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 μ ¬ A ( u ) = 1 − μ A ( u ) {\displaystyle \mu _{\lnot {A}}(u)=1-\mu _{A}(u)}

The complement is sometimes denoted by ∁A or A∁ instead of ¬A.

Standard intersection μ A ∩ B ( u ) = min { μ A ( u ) , μ B ( u ) } {\displaystyle \mu _{A\cap B}(u)=\min\{\mu _{A}(u),\mu _{B}(u)\}} Standard union μ A ∪ B ( u ) = max { μ A ( u ) , μ B ( u ) } {\displaystyle \mu _{A\cup B}(u)=\max\{\mu _{A}(u),\mu _{B}(u)\}}

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).[1] 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 .[2]

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. 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 (aka fuzzy intersection). The standard t-norm min is the only idempotent t-norm (that is, i (a1, a1) = a for all a ∈ [0,1]).[2]

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)]. 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 (aka 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]).[2]

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] 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.

• Fuzzy logic
• Fuzzy set
• T-norm
• Type-2 fuzzy sets and systems
• De Morgan algebra

• Klir, George J.; Bo Yuan (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall. ISBN 978-0131011717.

• L.A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965