Fuzzy set operations

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.