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Concept# Fuzzy set operations

Summary

Fuzzy set operations are a generalization of crisp set operations for fuzzy sets. There is in fact more than one possible generalization. The most widely used operations are called standard fuzzy set operations; they comprise: fuzzy complements, fuzzy intersections, and fuzzy unions.
Let A and B be fuzzy sets that A,B ⊆ U, u is any element (e.g. value) in the U universe: u ∈ U.
Standard complement
The complement is sometimes denoted by ∁A or A∁ instead of ¬A.
Standard intersection
Standard union
In general, the triple (i,u,n) is called De Morgan Triplet iff
i is a t-norm,
u is a t-conorm (aka s-norm),
n is a strong negator,
so that for all x,y ∈ [0, 1] the following holds true:
u(x,y) = n( i( n(x), n(y) ) )
(generalized De Morgan relation). This implies the axioms provided below in detail.
μA(x) is defined as the degree to which x belongs to A. Let ∁A denote a fuzzy complement of A of type c. Then μ∁A(x) is the degree to which x belongs to ∁A, and the degree to which x does not belong to A. (μA(x) is therefore the degree to which x does not belong to ∁A.) Let a complement ∁A be defined by a function
c : [0,1] → [0,1]
For all x ∈ U: μ∁A(x) = c(μA(x))
Axiom c1. Boundary condition
c(0) = 1 and c(1) = 0
Axiom c2. Monotonicity
For all a, b ∈ [0, 1], if a < b, 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]
c is a strong negator (aka fuzzy complement).
A function c satisfying axioms c1 and c3 has at least one fixpoint a* with c(a*) = a*,
and if axiom c2 is fulfilled as well there is exactly one such fixpoint. For the standard negator c(x) = 1-x the unique fixpoint is a* = 0.5 .
T-norm
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].
For all x ∈ U: μA ∩ B(x) = i[μA(x), μB(x)].
Axiom i1. Boundary condition
i(a, 1) = a
Axiom i2. Monotonicity
b ≤ d implies i(a, b) ≤ i(a, d)
Axiom i3. Commutativity
i(a, b) = i(b, a)
Axiom i4.

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Fuzzy set

In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].

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