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
:$\mu_(u) = 1 - \mu_A(u)$
The complement is sometimes denoted by ∁A or A^∁ instead of ¬A.

;Standard intersection
:$\mu_(u) = \min\$

;Standard union
:$\mu_(u) = \max\$

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'' ∈ , 1the following holds true:
:''u''(''x'',''y'') = ''n''( ''i''( ''n''(''x''), ''n''(''y'') ) )
(generalized De Morgan relation).Ismat Beg, Samina Ashraf
Similarity measures for fuzzy sets
at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016 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'' : ,1→ ,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'' ∈ , 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'' ∈ ,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 .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

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'': ,1� ,1→ ,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'' (''a''₁, ''b''₁) < ''i'' (''a''₂, ''b''₂) if ''a''₁ < ''a''₂ and ''b''₁ < ''b''₂

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'' (''a''₁, ''a''₁) = ''a'' for all ''a'' ∈ ,1.

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'': ,1� ,1→ ,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''
:''a''₁ < ''a''₂ and ''b''₁ < ''b''₂ implies ''u''(''a''₁, ''b''₁) < ''u''(''a''₂, ''b''₂)

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 ∈ ,1.

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'': ,1sup>''n'' → ,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 <''a''₁, ''a''₂, ..., ''a''_''n''> and <''b''₁, ''b''₂, ..., ''b''_''n''> of ''n''-tuples such that ''a''_''i'', ''b''_''i'' ∈ ,1for all ''i'' ∈ ''N''_''n'', if ''a''_''i'' ≤ ''b''_''i'' for all ''i'' ∈ ''N''_''n'', then ''h''(''a''₁, ''a''₂, ...,''a''_''n'') ≤ ''h''(''b''₁, ''b''₂, ..., ''b''_''n''); that is, ''h'' is monotonic increasing in all its arguments.

;Axiom h3. ''Continuity''
:''h'' is a continuous function.

See also

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

Further reading

*

References

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

{{DEFAULTSORT:Fuzzy Set Operations
Fuzzy logic