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]

v t e Non-classical logic
Intuitionistic	Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory
Fuzzy	Degree of truth Fuzzy rule Fuzzy set Fuzzy finite element Fuzzy set operations
Substructural	Structural rule Relevance logic Linear logic
Paraconsistent	Dialetheism
Description	Ontology (computer science) Ontology language
Many-valued	Three-valued Four-valued Łukasiewicz
Digital logic	Three-state logic Tri-state buffer Four-valued Verilog IEEE 1164 VHDL
Others	Dynamic semantics Inquisitive logic Intermediate logic Non-monotonic logic