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 an element in the U universe (e.g. value)

Standard complement

$\mu_A(u) = 1 - \mu_A(u)$

Standard intersection

Standard union

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 : [0,1] → [0,1]

c(A(x)) = cA(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]

Fuzzy intersections

Main article: T-norm

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].

(A ∩ B)(x) = i[A(x), B(x)] for all 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

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].

(A ∪ B)(x) = u[A(x), B(x)] for all 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

Axiom u7. Strict monotonicity

a₁ < 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:[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) = 1

Axiom h2. Monotonicity

For any pair <a₁, a₂, ..., a_n> and <b₁, b₂, ..., b_n> of n-tuples such that a_i, b_i ∈ [0,1] for all i ∈ N_n, if a_i ≤ b_i for all i ∈ N_n, then h(a₁, a₂, ...,a_n) ≤ h(b₁, b₂, ..., b_n); that is, h is monotonic increasing in all its arguments.

Axiom h3. Continuity

h is a continuous function.

See also

• Fuzzy logic
• Fuzzy set
• T-norm
• Type-2 fuzzy sets and systems

External References

• L.A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965