###### Degree | Informatics engineering

## Mathematics I

### Scientific Field:

Mathematics

### Duration:

Semester

### ECTS

6

### Contact Hours Theoretical Practices:

60h

### LEARNING OBJECTIVES

To complete this unit, students must acquire the knowledge and skills in mathematical analysis, which is an important tool in computer and multimedia engineering.

1. Assimilate and apply logical mathematical concepts and techniques.

2. Understand and apply concepts and techniques to develop in space two- and three-dimensional projects, by studying their mathematical foundations

3. Understand and apply the concepts and techniques of analysis of combinatorics and graph theory problems.

### PROGRAM

1. Mathematical logic

1.1. Propositional calculus

1.1.1. Boolean Operators

1.1.2. Propositional Formulas

1.1.3. Semantics of propositional calculus

1.1.4. Semantic tables

1.1.5. Deductive systems

1.2. Predicate calculus

1.2.1. First-Order Language

2.Semantics of predicate calculus

2.1.1. Semantic tables

2.1.2. Deductive systems

2.2. Logic applied to program verification.

2.2.1. The various meanings of the proof concept

2.2.2. Formal proof vs informal proof

2.2.3. Proof Strategies

2.2.4. Algorithmic problem solving

2.2.5. Program correction tests

3. Linear Algebra

3.1. Vectors in R and Rn

3.1.1 Vectors in R2

3.1.2 Vectors in R3

3.1.3 Vectors in Rn

3.1.4 Addition and scalar multiplication of vectors

3.1.5 Inner product of vectors

3.1.6 Complex numbers

3.1.7 Vectors in Cn

3.2 Algebra of matrices

3.2.1 Introduction to matrices

3.2.2 Addition of matrices

3.2.3 Scalar multiplication

3.2.4 Multiplication of matrices

3.2.5 Transposition of matrices

3.3 Linear equations systems

3.3.1 Concept of linear equation system

3.3.2 Methods for solving systems of linear equations

3.3.3 Matrix formulation of linear equations systems

3.4 Vector Spaces

3.4.1 Vector space concept

3.4.2 Examples of vector spaces

3.4.3 Linear Combinations

3.4.4 Subspaces

3.4.5 Dependency and Linear Independence

3.4.6 Matrix applications

4 Analysis and combinatorial calculation

4.2 Principles of Combinatorics

4.2.1 The addition principle

4.2.2 Multiplication principle

4.2.3 Subtraction principle

4.2.4 Division principle

4.2.5 Pigeonhole principle

4.2.6 Inclusion-exclusion principle

4.2.7 Derangements

4.3 Permutations and combinations

4.3.1 Linear permutations without repetitions

4.3.2 Linear permutations with repetitions

4.3.3 Circular permutations

4.3.4 Combinations without repetitions

4.3.5 Combinations with repetitions

4.3.6 Sampling and distribution models

4.4 Binomial coefficients

4.5 Graph Theory

### DEMONSTRATION OF COHERENCE BETWEEN SYLLABUS AND LEARNING RESULTS

This unit is composed of three domains in correspondence with the learning objectives. This unit covers content corresponding to three different learning objectives. First, the acquisition of concepts related to Logical-Mathematical, to support the development of software systems, covered by point 1 of the program. The ability to apply techniques and concepts to develop projects in two and three-dimensional space is covered by point 2. While point 3 covers the reasoning and techniques of problem analyses, combination models, permutations and graphs as a mathematical basis for the development of combinatorial algorithms.

### TEACHING METHODOLOGY AND ASSESSMENT

This unit has a theoretical-practical nature. In total 60 hours are planned for classroom teaching. The student’s total study time should be 162 hours. Around 50% of the total class time, will be practical classes.

Under ISTEC’s Regulation of Functioning, students are evaluated through a mandatory individual written exam. The student’s final classification may be positively affected by elements resulting from a continuous assessment process, such as tests, individual or group academic work, individual initiatives to participate in classes and learning resources provided by e-learning systems.

### DEMONSTRATION OF CONSISTENCY BETWEEN TEACHING METHODOLOGIES AND LEARNING RESULTS

This curricular unit teaching methodology assumes theoretical-practical characteristics appropriate to the nature of the subject. A spirit of reflection and discovery is encouraged for students to assimilate the theoretical knowledge gained in terms of practical application. Thus, achieving the defined objectives.

### BIBLIOGRAPHY

Fundamental:

Lover, Robert, Elementary Logic for Software Development, Springer-Verlag 2008

Hoffman, Kenneth, Ray Kunze, Linear Algebra, Prentice-Hall

Bóna, Miklos, A Walk Through Combinatorics – An Introduction to Enumeration and Graph Theory, World Scientific Publishing

Complementary:

Gonçalves, Ricardo (2018). Álgebra Linear Teoria e Prática (2.ª Edição revista e corrigida). Lisboa: Edições Sílabo.

Simões, Vasco (2013). Álgebra Linear. Resumo da Matéria + Problemas Resolvidos. Alfragide: Edições Orion.

Simões, Vasco (2009). Análise Matemática I. Resumo da Matéria + Problemas Resolvidos. Alfragide: Edições Orion.

INTERNET:

Access to specialist publications, free of charge, through the SPRINGER network:

https://link.springer.com/