Topics in Elementary Mathematics

Code:

M1024

Acronym:

M1024

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 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	105	Official Study Plan	1	-	9	84	243

Teaching language

Portuguese

Objectives

To introduce the basic concepts of mathematical logic and elementary set theory and to develop the study on integer arithmetic and the arithmetic of  polynomials.

Learning outcomes and competences

A student is expected to become familiar with deductive reasoning and the symbolic language of Mathematics, to explore mathematical methods of proof and to deepen his knowledge of some of the basic topics of mathematics.

Working method

Presencial

Program

1. Rudiments of logic and identification of their use in mathematical proofs. Mathematical language and basic mathematical symbolism.

2. Natural numbers and mathematical induction.

3. Elementary set theory; binary relations, equivalence relations; notions about functions. Notions of cardinality of infinite sets.

4. Arithmetic of integers: divisibility; division algorithm and the Euclidean algorithm; the fundamental theorem of Arithmetic; congruence module a positive integer; Fermat's Theorem and Euler's Theorem.

5. Arithmetic of the polynomials with coefficients in Q, R or C: divisibility; division algorithm; roots of polynomials; rational roots of polynomials with integer coefficients; Eisenstein's criterion; reference to the Fundamental Theorem of Algebra.

Mandatory literature

Ana Oliveira; Texto de apoio disponível na página da disciplina no Moodle.

Complementary Bibliography

K. Devlin; Sets, functions and logic, Chapman & Hall, 1992
C. Dodge; Sets, Logic and Numbers, Prindle, Weber and Schmidt, 1970
W. Sierpinski; Elementary Theory of Numbers, North-Holland, 1988
H. Stark; An introduction to number theory, MIT Press, 1991
Arjeh M. Cohen; Algebra interactive!. ISBN: 978-3-540-65368-4

Comments from the literature

-

Teaching methods and learning activities

The content of the syllabus is presented at the lectures, and proposed exercises are solved by the students at the practical classes.

keywords

Physical sciences > Mathematics > Mathematical logic
Physical sciences > Mathematics > Algebra > Set theory
Physical sciences > Mathematics > Number theory

Evaluation Type

Distributed evaluation without final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	159,00
Frequência das aulas	84,00
Total:	243,00

Eligibility for exams

No requirements.

Calculation formula of final grade

During the semester there will be two tests whose quotation sums 20. The second test will take place at the first exam season.

Course approval can be obtained 

1) at the first exam season, where the rating is the sum of the ratings obtained on the two tests.

2) at the second exam season, by taking the final exam.

The final exam consists of 2 parts, each corresponding to a test.

In the final exam, students who have not yet obtained approval (and only these) may choose not to solve one part of the exam, that would then get the grade of the corresponding test.

Special assessment (TE, DA, ...)

In any special evaluation season, the written exam might be preceded by an eliminatory oral test to assess whether the student satisfies minimum requirements to tentatively pass the written exam.

Classification improvement

Students wishing to undertake a classification improvement must solve all parts of the exam.

Observations

It may be asked to any student to take an extra oral or written examination if any doubt concerning his/her performance occurs on certain assessment pieces.