Fundamentals of Mathematics

• Teachers
• Weekly hours
• Competences
• Objectives
• Contents
• Activities
• Teaching methodology
• Evaluation methodology
• Bibliography
• Previous capacities

Teachers

Person in charge

• Rafel Farré Cirera ( )

Others

• Aitor Sort Nadal ( )
• Andreu Bellés Roca ( )
• Eric López Platón ( )
• Fernando Martínez Sáez ( )
• Francesc Tiñena Salvañà ( )
• Gemma Alsina Ruiz ( )
• Guillermo González Casado ( )
• Jaume Marti Farre ( )
• Lluis Vena Cros ( )
• Maria Isabel Gonzalez Perez ( )
• Mariona González Esteve ( )
• Mónica Sanchez Soler ( )
• Montserrat Maureso Sánchez ( )
• Roberto Gualdi ( )
• Victoria Graffigna ( )

Weekly hours

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

Teaching methodology

Evaluation methodology

Bibliography

Basic:

• Apunts de FONAMENTS MATEMÀTICS, part 1 - Farré, Rafel,
  
• Apunts de FONAMENTS MATEMÀTICS, part 2 - Farré, Rafel,
  
• Matemática discreta y sus aplicaciones - Rosen, Kenneth H; Pérez Morales, José Manuel, McGraw-Hill, cop. 2004. ISBN: 8448140737
  https://discovery.upc.edu/discovery/fulldisplay?docid=alma991002813919706711&context=L&vid=34CSUC_UPC:VU1&lang=ca

Complementary:

• Teoría y problemas de teoría de conjuntos y temas afines - Lipschutz, S, McGraw-Hill , 1970. ISBN: 007091625X
  https://discovery.upc.edu/discovery/fulldisplay?docid=alma991001934379706711&context=L&vid=34CSUC_UPC:VU1&lang=ca
• ¿La Dama o el tigre? y otros pasatiempos lógicos : incluyendo una novela matemática que presenta el gran descubrimiento de Gödel - Smullyan, R.M, Cátedra , 2017. ISBN: 9788437604145
  https://discovery.upc.edu/discovery/fulldisplay?docid=alma991004152489706711&context=L&vid=34CSUC_UPC:VU1&lang=ca
• 2000 problemas resueltos de matemática discreta - Lipschutz, S.; Lipson, M.L, McGraw-Hill , 2004. ISBN: 9788448142780
  https://discovery.upc.edu/discovery/fulldisplay?docid=alma991002929429706711&context=L&vid=34CSUC_UPC:VU1&lang=ca
• ¿Cómo se llama este libro?: el enigma de Drácula y otros pasatiempos lógicos - Smullyan, R.M, Cátedra , 2008. ISBN: 9788437602974
  https://discovery.upc.edu/discovery/fulldisplay?docid=alma991003838469706711&context=L&vid=34CSUC_UPC:VU1&lang=ca

Previous capacities