Lecturer: Prof. Vladimír Kvasnička (Faculty of
Informatics and IT,
Prof.. Jiří Pospíchal
(Faculty of Informatics and IT,
email: kvasnicka(at)fiit(·)stuba(·)sk
pospichal(at)fiit(·)stuba(·)sk
Lecture: Wednesday
1112.30 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.
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 1^{st} 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. (transparencie, examples)
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). (transparencies, examples)
1.
week. Calculus I (14. 11. 2007) – Function, sequences, limit of a
sequence, limit of a function and continuous functions (transparencies, examples, Second test)
2.
week. Calculus II (21. 11. 2007) –
Derivative, properties of a function with derivatives, plot of function
(transparencies, examples)
Results of the 2^{nd} test 

Vladimír Dziaban 
13.0 
Ľuboš Plavucha 
14.0 




3.
week. Calculus III – Real valued function of 2 variables, partial
derivatives, extremes of functions. (transparencies, examples)
(3. test)
4.
week. Calculus IV – Antiderivative – undefinite integral, definite
integral (transparencies, examples, examplessolution, Third test)
3^{rd}
test, 
Results of the 3^{rd} 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, 1^{st}
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:
[1]
V.
Kvasnička and J. Pospíchal: Informatics
for social sciences (in Slovak).
[2]
V.
Kvasnička and J. Pospíchal: Mathematical
logic (in Slovak). STU Press,
[3]
V.
Kvasnička and J. Pospíchal: Discrete
mathematics (in Slovak). STU Press,
http://www.fiit.stuba.sk/~kvasnicka/).