Fuzzy Set Theory

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Fuzzy Set Theory

Fuzzy sets: Fuzzy sets (FS) are sets whose elements have degrees of membership.

Fuzzy relations, which are used now in different areas, such as decision-making (Kuzmin, 1982), linguistics (De Cock, et al, 2000), and clustering (Bezdek, 1978), are very special cases of L-relations when L is the unit interval [0, 1] is it from start to end.

History

Lotfi A. Zadeh and Dieter Klaua introduce Fuzzy sets were in the year 1965 as an extension of the classical notion of set. At the same period, Salii (1965) defined a more general kind of structures L-relations, which were studied by Salii in an abstract algebraic context.

Fuzzy Set Theory

Fuzzy set theory defines set membership as a possibility distribution.
The general rule for this theory can expressed as:

&npsb;&npsb;&npsb;&npsb;f:[0,1]->[0,1]

where d some number of possibilities.

This basically states that we can take n possible events and us d to generate as single possible outcome.
This extends set of membership since we could have varying definitions for of, say, hot curries. One person might declare that only curries of “Vindaloo” strength or above are hot whilst another might say madras and above are hot We could allow for these variations definition by allowing both possibilities in definition of fuzzy.
Once set of membership has been redefined we can develop new logics based on combining of sets etc and reason effectively.

In classical set theory the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1] which denotes the start till the end. Fuzzy sets generalize classical sets since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets if the latter only take values 0 or 1. In fuzzy set theory, classical bivalent sets are usually called crisp sets The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise such as bioinformatics.

Classical sets

Classical sets are also called crisp (sets).
Lists: X = {apple, orange, cherrie, mango};
X = {a1,a2,a3 }
X = {2, 4, 6, 8, …}
Formulas: X = {a | a is an even natural number}
X = {a | a = 2n, n is a natural number}

Some Applications

Membership function for age calculation

young middle age and old

Temperature range measures

temperature in range

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