The Inclusion-Exclusion Principle On the Set of IF-sets Inclusion-Exclusion Principle On the Set of IF-sets Jana Kelemenová FacultyofNaturalSciences,MatejBelUniversity,Slovakia Abstract
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The Inclusion-Exclusion Principle On the Set ofIF-sets
Jana Kelemenová
Faculty of Natural Sciences, Matej Bel University, Slovakia
Abstract
P. Grzegorzewski [3] has worked the probability ver-sion of the inclusion-exclusion principle and made ageneralization for IF-events. He had applied twoversions of the generalized formula, correspondingto different t-conorms and so defined the union ofIF-events. This paper contains the generalization ofthe Grzegorzewski theorem. We prove it for map-pings from the set of IF sets to the unit interval([2],[1]). Similar generalizations are presented in [4] and[5].
K. Atanassov introduced in [1] the notion of an IF- set as a mapping
A = (µA, νA).
µA, νA : Ω→< 0, 1 >
are such that µa + νa ≤ 1, F is the set of all IF-sets such that A = (µA, νA). He considered alsothe following operations on F :
A ∩B = (µA ∧ µB , νA ∨ νB)
= (min(µA, µB),max(νA, νB))
A ∪B = (µA ∨ µB , νA ∧ νB)
= (max(µA, µB),min(νA, νB))
for any A = (µA, νA), B = (µB , νB) ∈ F .P. Grzegorzewski in [3] considers a classical prob-
ability space (Ω,S,P), when Ω is a non-empty set,S is a σ-algebra of subsets of Ω, and P : S → 〈0, 1〉is a probability measure, i.e. P is σ−additive andP(Ω) = 1. He works with IF-events, that are suchIF-sets A = (µA, νA) that µA, νA : Ω → 〈 0, 1〉are S- measureable, i.e. B ⊂ R,B is a Borel set⇒ µ−1
A (B) ∈ S, ν−1A (B) ∈ S.
P.Grzegorzewski considered the mapping m :F → 〈0, 1〉 , defined by the equality:
P(A) = P(µA, νA) =(∫
ΩµAdP, 1−
∫ΩνAdP
).
He extended the inclusion-exclusion principle forsuch mappings, i.e.
P
(n⋃i=1
Ai
)=n∑k=1
∑j1,...,jk
(−1)k+1m (Aj1 ∩ . . . ∩Ajk) ,
e.g.
P (A1 ∪A2) = P(A1) + P(A2)− P (A1 ∩A2) (I)
or
P (A1 ∪A2 ∪A3) = P(A1) + P(A2) + P(A3)−
−P (A1 ∩A2)− P (A1 ∩A3)− P (A2 ∩A3) +
+P (A1 ∩A2 ∩A3)
etc.In the paper, we prove the inclusion-exclusion
principle for any strongly additive mappings m :F → 〈0, 1〉 i.e. mappings satisfying (I). The resultis a generalization of the result of [3]. E.g. the in-ex principle works for the mappings m[(A),m](A) :F → 〈0, 1〉defined by
m[(A) = 12
∫ΩµAdP + 1
2
∫ΩνAdP,
m](A) = 34
∫ΩµAdP + 1
4
∫ΩνAdP,
hence also for P : F → 〈0, 1〉 × 〈0, 1〉 defined by theequality
P(A) =(m[(A),m](A)
).
Of course, the mapping cannot be covered by theGrzegorzewski result. Recall that another general-izations of [1] will be published in [4] and [5].
2. Inclusion-exclusion principle for IF-sets
Theorem 1 Let F be the set of pairs A = (µA, νA);
A ≤ B
µA ≤ µB , νA ≥ νB0 = (0,1)
µA, νA : Ω→< 0, 1 >,µA+νA ≤ 1. Let the mappingm : F −→ 〈0, 1〉 be strongly additive, that is
and similarly for any m (a1 ∪ a2 ∪ . . . ∪ an) . Inthis paper we generalize the principle for stronglyadditive states defined on the set of IF-sets.
References
[1] K. Atanassov,Intuitionistic Fuzzy Sets: Theoryand Applications, Physica- Verlag, New York,1999.
[2] B. Riečan, D. Mundici, Probability on MV alge-bras, Handbook of Measure Theory, Amsterdam,New York, pages 869 - 909, 2002.
[3] P. Grzegorzewski, The Inclusion-Exclusion Prin-ciple for IF - Events, Information Sciences, Vol-ume 181, Issue 3, pages 536-546, 2011.
[4] M. Kuková, The Inclusion-Exclusion Principlefor IF-events, to appear in Information Sciences,2011.
[5] J. Kelemenová, The Inclusion-Exclusion Princi-ple in semigroups. To appear in Developmentsin Fuzzy Sets, Intuitionistic Fuzzy Sets, Gen-eralized Nets and Related Topics, proceedingsof the 9th international workshop on intuition-istic fuzzy sets and generalized nets (IWIFSGN2010), IBS PAN - SRI PAS, Warsaw, 2011.