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.

Standard fuzzy set operations 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 ( 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).Ismat Beg, Samina Ashraf: Similarity measures for fuzzy sets, at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016 This implies the axioms provided below in detail.

Fuzzy complements 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))

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 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 (a.k.a. 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 .Günther Rudolph: Computational Intelligence (PPS), TU Dortmund, Algorithm Engineering LS11, Winter Term 2009/10. Note that this power point sheet may have some problems with special character rendering

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].

For all x ∈ U: μA ∩ B(x) = i[μA(x), μB(x)].

Axioms for fuzzy intersection 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. 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 for all 0 < a < 1

Axiom i7. Strict monotonicity i (a1, b1) < i (a2, b2) if a1 < a2 and b1 < b2

Axioms i1 up to i4 define a t-norm (a.k.a. fuzzy intersection). The standard t-norm min is the only idempotent t-norm (that is, i (a1, a1) = a for all a ∈ [0,1]).

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].

For all x ∈ U: μA ∪ B(x) = u[μA(x), μB(x)].

Axioms for fuzzy union Axiom u1. Boundary condition u(a, 0) =u(0 ,a) = a

Axiom u2. Monotonicity b ≤ d 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. Superidempotency u(a, a) > a for all 0 < a < 1

Axiom u7. Strict monotonicity a1 < a2 and b1 < b2 implies u(a1, b1) < u(a2, b2)

Axioms u1 up to u4 define a t-conorm (a.k.a. s-norm or fuzzy union). The standard t-conorm max is the only idempotent t-conorm (i. e. u (a1, a1) = a for all a ∈ [0,1]).

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) = one

Axiom h2. Monotonicity For any pair <a1, a2, ..., an> and <b1, b2, ..., bn> of n-tuples such that ai, bi ∈ [0,1] for all i ∈ Nn, if ai ≤ bi for all i ∈ Nn, 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 Fuzzy logic Fuzzy set T-norm Type-2 fuzzy sets and systems De Morgan algebra

Fuzzy set operations

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