Fuzzy set operations


A fuzzy set operation is an operation 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: 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 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 is called De Morgan Triplet iff
so that for all x,y ∈ the following holds true:
. This implies the axioms provided below in detail.

Fuzzy complements

μA 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 is the degree to which x belongs to ∁A, and the degree to which x does not belong to A. Let a complement ∁A be defined by a function

Axioms for fuzzy complements

;Axiom c1. Boundary condition
;Axiom c2. Monotonicity
;Axiom c3. Continuity
;Axiom c4. Involutions
c is a strong negator.
A function c satisfying axioms c1 and c2 has at least one fixpoint a* with c = a*,
and if axiom c3 is fulfilled as well there is exactly one such fixpoint. For the standard negator c = 1-x the unique fixpoint is a* = 0.5.

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

Axioms for fuzzy intersection

;Axiom i1. Boundary condition
;Axiom i2. Monotonicity
;Axiom i3. Commutativity
;Axiom i4. Associativity
;Axiom i5. Continuity
;Axiom i6. Subidempotency
;Axiom i7. Strict monotonicity
Axioms i1 up to i4 define a t-norm. The standard t-norm min is the only idempotent t-norm.

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

Axioms for fuzzy union

;Axiom u1. Boundary condition
;Axiom u2. Monotonicity
;Axiom u3. Commutativity
;Axiom u4. Associativity
;Axiom u5. Continuity
;Axiom u6. Superidempotency
;Axiom u7. Strict monotonicity
Axioms u1 up to u4 define a t-conorm. The standard t-conorm max is the only idempotent t-conorm.

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 is defined by a function

Axioms for aggregation operations fuzzy sets

;Axiom h1. Boundary condition
;Axiom h2. Monotonicity
;Axiom h3. Continuity