Fuzzy set operations are a generalization ofcrisp setoperations forfuzzy sets. There is in fact more than one possible generalization. The most widely used operations are calledstandard fuzzy set operations; they comprise:fuzzy complements,fuzzy intersections, andfuzzy unions.

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

${\displaystyle {\mu }_{\mathrm{\neg }A}(u)=1-{\mu }_A(u)}$

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 calledDe Morgan Tripletiff

• i is at-norm,
• u is at-conorm (aka s-norm),
• n is astrong negator,

so that for allx,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.

μ_A(x) is defined as the degree to whichx belongs toA. Let∁A denote a fuzzy complement ofA of typec. Thenμ_∁A(x) is the degree to whichx belongs to∁A, and the degree to whichx does not belong toA. (μ_A(x) is therefore the degree to whichx does not belong to∁A.) Let a complement∁A be defined by a function

c : [0,1] → [0,1]

For allx ∈U:μ_∁A(x) =c(μ_A(x))

Axiom c1.Boundary condition

c(0) = 1 andc(1) = 0

Axiom c2.Monotonicity

For alla,b ∈ [0, 1], ifa <b, thenc(a) >c(b)

Axiom c3.Continuity

c is continuous function.

Axiom c4.Involutions

c is aninvolution, which means thatc(c(a)) =a for eacha ∈ [0,1]

c is astrongnegator (akafuzzy 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]

The intersection of two fuzzy setsA andB 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 allx ∈U:μ_{A ∩B}(x) =i[μ_A(x),μ_B(x)].

Axiom i1.Boundary condition

i(a, 1) =a

Axiom i2.Monotonicity

b ≤d impliesi(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) ifa₁ <a₂ andb₁ <b₂

Axioms i1 up to i4 define at-norm (akafuzzy intersection). The standard t-norm min is the only idempotent t-norm (that is,i (a₁,a₁) =a for alla ∈ [0,1]).^[2]

The union of two fuzzy setsA andB is specified in general by a binary operation on the unit interval function of the form

u:[0,1]×[0,1] → [0,1].

For allx ∈U:μ_{A ∪B}(x) =u[μ_A(x),μ_B(x)].

Axiom u1.Boundary condition

u(a, 0) =u(0 ,a) =a

Axiom u2.Monotonicity

b ≤d impliesu(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 andb₁ <b₂ impliesu(a₁,b₁) <u(a₂,b₂)

Axioms u1 up to u4 define at-conorm (akas-norm orfuzzy 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 on fuzzy sets are operations by which several fuzzy sets are combined in a desirable way to produce a single fuzzy set.

Aggregation operation onn fuzzy set (2 ≤n) is defined by a function

h:[0,1]ⁿ → [0,1]

Axiom h1.Boundary condition

h(0, 0, ..., 0) = 0 andh(1, 1, ..., 1) = one

Axiom h2.Monotonicity

For any pair <a₁,a₂, ...,a_n> and <b₁,b₂, ...,b_n> ofn-tuples such thata_i,b_i ∈ [0,1] for alli ∈N_n, ifa_i ≤b_i for alli ∈N_n, thenh(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.

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

• Klir, George J.; Bo Yuan (1995).Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall.ISBN 978-0131011717.

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

• Fuzzy logic

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