Fuzzy set operations

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

¬ μ A ( u ) = 1 − μ 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

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.