Fuzzy Set Operations

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A fuzzy set operation is an operator 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 Standard complementcA(x) = 1 − A(x)

Standard intersection(A ∩ B)(x) = min B(x)

Standard union(A ∪ B)(x) = max B(x)

Fuzzy complements A(x) is defined as the degree to which x belongs to A. Let cA denote a fuzzy complement of A of type c. Then cA(x) is the degree to which x belongs to cA, and the degree to which x does not belong to A. (A(x) is therefore the degree to which x does not belong to cA.) Let a complement cA be defined by a function

c : →

c(A(x)) = cA(x)

Axioms for fuzzy complements Axiom c1. Boundary conditionc(0) = 1 and c(1) = 0

Axiom c2. MonotonicityFor all a, b ∈ 1, if a ≤ b, then c(a) ≥ c(b)

Axiom c3. Continuityc is continuous function.

Axiom c4. Involutionsc is an involution, which means that c(c(a)) = a for each a ∈

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:× → .

(A ∩ B)(x) = i B(x) for all x.

Axioms for fuzzy intersection Axiom i1. Boundary conditioni(a, 1) = a

Axiom i2. Monotonicityb ≤ d implies i(a, b) ≤ i(a, d)

Axiom i3. Commutativityi(a, b) = i(b, a)

Axiom i4. Associativityi(a, i(b, d)) = i(i(a, b), d)

Axiom i5. Continuityi is a continuous function

Axiom i6. Subidempotencyi(a, a) ≤ a

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:× → .

(A ∪ B)(x) = u B(x) for all x

Axioms for fuzzy union Axiom u1. Boundary conditionu(a, 0) = a

Axiom u2. Monotonicityb ≤ d implies u(a, b) ≤ u(a, d)

Axiom u3. Commutativityu(a, b) = u(b, a)

Axiom u4. Associativityu(a, u(b, d)) = u(u(a, b), d)

Axiom u5. Continuityu is a continuous function

Axiom u6. Superidempotencyu(a, a) > a

Axiom u7. Strict monotonicitya1 < a2 and b1 < b2 implies u(a1, b1) < u(a2, b2)

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:n →

Axioms for aggregation operations fuzzy sets Axiom h1. Boundary conditionh(0, 0, ..., 0) = 0 and h(1, 1, ..., 1) = 1

Axiom h2. MonotonicityFor any pair and of n-tuples such that ai, bi ∈ 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. Continuityh is a continuous function.

See also

Fuzzy logic
Fuzzy set
T-norm