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

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

μ_A(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. (μ_A(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(μ_A(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 .^[2]

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: μ_{A ∩ B}(x) = i[μ_A(x), μ_B(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]).^[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: μ_{A ∪ B}(x) = u[μ_A(x), μ_B(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]).^[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, ..., 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.

References

^ Ismat Beg, Samina Ashraf: Similarity measures for fuzzy sets, at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016
^ ^a ^b ^c Günther Rudolph: Computational Intelligence (PPS), TU Dortmund, Algorithm Engineering LS11, Winter Term 2009/10. Note that this power point sheet may have some problems with special character rendering

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

[1] Ismat Beg, Samina Ashraf: Similarity measures for fuzzy sets, at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016

[GR2009-2010-2] Günther Rudolph: Computational Intelligence (PPS), TU Dortmund, Algorithm Engineering LS11, Winter Term 2009/10. Note that this power point sheet may have some problems with special character rendering

[1]

[2]