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 any element (e.g. value) in the U universe: u ∈ U.

Standard complement
\( {\displaystyle \mu _{\lnot {A}}(u)=1-\mu _{A}(u)} \)

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

Standard intersection
\( {\displaystyle \mu _{A\cap B}(u)=\min\{\mu _{A}(u),\mu _{B}(u)\}} \)

Standard union
\( {\displaystyle \mu _{A\cup B}(u)=\max\{\mu _{A}(u),\mu _{B}(u)\}} \)

In general, the triple (i,u,n) is called De Morgan Triplet iff

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).[1] 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 c2 has at least one fixpoint a* with c(a*) = a*, and if axiom c3 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 .[2]
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].

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

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]).[2]
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

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 intersection). The standard t-conorm max is the only idempotent t-conorm (i. e. u (a1, a1) = a for all a ∈ [0,1]).[2]
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 <a1, a2, ..., an> and <b1, b2, ..., bn> 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.