Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OB | 2 | 1 |
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BMath first year courses Linear Algebra-100088 and Fundamental Mathematics-100089.
Discrete Mathematics is a field of mathematics that studies finite mathematical objects. It covers topics such as combinatorics, graph theory, cryptography, error-correcting codes, combinatorial design theory, game theory, logic, optimization, or algorithm design and analysis that can be applied to solve problems in any of the above-mentioned branches. Most of discrete mathematics has evolved relatively recently, motivated by the challenges mainly in computer science and operations research. The chapters of this introductory course are quite independent from one another. A prior knowledge of linear algebra, modular arithmetic, basic combinatorics, and -fundamentally- mathematical language and reasoning should be adequate to understand the subject matter of each chapter.
The first chapter addresses the topic of generating functions and recurrent sequences, this being a natural next step following the elementary combinatorics studied in the first year course Fundamental Mathematics. Once again, problems in combinatorics require training the skill of formulating a problem in terms of a mathematical statement. The second chapter deals with graphs, which are a fundamental tool for problem solving in quite different settings, ranging from the most abstract mathematics up to operations research. In some cases, a graph-theoretical formulation of the problem turns out to be, in itself, illuminating and a highly efficient step towards the solution.
The last chapter deals with the topic combinatorial optimisation, which deals with combinatoric questions in which, rather than counting objects of a particular type, one is interested in finding "the optimal" ones with respect to some criterion. The answers in this case won't be given by formulas but by algorithms to find or approach such optimal objects. The theory used involves basic linear algebra techniques (for linear programming) and matroids.
In summary, throughout this course in discrete mathematics, a variety of examples and applications will be explained, where by means of relatively simple tools and a clever approach, we get to solve interesting and difficult problems. Moreover, by their work on problem sets, students will be able to practice the first step in the process of mathematical modeling, namely, understanding a problem and finding a suitable mathematical formulation that leads to the solution.
1. Generating functions and recurrent sequences.
- Generating function definition. Computation techniques. Solving combinatorial problems using generating functions.
- Recurrent sequences. First and second order linear recurrence relations.
- Solving recurrence relations with generating functions.
2. Graph Theory.
- Definition and examples of mathematical modeling using graph theory.
- Basic concepts. Some graph families.
- Graph representation, graph isomorphism.
- Paths and cycles.
- Trees.
3. Combinatorial optimization.
- Introduction. Some examples.
- Linear Programming. The simplex method.
- Matroids
4. Brief Introduction to other topics in Discrete Mathematics(Student-presented Seminar).
Topics may include combinatorial designs, cryptography, planar graphs, Hamiltonian graphs, Random graphs, Coding theory, Ramsey theory, Game theory, etc...
Classroom activities will consist of:
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Computer Lab | 8 | 0.32 | |
Lectures | 28 | 1.12 | |
Problem Set meetings | 16 | 0.64 | 7 |
Type: Supervised | |||
Meeting with instructor for feedback on the seminar project | 1 | 0.04 | 10, 9, 8 |
Type: Autonomous | |||
Group meetings for study/preparation for the Student-presented Seminar | 14 | 0.56 | 10, 9, 8 |
Individual practice on computer solving of exercises | 8 | 0.32 | 7 |
Self-study of lecture material | 26 | 1.04 | 10, 9, 7 |
Working on problem sets | 36 | 1.44 | 7 |
There are four graded tasks: a midterm exam, a computer lab test, a seminar project, and a final exam.
Course grade is calculated according to the following formula:
0.25 midterm exam score + 0.2 computer lab test score + 0.2 seminar project score+ 0.35 final exam score
Exam Retake Option: There is only one possible retake for the written exams (60%). To be eligible to retake the final exam you should have participated, at least, in three graded tasks.
Participating in less than three graded tasks and not taking the final exam will result in a "Non-assessable" course grade.
After the final exam, course grades with distinction/honors are granted to those students whose work clearly merits it. Students retaking the final exam to improve their score may also apply for consideration for distinction.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Computer Lab test | 0.20 | 2 | 0.08 | 3, 4, 12 |
Final exam | 0.35 | 4 | 0.16 | 11, 3, 4, 5 |
Grading of the student-presented seminar | 0.20 | 1 | 0.04 | 2, 1, 10, 9, 8, 7, 6 |
Midterm exam | 0.25 | 2 | 0.08 | 11, 3, 4, 5, 7 |
Retake exam | 0.60 | 4 | 0.16 | 11, 3, 4, 5 |
Recommended textbooks:
Basart, J.M. , Rifà, J, and Villanueva, M. "Fonaments de matemàtica discreta. Elements de combinatòria i d'aritmètica". Col. Materials de la UAB, n. 36. 1997.
Graham, R.L. , Knuth, D. E. , and Patashnik, O. "Concrete mathematics: a foundation for computer science". Addison-Wesley. 1990.
Grimaldi, Ralph P. "Discrete and combinatorial mathematics: an applied introduction". 5th ed. Pearson.Addison-Wesley. 2004.
Rosen, Kenneth H. "Discrete mathematics and its applications". 6th ed. McGraw-Hill. 2007.
Lawler, Eugene. Combinatorial Optimization: Networks and Matroids. Dover. ISBN 0-486-41453-1. (2001)
Suggested Readings on Graph Theory:
Bondy, J.A. , and Murty, U.S.R. "Graph Theory". Springer. 2008.
Wilson, R.J. , and Watkins, J. "Graphs: an introductory approach: a first course in discrete mathematics". Wiley, cop. New York. 1990.
Suggested Readings on Linear Programming:
Alabert, A. , and Camps, R. "Programació Lineal, una introducció a la presa de decisions racional".
Basart, J.M. "Programació lineal". Col. Materials de la UAB, n. 58.. 1998.
Luenberger, D. , and Ye, Y. "Linear and Nonlinear Programming". 3rd ed. Springer. 2008.
Python, SageMath.