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
 * $$\mu_{\lnot{A}}(u) = 1 - \mu_A(u)$$

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


 * Standard intersection
 * $$\mu_{A \cap B}(u) = \min\{\mu_A(u), \mu_B(u)\}$$


 * Standard union
 * $$\mu_{A \cup B}(u) = \max\{\mu_A(u), \mu_B(u)\}$$

In general, the triple (i,u,n) is called De Morgan Triplet iff so that for all x,y ∈ [0, 1] the following holds true:
 * i is a t-norm,
 * u is a t-conorm (aka s-norm),
 * n is a strong negator,
 * u(x,y) = n( i( n(x), n(y) ) )

(generalized De Morgan relation). 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 (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: 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 (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]).

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 (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]).

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  and  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.