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# Mathematics for Cognitive Science 2-IKVa-102

The lectures will provide students with basics of propositional and predicate logic, linear algebra, mathematical analysis, and probability that are important for the study of informatics and its role in (computational) cognitive science. At the same time, students will learn about mathematical culture, notation, way of thinking and expressing oneself.

## Course schedule

Type Day Time Room Lecturer
Lecture/Excercise Thursday 09:50 M-112 Martina Koronci Babinská
Excercise/Lecture Thursday 14:50 M-V Martina Koronci Babinská

## Syllabus

Date Topic References
03.10. Introduction. The basics of logic and proving methods: propositional logic. Discrete and combinatorial mathematics: An applied introduction / Ralph P. Grimaldi.

05.10. The basics of logic and proving methods: predicate logic. Discrete and combinatorial mathematics: An applied introduction / Ralph P. Grimaldi.

12.10. The basics of logic and proving methods: Sets (sets of numbers, set theory, set operations, the laws of set theory) Discrete and combinatorial mathematics: An applied introduction / Ralph P. Grimaldi.

12.10. The basics of logic and proving methods: Proofs (constructive, direct, contrapositive, contradiction, biconditional, mathematical induction) Discrete and combinatorial mathematics: An applied introduction / Ralph P. Grimaldi.

Rose-Hulman Institute of Technology: Pearson, 2004; Download here; chap. 2, 3, 4.1

19.10. The basics of mathematical analysis: Functions (definition, graph, characteristics, basic mathematical functions, functions in the real word)
26.10. I The basics of mathematical analysis: Basic function's characteristics in the definitions (domain, range, monotonicity, boundaries, extremes, continuity, infimum, supremum).

26.10. II The basics of mathematical analysis: Limit of a function (the real word problem, definition, one-side limit) Calculus / Gilbert Strang. Massachusetts Institute of Technology: Wellesley-Cambridge Press. Download here; chap. 1.1, 1.2, 1.3, 2.1
02.11. The basics of mathematical analysis: The rate of change - derivative (the real word problem, definition, basic functions derivatives, formulas and calculating rules) Calculus / Gilbert Strang. Massachusetts Institute of Technology: Wellesley-Cambridge Press. Download here; chap. 2.2, 2.3, 2.4, 2.5
09.11. I The basics of mathematical analysis: Minimum/maximum problem and Convex/Concave problem in case of derivatives Calculus / Gilbert Strang. Massachusetts Institute of Technology: Wellesley-Cambridge Press. Download here; chap. 3.2, 3.3, 3.4
09.11. II The basics of mathematical analysis: The Chain Rule Calculus / Gilbert Strang. Massachusetts Institute of Technology: Wellesley-Cambridge Press. Download here; chap. 4.1
16.11. I The basics of mathematical analysis: The Chain Rule + Minimum/maximum problem - counting exercise Calculus / Gilbert Strang. Massachusetts Institute of Technology: Wellesley-Cambridge Press. Download here; chap. 4.1
16.11. II The basics of linear algebra: Linear equations - the basic problem of linear algebra, Matrix and Vector definition Fundamentals of Linear Algebra / James B. Carrell. Canada: University of British Colombia, 2005. Download here; chap. 2.1, 2.2, 3.1
23.11. I Repeating class
23.11. II Middle term writing test
30.11. I - II The basics of linear algebra: Matrix and Vector operations, Gaussian Reduction - introduction Fundamentals of Linear Algebra / James B. Carrell. Canada: University of British Colombia, 2005. Download here; chap. 2.2, 2.3, 2.4
07.12. I The basics of linear algebra: Gaussian Reduction practice - various number of solutions Fundamentals of Linear Algebra / James B. Carrell. Canada: University of British Colombia, 2005. Download here; chap. 2.4
07.12. II The basics of linear algebra: Matrix operations - transposition, multiplication, inverse matrix, determinants Fundamentals of Linear Algebra / James B. Carrell. Canada: University of British Colombia, 2005. Download here; chap. 3.1, 8.1

## Homework

Date Homework Points References
05.10. 1. Choose 1 Exercise from the Exercise 2.1 (page 54).

2. Is 0/0 = 0 statement or not? Why?

1 point

1 point

Discrete and combinatorial mathematics: An applied introduction / Ralph P. Grimaldi.

12.10. 1. Choose 2 Exercises from the Exercise 2.4 (page 100). 1 point / exercise Discrete and combinatorial mathematics: An applied introduction / Ralph P. Grimaldi.

19.10 Find/create the open statement which is possible to prove by mathematical induction.

Prove the statement and use the proof to explain the principle of mathematical induction.

3 points
26.10 1. Take the FEV1 Spirometry graph in time (Morning Lecture, slide 27). Based on that graph write as many information as you can.

2. Find the quadratic function characteristics.

2 - 4 points

2 points

02.11 Find derivatives:

1. y = (x + 5)*x^2

2. y = log(2)x – 5x + 7x^5

3. y = sin x / cos x

4. y = (3x + 5^x – 6)/(ln x + e^x)

4 points
09.11 Find the local and global extremes of a function. Create the graph of that function: y = x2 / (1 – x2)

4 points Calculus / Gilbert Strang. Massachusetts Institute of Technology: Wellesley-Cambridge Press. Download here;
30.11 Find non-homogeneous linear system of 5 equations with 5 variables.

Use Gaussian Reduction to find the solution of your system.

4 points Fundamentals of Linear Algebra / James B. Carrell. Canada: University of British Colombia, 2005. Download here;
07.12 Try to find (google or book(s)) the explanation of the mathematical term: eigenvalue.

Choose any matrix and find its eigenvalues.

4 points Fundamentals of Linear Algebra / James B. Carrell. Canada: University of British Colombia, 2005. Download here;

## References

• Discrete and combinatorial mathematics: An applied introduction / Ralph P. Grimaldi.

• Calculus / Gilbert Strang. Massachusetts Institute of Technology: Wellesley-Cambridge Press. Download here;
• Fundamentals of Linear Algebra / James B. Carrell. Canada: University of British Colombia, 2005. Download here;
• Artificial Intelligence: A Modern Approach / Stuart Russell and Peter Norvig. The USA: Pearson, 2010. Download here;

To be classified student has to achieve at least 50% of every activity:

PROJECT

• form: essay, presentation, song or movie
• topic: What does mathematics mean for me? What am I expecting from this course?
• term: 27.10.2017
• goal: self-study motivation
• weight: 15%

WEEKLY EXAMS

• form: 10-15 minutes writing tests
• term: every Thursday at the beginning of the exercise
• goal: regular preparation
• weight: 40%
• note: student can also achieve extra (bonus) points for: weekly homeworks, class work (solving problems and schoolmate’s help) and/or self-activity (lecture preparation… )

MIDDLE TERM EXAM

• form: 90 minutes writing test (student can choose from the offered task sets)
• term: 23.11.2017
• goal: progress definition
• weight: 15%

FINAL EXAM

• form: 90 minutes writing test
• term: January, February 2018
• goal: course output
• weight: 30%

OVERALL GRADING: A > 90%, B > 80%, C> 70%, D > 60%, E > 52% points.