Fundamentals of Mathematics

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

Others

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)
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    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)
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Partial exam


    Objectives: 4 6 7
    Week: 15 (Outside class hours)
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    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 midterm exams, not in class time, (40% 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)

    * The continuous evaluation mark AC is obtained as follows:

    AC= 0.4*P1+0.4*P2+0.2*L

    *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