Topics in Elementary Mathematics

Code:

M1024

Acronym:

M1024

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:M	92	Official Study Plan	1	-	9	84	243

Teaching language

Suitable for English-speaking students

Objectives

To introduce the basic concepts of mathematical logic, elementary set theory, combinatorics, and elementary number theory, that form the basis of much of what is taught in other courses of this degree. At the same time, one aims to familiarize the students with the logical-deductive reasoning of mathematics, and deepen their knowledge of the numbers, from the integers to the complex.

Learning outcomes and competences

It is intended that the students become familiar with the deductive reasoning and symbolic mathematical language, that they deepen their knowledge of some of the basic topics of mathematics, and explore basic techniques of mathematical proof.

Working method

Presencial

Program

• Some observations on mathematical language, mathematical symbolism and basic algebraic manipulations. Rudiments of logic. Examples of proofs.


• Elementary set theory. Some cardinality notions on finite and infinite sets. Binary relations. Equivalence relations. Functions and permutations.


• Some counting techniques. The binomial theorem and combinations.


• Integers and mathematical induction. Divisibility and prime numbers. The algorithm of division and the Euclidean algorithm. The fundamental theorem of arithmetic. Congruences.


• Rational numbers. Numerabilidade the set of rational numbers. Existence of a rational between any two rational numbers. Finite and infinite, periodic and non-periodic decimals. Irrational numbers, algebraic and transcendental numbers. Complex numbers: their historical genesis and geometric interpretation.

Mandatory literature

António Machiavelo; Apontamentos disponibilizados na página da UC

Complementary Bibliography

Devlin Keith; Sets, functions and logic. ISBN: 0-412-45970-1
Halmos Paul Richard; Naive set theory. ISBN: 0-387-90092-6
Vinogradov I. M.; Elements of number theory. ISBN: 0-486-60259-1
Hahn Liang-Shin; Complex numbers and geometry. ISBN: 0-88385-510-0
Aigner Martin; Proofs from the book. ISBN: 3-540-63698-6

Teaching methods and learning activities

Lecturing on the various subjects in class, supported by notes made available at the course's page on Sigarra. Solving of exercises in practical classes, with the support of the respective teachers.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Calculation formula of final grade

During the semester there will be two tests, but distributed into mini-tests to be carried out at the practical classes, each corresponding to a question of one of those tests. Thus there will be a small test per week, with rare exceptions. Each of these two tests has a weight of 6 points towards the final grade.

The approval of the course is made at the final exam.

The final examination will have 3 parts corresponding to two of them to the two tests. The student may choose not to solve one or both of these parts of the exam, in which case (s)he will get the mark obtained in the corresponding test.