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Encyclopedia > Fuzzy set operations

Fuzzy set operations are a set of operations performed 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. There are three operations, they are fuzzy complements, fuzzy intersections, and fuzzy unions. The word operation can mean any of several things: The method, act, process, or effect of using a device or system. ... Fuzzy sets are an extension of the classical set theory used in Fuzzy logic. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ...


Standard fuzzy set operations

Standard complement
cA(x) = 1 − A(x)
Standard intersection
(AB)(x) = min [A(x), B(x)]
Standard union
(AB)(x) = max [A(x), B(x)]

Fuzzy complements

A(x) is defined as the degree to which x belongs to A. Let cA denote a fuzzy complement of A of type c. Then cA(x) is the degree to which x belongs to cA, and the degree to which x does not belong to A. (A(x) is therefore the degree to which x does not belong to cA.) Let a complement cA be defined by a function

c : [0,1] → [0,1]
c(A(x)) = cA(x)

Axioms for fuzzy complements

Axiom c1. Boundary condition
c(0) = 1 and c(1) = 0
Axiom c2. Monotonicity
For all a, b ∈ [0, 1], if ab, then c(a) ≥ c(b)
Axiom c3. Continuity
c is continuous function.
Axiom c4. Involutions
c is an involution, which means that c(c(a)) = a for each a ∈ [0,1]

In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...

Fuzzy intersections

The intersection of two fuzzy sets A and B is specified in general by a binary operation on the unit interval, a function of the form

i:[0,1]×[0,1] → [0,1].
(AB)(x) = i[A(x), B(x)] for all x.

Axioms for fuzzy intersection

Axiom i1. boundary condition
i(a, 1) = a
Axiom i2. Monotonicity
bd implies i(a, b) ≤ i(a, d)
Axiom i3. Commutativity
i(a, b) = i(b, a)
Axiom i4. Associativity
i(a, i(b, d)) = i(i(a, b), d)
Axiom i5. Continuity
i is a continuous function
Axiom i6. Subidempotency
i(a, a) < a

Fuzzy unions

The union of two fuzzy sets A and B is specified in general by a binary operation on the unit interval function of the form

u:[0,1]×[0,1] → [0,1].
(AB)(x) = u[A(x), B(x)] for all x

Axioms for fuzzy union

Axiom u1. Boundary condition
u(a, 0) = a
Axiom u2. Monotonicity
bd implies u(a, b) ≤ u(a, d)
Axiom u3. Commutativity
u(a, b) = u(b, a)
Axiom u4. Associativity
u(a, u(b, d)) = u(u(a, b), d)
Axiom u5. Continuity
u is a continuous function
Axiom u6. Subidempotency
u(a, a) > a
Axiom u7. strict monotonicity
a1 < a2 and b1 < b2 implies u(a1, b1) < u(a2, b2)

Aggregation operations

Aggregation operations on fuzzy sets are operations by which several fuzzy sets are combined in a desirable way to produce a single fuzzy set.

Aggregation operation on n fuzzy set (2 ≤ n) is defined by a function

h:[0,1]n → [0,1]

Axioms for aggregation operations fuzzy sets

Axiom h1. boundary condition
h(0, 0, ..., 0) = 0 and h(1, 1, ..., 1) = 1
Axiom h2. monotonicity
For any pair <a1, a2, ..., an> and <b1, b2, ..., bn> of n-tuples such that ai, bi ∈ [0,1] for all iNn, if aibi for all iNn, then h(a1, a2, ...,an) ≤ h(b1, b2, ..., bn); that is, h is monotonic increasing in all its arguments.
Axiom h3. Continuity
h is a continuous function.

See also

  Results from FactBites:
NationMaster - Encyclopedia: Fuzzy set operations (661 words)
Another possible way of defining a set is by setting a "characteristic function," which assigns to elements of the universe of discourse a value of either 1 (which indicates membership in the set) or 0 (which indicates non-membership), thereby declaring which elements of the universe are members of the characterized set.
Fuzzy models of data constructed upon fuzzy set theory are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious "world" that may not be captured by observation and measurement.
A fuzzy measure on a universe and a family of its subsets is a monotonic function from this family to [0,1] that assigns 0 to the empty set and 1 to the universe.
Fuzzy Sets and Operations (885 words)
The Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra.
The Intersection operation in Fuzzy set theory is the equivalent of the AND operation in Boolean algebra.
The Complement operation in Fuzzy set theory is the equivalent of the NOT operation in Boolean algebra.
  More results at FactBites »



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