Subject: Mathematics

Scientific Area:

Mathematics

56 Hours

Number of ECTS:

5,5 ECTS

Language:

Portuguese

# Overall objectives:

1 - Review of some mathematical concepts that are essential for understanding the discipline;
2 - In the domain of Logic and Set Theory, explore operations on propositions, aiming at a bivalent logic in which the Principles of non-contradiction and the excluded third party apply. Understand and apply the concept of universal and existential quantification of conditions and the relationship between operations on conditions and on the respective solution sets. It is intended that students are able to translate in a very language of the theories developed on this theme some demonstration techniques with and without the use of truth tables.
3 - In the domain of Graphs, develop the ability to create models, schemes and mathematical structures capable of representing problems. Provide students with tools capable of representing real situations/problems and solving them with scientific rigor.

# Syllabus:

1 - Introduction to bivalent logic and set theory
1.1 - Propositions
1.1.1 - Logical value of a proposition; Principle of non-contradiction;
1.1.2 - Proposition operations: negation, conjunction, disjunction, implication and equivalence;
1.1.3 - Priorities of logical operations;
1.1.4. - Logical relationships between the different operations; property of double negation; Principle of the excluded third; Double implication principle;
1.1.5 - Commutative and associative, disjunction and conjunction properties and distributive properties of the conjunction in relation to the disjunction and disjunction in relation to the conjunction;
1.1.6 - De Morgan's Laws;
1.1.7 - Counter-reciprocal implication;
1.2 - Conditions and Sets
1.2.1 - Propositional expression or condition; universal quantifier, existential quantifier and De Morgan's second laws; counterexamples;
1.2.2 - Set defined by a condition; Equality between sets; sets defined in extension; Union (or gathering), intersection and difference of sets and complementary set;
1.2.3 - Inclusion of sets;
1.2.4 - Relationship between logical operations on conditions and operations on the sets they define; - Principle of double inclusion and demonstration of equivalences by double implication;
1.2.5 - Denial of a universal implication; counter-reciprocal demonstration;
2 - Graphs
2.1 - Graph theory: basic concepts
2.2 - Eulerian paths and circuits
2.3 - Graph eulerization: problem of the Chinese postman
2.4 - Hamiltonian circuits: traveling salesman problem
2.5 - Graph coloring
2.6 - Comprehensive trees
2.7 - Critical paths

# Literature/Sources:

Makinson, D. , 2012 , Sets, Logic and Maths for Computing , Springer
Campos Ferreira, J. , 2001 , Elementos de Lógica Matemática e Teoria dos Conjuntos , Dep. de Matemática, Instituto Superior Técnico
Magalhães, L.T. , 1998 , Álgebra Linear como Introdução à Matemática Aplicada , Texto Editora
Bondy, J. A.; Murty, U.S.R. , 1976 , Graph theory with applications , London: Macmillan
Apostol, T. M. , 1998 , Cálculo , Reverté
Jungnickel, D. , 2008 , Graphs, networks and algorithms. 3rd ed. , Berlin: Springer, cop
Grimaldi, R. P. , 2003 , Discrete and Combinatorial Mathematics: an Applied Introduction , Pearson

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
In accordance with the expected for the Professional Higher Technical Courses.