# Applications Of Set Theory

Fuzzy set theory was introduced by Zadeh in 1965. Fuzzy set theory is a sweeping statement of set theory. Fuzzy sets are created to deal with a particular kind of uncertainty- degree vagueness- which results when we have a property that can be possessed by objects to changeable degrees. The fuzzy set theory is used in areas where the information is not precise or incomplete like in Bioinformatics. Some other applications of fuzzy set theory are: in artificial intelligence, digital image processing, washing machines, air conditioners, dishwashers, rice cookers, pattern recognition, in some microprocessors and microcontrollers, cameras etc.

Combinatorial set theory involves extensions of finite combinatorics to infinite sets. This includes the study of extensions of Ramsey Theory and Cardinal Arithmetic.

In Axiomatic set theory, axioms are used to describe sets. Generally Zermelo-Fraenkl axioms (ZF) are included in sets of axioms for set theory. Axiomatic set theory is used to define the natural numbers, the real numbers etc. and hence this theory is mainly used in performing ordinary mathematics. With the help of axiomatic set theory- vector spaces, manifolds, graphs, rings can be defined as sets having axiomatic properties.

From the axioms and definitions of set theory theorems for discrete mathematics, abstract algebra, topology and analysis can also be derived. Hence set theory acts as a strong base for these areas of mathematics. Integral calculus and differential calculus are the main constituents of "Analysis". Limit points and continuity of function both are derived from set theory. These operations result in Boolean algebra in which the operations of union, intersection and difference are recognized as equivalent to the logical operations "or", "and" and "not" respectively. Boolean algebra is used in making calculators and PCs.

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