# Mathematics for Cognitive Science (school year 2007-2008)

Lecturer:        Prof. Vladimír Kvasnička (Faculty of Informatics and IT, Slovak Tech. University)

Prof.. Jiří Pospíchal (Faculty of Informatics and IT, Slovak Tech. University)

email: kvasnicka(at)fiit(·)stuba(·)sk

pospichal(at)fiit(·)stuba(·)sk

Lecture: Wednesday 11-12.30 a. m., FII STU, conference room D125.

Goals of lecture

Familiarize students of Cognitive science with basic mathematical structures that are needed for a deeper understanding of Cognitive science and capability of formal mathematical thinking when Cognitive science problems are solved and formulated.

Annotation of lecture:

1.                  Mathematical logic – propositional logic, predicate logic, natural deduction, basic types of proofs, verification and falsification, inductive proofs.

2.                  Theory of sets  - basic algebraic operations, set theory algebra, functions and relations.

3.                  Theory of graphs – graph definition, basic types of graphs, valence of vertices, path, cycles, isomorphism.

4.                  Linear algebra – matrices, operations with matrices, rank, inverse matrix, system of linear equations, Gauss elimination method, determinants.

5.                  Basic calculus – function, continuity, derivative, properties of functions with derivative, partial derivative, antiderivative, definite integrals and their calculation.

Key words:

Logic, prepositional logic, predicate logic, proof, natural deduction, inductive generalization, falsification, set, graph, properties of graphs, matrix, system of linear equation, Gauus elimination method, determinant, function, derivative, partial derivative, antiderivative, definite integral

Requirements for successful passing:

(1)    During the course of semester to gain at least 20 points (from 45 points) from 3 tests and no unexcused absence from seminars.

(2)    Exam – achieve a total score of at least 56 points from total number of possible 100 points:

(a)    max 45 points for 3 tests (3x15=45) during the term

(b)   max 55 points from written exam test.

### Syllabi of lecture:

1.      week. Mathematical logic I. An introduction to propositional and predicate logic, natural deduction, methods of mathematical proofs, deductive proofs  (transparencies)

2.      week. Mathematical logic II. Basic rules of logical reasoning, direct proof, indirect proof, proof with contradiction, different cases of generalizations in predicate logic, falsification (counterexample), mathematical induction (transparencies)

3.      week.  Theory of sets I – set, subset, set operations, set algebra, cardinality and enumeration  (counting), Cartesian product.. (transparencies)

4.      week (17. 10. 2007). Theory of sets II – relations, operations over relations, equivalence relation, partial ordered sets, Hasse diagrams. Function, composite function, inverse function. (transparencies) (First test)

 Results of the 1st test Vladimír Dziaban 13.5 Ľuboš Plavucha 15.0

5.      week (24. 10. 2007). Graph theory – definition of graph, basic types of graphs, vertex degree, path, cycle, isomorphism. (transparencies)

6.      week. Matrix algebra I (31. 10. 2007) – definition of matrix, special types of matrices, operations with matrices, matrix algebra, rank of matrix, inverse matrix. ()

7.      week. Matrix algebra II (7. 11. 2007) – system of linear equations, Frobenius theorem, Gauss elimination method. Determinants, basic properties, calculation of determinants, using determinants to solve systems of linear equations (Cramer’s rule). ()

1.      week. Calculus I (14. 11. 2007) – Function, sequences, limit of a sequence, limit of a function and continuous functions  ()

2.      week. Calculus II (21. 11. 2007)  Derivative, properties of a function with derivatives, plot of function ()

 Results of the 2nd  test Vladimír Dziaban 13.0 Ľuboš Plavucha 14.0

3.      week. Calculus III – Real valued function of 2 variables, partial derivatives, extremes of functions. ()

(3. test)

4.      week. Calculus IV – Antiderivative – undefinite integral, definite integral ()

 3rd test,

 Results of the 3rd   test Vladimír Dziaban 14.0 Ľuboš Plavucha 15.0

 Summarized results Vladimír Dziaban 40.5 Ľuboš Plavucha 44.0

 Final exam will be on January 23. 2008 at 12.30 o´clock p.m. at Faculty of Informatics and IT STU, conference room D124, 1st floor, block D, Test will composed of similar exercises that were used in the previous written exams. Test is composed of 11 exercises, each potentially evaluated by  a maximum of 5 points, e. i. max. number of points for the test is 55. Test (pdf)

 Exam results overall sum Degree Vladimír Dziaban 48.5 89 B Ľuboš Plavucha 41 85 B

 Grade numerical evaluation number of points A (excellent) 1.0 <94,100> B (very good) 1.5 <84,94) C (good) 2.0 <72,84) D (fair) 2.5 <62,72) E (poor) 3.0 <56,62) FX (failed) 4.0 <0,56)

Literature:

                           V. Kvasnička and J. Pospíchal: Informatics for social sciences (in Slovak). Comenius University Press, Bratislava, 2004.

                           V. Kvasnička and J. Pospíchal: Mathematical logic (in Slovak). STU Press, Bratislava, 2006.

                           V. Kvasnička and J. Pospíchal: Discrete mathematics (in Slovak). STU Press, Bratislava, in press (available on the internet address

http://www.fiit.stuba.sk/~kvasnicka/).