The Engineer's Blog: Fuzzy Set Operations

LEOPRABU

Fuzzy Set Operations

Definitions.

Universe of Discourse

The Universe of Discourse is the range of all possible values for an input to a fuzzy system.

Fuzzy Set

A Fuzzy Set is any set that allows its members to have different grades of membership (membership function) in the interval [0,1].

Support

The Support of a fuzzy set F is the crisp set of all points in the Universe of Discourse U such that the membership function of F is non-zero.

Crossover point

The Crossover point of a fuzzy set is the element in U at which its membership function is 0.5.

Fuzzy Singleton

A Fuzzy singleton is a fuzzy set whose support is a single point in U with a membership function of one.

Fuzzy Set Operations.

Union

The membership function of the Union of two fuzzy sets A and B with membership functions and respectively is defined as the maximum of the two individual membership functions

The Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra.

Complement

The membership function of the Complement of a Fuzzy set A with membership function is defined as

The following rules which are common in classical set theory also apply to Fuzzy set theory.

De Morgans law

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Associativity

Commutativity

Distributivity

REFERENCES
[1] Daniel Mcneil and Paul Freiberger " Fuzzy Logic"

[2] http://www.ortech-engr.com/fuzzy/reservoir.html

[3] http://www.quadralay.com/www/Fuzzy/FAQ/FAQ00.html

[4] http://www.fll.uni.linz.ac.af/pdhome.html

[5] http://soft.amcac.ac.jp/index-e.html

[6] http://www.abo.fi/~rfuller/nfs.html

[7] L.A.Zadeh,"Making computer think like people, IEEE spectrum, 8/1984, pp 26-32 [8] S.Haack, " Do we need fuzzy logic? " Int .Jr nl .of Man-Mach.stud , vol.11, 1979, pp 437-445