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Membership function (mathematics)
Encyclopedia
The membership function of a fuzzy set
is a generalization of the indicator function in classical sets. In fuzzy logic
, it represents the degree of truth as an extension of valuation
. Degrees of truth are often confused with probabilities
, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition. Membership functions were introduced by Zadeh
in the first paper on fuzzy sets (1965).
, a membership function on 
is any function from 
to the real unit interval [0,1].
Membership functions on
represent fuzzy subsets
of
. The membership function which represents a fuzzy set 
is usually denoted by 
For an element 
of 
, the value 
is called the membership degree of 
in the fuzzy set 
The membership degree 
quantifies the grade of membership of the element 
to the fuzzy set 
The value 0 means that 
is not a member of the fuzzy set; the value 1 means that 
is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially.
Membership function of a fuzzy set
Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure
; usually it is required that 
be at least a poset or lattice
. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions.
.
In decision theory, a capacity is defined as a function,
from S, the set of subsets of some set, into 
, such that 
is set-wise monotone and is normalized (i.e. 
Clearly this is a generalization of a probability measure
, where the probability axiom of countability is weakened. A capacity is used as a subjective measure of the likelihood of an event, and the "expected value
" of an outcome given a certain capacity can be found by taking the Choquet integral
over the capacity.
Fuzzy set
Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced simultaneously by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to...
is a generalization of the indicator function in classical sets. In fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
, it represents the degree of truth as an extension of valuation
Valuation (logic)
In logic and model theory, a valuation can be:*In propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional formulas with those variables....
. Degrees of truth are often confused with probabilities
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition. Membership functions were introduced by Zadeh
Lotfi Asker Zadeh
Lotfali Askar Zadeh , better known as Lotfi A. Zadeh, is a mathematician, electrical engineer, computer scientist, artifical intelligence researcher and professor emeritus of computer science at the University of California, Berkeley...
in the first paper on fuzzy sets (1965).
Definition
For any set, a membership function on is any function from to the real unit interval [0,1].Membership functions on
represent fuzzy subsetsFuzzy set
Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced simultaneously by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to...
of
. The membership function which represents a fuzzy set is usually denoted by For an element of , the value is called the membership degree of in the fuzzy set The membership degree quantifies the grade of membership of the element to the fuzzy set The value 0 means that is not a member of the fuzzy set; the value 1 means that is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially.Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure
; usually it is required that be at least a poset or latticeLattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions.
Capacity
One application of membership functions is as capacities in decision theoryDecision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...
.
In decision theory, a capacity is defined as a function,
from S, the set of subsets of some set, into , such that is set-wise monotone and is normalized (i.e. Clearly this is a generalization of a probability measureProbability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
, where the probability axiom of countability is weakened. A capacity is used as a subjective measure of the likelihood of an event, and the "expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
" of an outcome given a certain capacity can be found by taking the Choquet integral
Choquet integral
In decision theory, a Choquet integral is a way of measuring the expected utility of an uncertain event. It is applied specifically to membership functions and capacities. In imprecise probability theory, the Choquet integral is also used to calculate the lower expectation induced by a 2-monotone...
over the capacity.
See also
- DefuzzificationDefuzzificationDefuzzification is the process of producing a quantifiable result in fuzzy logic, given fuzzy sets and corresponding membership degrees. It is typically needed in fuzzy control systems. These will have a number of rules that transform a number of variables into a fuzzy result, that is, the result...
- Fuzzy measure theoryFuzzy measure theoryFuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership function and the classical probability measures...
- fuzzy set operationsFuzzy set operationsA fuzzy set operation is an operation on fuzzy sets. These operations are generalization of crisp set operations. There is more than one possible generalization. The most widely used operations are called standard fuzzy set operations...
- Rough setRough setIn computer science, a rough set, first described by a Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set in terms of a pair of sets which give the lower and the upper approximation of the original set...