Mathematics 6F - Requirements for Examination

Types of uncertainty (probabilistic, fuzzy, quantum), their origin and distinction, mathematical description.


(We assume the knowledge of the respective parts of probability theory, not only of those emphasized here.)
Random sample, sample size, observation, preperation of experiments, the notion of statistics.
Characteristics of random variables: mean, variance, standard deviation, general and central moments.
Estimators (and their properties): sample mean, sample variance, sample standard deviation, sample general moments.
Properties of estimators: unbiased, efficient, best unbiased, consistent, etc.
Chebyshev Inequality, Central Limit Theorem.
Moment method, maximum likelihood method.
Interval estimators.
Tests of significance (of differences of means and variations).
t-distribution, chi2-distribution, F-distribution (definition, motivation, properties and applications).
chi2 goodness-of-fit test, tests of equality and independence of two distributions.

Fuzzy logic:

The notion of fuzzy set, height, support, core, cardinality, etc.
System of cuts of a fuzzy set, theorem on representation of fuzzy sets by cuts, conversion between vertical and horizontal representation.
Fuzzy inclusion.
Fuzzy propositional operations: fuzzy negations, fuzzy conjunctions (t-norms), fuzzy disjunctions (t-conorms), fuzzy implications and biimplications (including representation theorems for fuzzy negations, strict and nilpotent fuzzy conjunctions and nilpotent fuzzy disjunctions).
Operations with fuzzy sets, their definition using fuzzy propositional operations, calculation in horizontal and vertical representation.
Properties of fuzzy propositional and set operations.
Fuzzy relations, their composition, fuzzy equivalence (similarity), fuzzy partial order, hereditary (consistent) properties.
Projection of a fuzzy relation.
Cylindric extension (cartesian product) of fuzzy sets.
Extension principle for unary operations (functions), its properties.
Extension principle for binary operations, arithmetic of fuzzy numbers and intervals.
Fuzzy numbers and fuzzy intervals, basic arithmetical operations with them, their properties.
Basic principles of fuzzy control, Takagi-Sugeno and Mamdani-Assilian fuzzy controller.
Comparison of fuzzy control to other approaches.