Fuzzy set operations#Fuzzy 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

:$\backslash mu\_\{\backslash lnot\{A\}\}(u)\; =\; 1\; -\; \backslash mu\_A(u)$

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

;Standard intersection

:$\backslash mu\_\{A\; \backslash cap\; B\}(u)\; =\; \backslash min\backslash \{\backslash mu\_A(u),\; \backslash mu\_B(u)\backslash \}$

;Standard union

:$\backslash mu\_\{A\; \backslash cup\; B\}(u)\; =\; \backslash max\backslash \{\backslash mu\_A(u),\; \backslash mu\_B(u)\backslash \}$

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).Ismat Beg, Samina Ashraf: [https://www.researchgate.net/publication/228744370_Similarity_measures_for_fuzzy_sets 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.

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))

;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 .Günther Rudolph: [https://ls11-www.cs.tu-dortmund.de/people/rudolph/teaching/lectures/CI/WS2009-10/lec06.pps 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

{{main|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: μ_{A ∩ B}(x) = i[μ_A(x), μ_B(x)].

;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]).

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)].

;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]).

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]

;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.

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

• {{cite book |last1=Klir |first1=George J. |authorlink1=George Klir |author2=Bo Yuan |title=Fuzzy Sets and Fuzzy Logic: Theory and Applications |isbn=978-0131011717 |date=1995 |publisher=Prentice Hall}}

{{Reflist}}

:* [https://web.archive.org/web/20071127005930/http://www-bisc.cs.berkeley.edu/Zadeh-1965.pdf L.A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965]

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Category:Fuzzy logic