shall, by the second week of the beginning of the semester, contact the person responsible for the curricular unit, in person or by e-mail patricia.ribeiro@estsetubal.ips.pt, in order to present their pertinent specificities, under the terms established in the respective diplomas, failing which they can not be executed due to lack of objective conditions.

Assement and Attendance registers

Primary Bibliography

Secondary Bibliography

Observations

Clarifications:

Prof. Artur Cruz (office E316)
Monday from 9am to 12am


Prof. Patrícia Ribeiro (office E316)
Tuesday from 10am to 11am;
Friday from 10am to 12am