Fundamentals of Mathematics

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Credits
7.5
Types
Compulsory
Requirements
This subject has not requirements, but it has got previous capacities
Department
MAT
This subject has two parts. In the first part we focus on propositional and predicate logic and mathematical reasoning that are needed for the curriculum. The second part deals with the basic concepts of integer arithmetic, such as divisibility and congruence relations.

Teachers

Person in charge

  • Rafel Farré Cirera ( )

Others

  • Fernando Martínez Sáez ( )
  • Gemma Alsina Ruiz ( )
  • Jose Luis Ruiz Muñoz ( )

Weekly hours

Theory
3
Problems
0
Laboratory
2
Guided learning
0
Autonomous learning
7.5

Competences

Technical Competences

Common technical competencies

  • CT1 - To demonstrate knowledge and comprehension of essential facts, concepts, principles and theories related to informatics and their disciplines of reference.
    • CT1.2A - To interpret, select and value concepts, theories, uses and technological developments related to computer science and its application derived from the needed fundamentals of mathematics, statistics and physics. Capacity to solve the mathematical problems presented in engineering. Talent to apply the knowledge about: algebra, differential and integral calculus and numeric methods; statistics and optimization.
    • CT1.2C - To use properly theories, procedures and tools in the professional development of the informatics engineering in all its fields (specification, design, implementation, deployment and products evaluation) demonstrating the comprehension of the adopted compromises in the design decisions.

Transversal Competences

Reasoning

  • G9 [Avaluable] - Capacity of critical, logical and mathematical reasoning. Capacity to solve problems in her study area. Abstraction capacity: capacity to create and use models that reflect real situations. Capacity to design and perform simple experiments and analyse and interpret its results. Analysis, synthesis and evaluation capacity.
    • G9.1 - Critical, logical and mathematical reasoning capacity. Capacity to understand abstraction and use it properly.

Objectives

  1. To understand the importance of language in scientific communication and the need to refine it and define it to avoid, as far as possible, the ambiguity.
    Related competences: G9.1,
  2. To understand what a mathematical proof is and to know the main types of proofs which the student may meet
    Related competences: G9.1,
  3. Understanding the language of sets as an essential tool in mathematical communication and also as an instrument
    Related competences: G9.1,
  4. Understanding the language of mappings as a way to define and to study correspondences and rules
    Related competences: G9.1,
  5. To understand that we cannot prove that a certain property is valid for infinitely many numbers by testing the property one number at a time but that we must use some principle that makes possible the proof
    Related competences: G9.1,
  6. To understand the properties of the divisibility of integers, to calculate the greatest common divisor using Euclid's algorithm and to write Bézout's identity of two integers. To calculate small prime numbers and to understand the difficulty of performing integer factorization.
    Related competences: G9.1, CT1.2A, CT1.2C,
  7. To u nderstand the concept of congruence and to be able of computing with congruences. To apply the language of congruences to solve arithmetic problems.
    Related competences: G9.1, CT1.2A, CT1.2C,

Contents

  1. Reasoning
    Sentences, statements and propositions. Formal propositional calculus. Proofs. Predicate logic.
  2. The principle of induction
    Induction. Complete induction.
  3. Sets
    Sets and elements, the membership relation. Elementary operations with sets. Relations. Equivalence relations and quotient set.
  4. Functions
    Functions. Injectivity and surjectivity. Inverse function. Image and pre-imaging. Composition.
  5. Divisibility of integers
    The divisibility relation in the set of integers. The division theorem. Primes. Infinitude of primes. Sieve of Eratosthenes. Greatest common divisor and least common multiple. Euclidean algorithm. Bézout's identity. Gauss Lemma.
  6. Congruences of integers
    The relation of congruence. Operations with congruences. Modular inverse: calculation. Congruence classes and the quotient group Zn. Operations with congruence classes.
  7. Applications of congruences
    Modular exponentiation. Linear equations in congruences. The chinese remainder theorem. The RSA cryptography systgem.

Activities

Activity Evaluation act


Reasoning

Logic formalism
Objectives: 1
Contents:
Theory
6h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
16h

Sets

Sets and proofs about sets
Objectives: 2 3
Contents:
Theory
7h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
21h

Mappings

Set mappings
Objectives: 4
Contents:
Theory
6h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
14h

The Induction Principle

The Induction Principle
Objectives: 5
Contents:
Theory
4h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
14h

Partial exam

Partial exam
Objectives: 1 2 3 5
Week: 10 (Outside class hours)
Type: theory exam
Theory
2h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
2h

Divisibility

Divisibility of integers
Objectives: 2 5 6
Contents:
Theory
5h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
14h

Congruences

Congruences of integers
Objectives: 2 6 7
Contents:
Theory
5h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
14h

Applications of congruences

Some applications of congruences
Objectives: 4 6 7
Contents:
Theory
2h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
7h

Review

Review of the main contents and problem solution

Theory
3h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
7h

Final exam

Final exam
Objectives: 1 2 3 4 5 6 7
Week: 15 (Outside class hours)
Type: theory exam
Theory
3h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
3h

Partial exam


Objectives: 4 6 7
Week: 15 (Outside class hours)
Type: theory exam
Theory
2h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
0.5h

Teaching methodology

In theoretical classes the theoretical content of the course is taught and illustrated with examples. In workshops students, guided by the teacher, will work topics explained in theoretical classes.

Evaluation methodology

* There are two midterm exams, not in class time, (35% each one). Rating: P1 and P2 (both out of 10).

* The goal achievements in the laboratory sessions will be also considered (20%). Rating: L (out of 10)

*There will be several ATENEA tests (10%). Rating: A (out of 10)

* The continuous evaluation mark AC is obtained as follows:

AC= 0.35*P1+0.35*P2+0.2*L+0.1*A

*Students can take the final exam (100%). Rating :F (out of 10)

* The course grade is the maximum between AC and F: max (AC, F)


Due to the particularities of the subject, the grade for the cross competition will be calculated from the course grade as follows:

* between 0 and 4.9 : D
* between 5 to 6.9 : C
* between 7 and 8.4 : B
* between 8.5 and 10 : A

Bibliography

Basic:

Complementary:

Previous capacities

The kind of abilities that a student that has succesfully passed his/her secondary studies is supposed to have