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

:<math>\mu_{\lnot{A(u) = 1 - \mu_A(u)</math>

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

;Standard intersection

:<math>\mu_{A \cap B}(u) = \min\{\mu_A(u), \mu_B(u)\}</math>

;Standard union

:<math>\mu_{A \cup B}(u) = \max\{\mu_A(u), \mu_B(u)\}</math>

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.

Fuzzy complements

μ_A(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. (μ_A(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(μ_A(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 (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 .

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: μ_{A ∩ B}(x) = i[μ_A(x), μ_B(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 (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]).