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

  • Carlos Seara Ojea ( )
  • Daniel Gil Muñoz ( )
  • Eloy Cabezas Cardenas ( )
  • Fernando Martínez Sáez ( )
  • Francesc Prats Duaygues ( )
  • Francesc Tiñena Salvañà ( )
  • German Saez Moreno ( )
  • Guillermo González Casado ( )
  • Jaume Marti Farre ( )
  • Luis Valencia i Lopez ( )
  • Nuria Mira Gómez ( )

Weekly hours

Theory
3
Problems
0
Laboratory
2
Guided learning
0.5
Autonomous learning
7

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 - 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
14h

Sets

Sets and proofs about sets
Objectives: 2 3
Contents:
Theory
9h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
21h

Partial exam

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

Mappings

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

The Induction Principle

The Induction Principle
Objectives: 5
Contents:
Theory
6h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
14h

Partial exam

Partial exam

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

Divisibility

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

Congruences

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

Applications of congruences

Some applications of congruences
Objectives: 4 6 7
Contents:
Theory
3h
Problems
0h
Laboratory
4h
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: final exam
Theory
0h
Problems
0h
Laboratory
0h
Guided learning
3h
Autonomous learning
0h

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 partial exams, not in classtime, (20% each one). Rating: P1 and P2 (both over 10).

* There is be a final exam, not in classtime, (40%). Rating: F (over 10)

* The achievement of objectives in the workshop sessions will be also considered (20%). Rating: T (over 10)

Moreover, there is the possibility to improve the partial marks during the final exam by answering and additional task. Say that RP1 and RP2 are the marks corresponding to those additional parts. Then the mark for the course is computed as:

0.20 * T + 0.20*max(P1,RP1) + 0.20*max(P2,RP2) + 0.40*F

All the students who want to do the additional task in the final exam to improve the partial marks have to say so in advance.

The participation in less than 30% by weight of evaluating sessions will be considered as non evaluated.

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

Addendum

Contents

L'unic canvi, per raons de calendari, és que donarem menys pes a la inducció completa.

Teaching methodology

Algunes classes de teoria es faran de manera no presencial mitjançant sessions de Meet. Cada setmana farem una de les tres hores de teoria amb Meet. La resta de la docència no canvia.

Evaluation methodology

Hi haurà dues proves de taller que puntuaran un 10% cada una. Hi haurà un primer parcial fora d'hores de classe que puntuarà un 25%. Hi haurà un segon parcial fora d'hores de classe que puntuarà un 45%. La resta de la nota, un 10% s'obtindrà responent diversos qüestionaris Atenea. La nota per parcials N s'obté amb la fórmula: N=0.1T1+0.1T2+0.25P1+0.45P2+0.1A Els que suspenguin per parcials, podran optar a un final i aprovar el curs amb aquest examen. La nota de curs es: max( N, F ) T1,T2 tallers. P1,P2 parcials. A Atenea. F final. Els exàmens parcials i el final tindran una part amb Qüestionaris Atenea i una part escrita.

Contingency plan

Totes les classes es faran amb Meet, respectant els grups i els horaris. Les avaluacions dels parcials es faran mitjançant qüestionaris Atenea, respectant continguts, pesos de les proves i horaris en la mesura del possible.