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 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

i is a t-norm,
u is a t-conorm,
n is a strong negator,

so that for all x,''y'' ∈ the following holds true:
. This implies the axioms provided below in detail.

Fuzzy complements

μ_A 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 c3 has at least one fixpoint a^* with c = a^*,
and if axiom c2 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''