Fundamentals of Mathematics

Credits
6
Types
Compulsory
Requirements
This subject has not requirements , but it has got previous capacities
Department
MAT
Web
https://web.mat.upc.edu/fib/fm-gia/
The subject of Mathematical Fundamentals introduces mathematical notation and terminology, and will see the main methods of demonstration needed to successfully follow a mathematics course. In addition, set theory is introduced, emphasizing the elements related to combinatorics. Finally, a brief introduction to graph theory is given.

Teachers

Person in charge

Weekly hours

Theory
2
Problems
2
Laboratory
0
Guided learning
0
Autonomous learning
6

Competences

Transversal Competences

Transversals

  • CT6 [Avaluable] - Autonomous Learning. Detect deficiencies in one's own knowledge and overcome them through critical reflection and the choice of the best action to extend this knowledge.

    • Generic Technical Competences

    Generic

  • CG4 - Reasoning, analyzing reality and designing algorithms and formulations that model it. To identify problems and construct valid algorithmic or mathematical solutions, eventually new, integrating the necessary multidisciplinary knowledge, evaluating different alternatives with a critical spirit, justifying the decisions taken, interpreting and synthesizing the results in the context of the application domain and establishing methodological generalizations based on specific applications.

    • Objectives

    1. Know how to use the summation notation. Be able to manipulate expressions with sums.
      Related competences: CG4,
      Subcompetences
      • Be able to manipulate expressions with double sums.
      • Be able to identify arithmetic progressions and geometric progressions. Be able to calculate the general term and the sum of consecutive terms of these progressions.
      • Know the summation symbol.
      • Be able to manipulate expressions with summations.
    2. Know and be able to use formal language and mathematical reasoning. Be able to understand and make demonstrations.
      Related competences: CT6, CG4,
      Subcompetences
      • Know the universal and existential quantifiers.
      • Know the main demonstration methods.
      • Know the main logical connectives.
      • Know the Induction Principle.
      • Be able to perform simple mathematical demonstrations.
    3. Know the language of set theory.
      Related competences: CG4,
      Subcompetences
      • Know that the set of parts of a set with intersection and union operations has Boolean algebra structure.
      • Know the main operations of sets (union, intersection, difference, complementary, set of parts, Cartesian product).
      • Know what the binomial numbers are and some of its properties.
      • Know what the cardinal of a set is.
    4. Know the equivalence relations.
      Related competences: CG4,
      Subcompetences
      • Know what a binary relationship is.
      • Know how to identify binary relations that are equivalence relations.
      • Know what equivalence classes are and what a partition is. Know the relationship between equivalence classes and partitions.
    5. Know what is a map and its properties.
      Related competences: CG4,
      Subcompetences
      • Be able to identify a map.
      • Be able to calculate images and anti-images for a map.
      • Be able to identify an injection, surjection, bijection.
      • Be able to compose maps.
      • Know what the inverse of an application is. Be able to calculate the inverse, when it exists, in simple cases.
    6. Know the main objects of combinatorics.
      Related competences: CG4,
      Subcompetences
      • Know the Pigeonhole Principle.
      • Know the definition of cardinality of a set and the main difference between finite and infinite sets.
      • Know and be able to calculate the number of ways to select objects whether or not the order of the elements is taken into account, and whether or not repetition of objects is allowed.
      • Be able to calculate the number of configurations with certain properties.
    7. Know the language of graph theory.
      Related competences: CT6, CG4,
      Subcompetences
      • Be able to identify graphs as a binary relation.
      • Be able to identify a tree. Know the characterization and main properties of trees.
      • Know the main terminology of Graph Theory.
      • Know the Handshake Lemma. Be able to use it to deduce properties of graphs.
      • Know the different types of walks in a graph. Be able to calculate the distance between two vertices. Be able to calculate the diameter and radius of a graph. Be able to find the connected components of a graph. Know what a cut vertices and bridges are.

    Contents

    1. Formalism and proofs.
      Summation notation. Summation manipulation. Double sums. Arithmetic and geometric progressions.Propositions. Logical connectives. Tables of truth. Quantifiers. Demonstration methods. Induction principle.
    2. Set Theory
      Sets. Cardinalities. Subsets. Representation of a subset as a binary word. Binomial numbers. Operations with sets: union, intersection, difference, complementary, Cartesian product. Power set.
      Binary relations. Equivalence relations. Equivalence classes. Partitions Quotient set.
      Maps. Images and anti-images. Composition. Injective, exhaustive and bijective maps. Inverse.
    3. Combinatorics
      Cardinalities. Finite and infinite sets. Pigeonhole Principle. Permutations and combinations with and without repetition. Binomial numbers. Permutations of multisets. Multinomial numbers. Inclusion-exclusion principle.
    4. Graphs
      Graphs. Representation of graphs. Degrees. Adjacency matrix. Handshaking Lemma. Isomorphisms. Operations with graphs. Walks. Connected graphs. Distance. Cut vertices and bridges. Trees. Spanning trees.

    Activities

    Activity Evaluation act


    Summation

    The student must study and assimilate the concepts explained in theory class and apply them to do the exercises indicated and that will be solved in problem classes.
    Objectives: 1
    Contents:
    Theory
    2h
    Problems
    2h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    6h

    Reasoning

    The student must study and assimilate the concepts explained in theory class and apply them to do the exercises indicated and that will be solved in problem classes.
    Objectives: 2
    Theory
    4h
    Problems
    4h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    12h

    Sets

    The student must study and assimilate the concepts explained in theory class and apply them to do the exercises indicated and that will be solved in problem classes.
    Objectives: 3
    Contents:
    Theory
    2h
    Problems
    4h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    12h

    Equivalence relations

    The student must study and assimilate the concepts explained in theory class and apply them to do the exercises indicated and that will be solved in problem classes.
    Objectives: 4
    Theory
    2h
    Problems
    2h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    6h

    Maps

    The student must study and assimilate the concepts explained in theory class and apply them to do the exercises indicated and that will be solved in problem classes.
    Objectives: 5
    Theory
    2h
    Problems
    2h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    6h

    Midterm exam

    Partial exam corresponding to the first part of the course.
    Objectives: 1 2 3 4
    Week: 8 (Outside class hours)
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Combinatorics

    The student must study and assimilate the concepts explained in theory class and apply them to do the exercises indicated and that will be solved in problem classes.
    Objectives: 6
    Contents:
    Theory
    5h
    Problems
    8h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    12h

    Graphs

    The student must study and assimilate the concepts explained in theory class and apply them to do the exercises indicated and that will be solved in problem classes.
    Objectives: 7
    Contents:
    Theory
    8h
    Problems
    8h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    16h

    Final exam

    Final exam on the contents of the second part of the course, but which may require knowledge and application of the methods seen in the first part of the course,
    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

    Teaching methodology

    The theory classes consist of exposing the theoretical contents together with examples and possible applications.

    In problem classes, exercises from a list published list will be solved. Students have to prepare them in advance.

    Evaluation methodology

    The grade for the course will be obtained from:
    -a midterm exam, P;
    -a final exam, F;
    -valuation of work and achievement of objectives throughout the course, C.

    In the midterm exam, the contents of the first part of the subject will be evaluated.

    In the final exam will be evaluated mainly the contents of the second part of the subject, but it may be necessary to apply knowledge and methods previously seen .

    The midterm and final exams will take place outside of class hours.

    In addition, the continuous work of the student will be evaluated through questionnaires and/or delivery of exercises carried out in class or outside class hours.

    The assessment of the transversal competence is included in the grades above, since the application of the competence is required to achieve the objectives of the subject.

    The final grade of the course will be:
    max(0.30*P+0.50*F +0.20*C,F)
    where P, F and C are the marks, up to 10, of the partial exam, the final exam and of the continued work, respectively.

    The grade of not presented (NP) will be awarded to students who have not taken either the partial exam or the final exam.

    The grade of the transversal competence will be in terms of the final grade:
    A: final grade from 8 to 10
    B: final grade from 6.5 to 7.9
    C: final grade from 5 to 6.4
    D: final grade from 0 to 4.9
    NA: final grade NP

    Reassessment: Only those who have failed the final exam may take the reassessment. The maximum grade that can be obtained in the reassessment is 7.

    Bibliography

    Basic

    Complementary

    Web links

    Previous capacities

    It is assumed that the student has achieved the objectives and knowledge of high school mathematics prior to college.