The Inclusion-Exclusion Principle for IF-States

Document Type : Research Paper


1 Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419, USA

2 Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovskeho 40, Banska Bystrica, Slovakia


Applying two definitions of the union of IF-events, P. Grzegorzewski gave two generalizations of the inclusion-exclusion principle for IF-events.
In this paper we prove an inclusion-exclusion principle for IF-states based on a method which can also be used to prove Grzegorzewski's inclusion-exclusion principle for probabilities on IF-events.
Finally, we give some applications of this principle by extending some results regarding the classical probabilities to the case of the IF-states.


K. Atanassov, {it Intuitionistic fuzzy sets}, Fuzzy Sets and Systems, {bf 20}textbf{(1)} (1986), 87-96.
K. Atanassov, {it Intuitionistic fuzzy sets: theory and applications}, Physica Verlag,
New York, (1999).
K. Atanassov and B. Riev can, {it On two operations over intuitionistic fuzzy sets}, Journal of Applied Mathematics, Statistics and Informatics, {bf 2}textbf{(2)} (2006), 145-148.
L. C. Ciungu and B. Riev can, {it General form of probabilities on IF-Sets}, In: Proc. WILF 2009, Palermo, Italy. Lecture Notes in Computer Science 5571, Springer, Berlin, (2009), 101-107.
L. C. Ciungu and B. Riev can, {it Representation theorem for probabilities on IFS-events},
Information Sciences, {bf 180}textbf{(5)} (2010), 793-798.
P. Grzegorzewski and E. Mr' owka, {it Probability of intuitionistic fuzzy events}, In: Soft Methods in Probability, Statistics and Data Analysis (P. Grzegorzewski et al., eds.) Physica Verlag, New York, (2002), 105-115.
P. Grzegorzewski, {it The inclusion-exclusion principle on IF-events}, Information Sciences,
{bf 181}textbf{(3)} (2011), 536-546.
W. L. Hung and M. S. Yang, {it On the J-divergence of intuitionistic fuzzy sets with its application to pattern recognition}, Information Sciences, {bf 178}textbf{(6)} (2008), 1641-1650.
M. Kukov'a, {it The inclusion-exclusion principle for $L$-states and IF-events}, Information Sciences,
{bf 224}textbf{(1)} (2013), 165-169.
M. Kukov'a and M. Navara, {it Principles of inclusion and exclusion for fuzzy sets}, Fuzzy Sets and Systems,, 2013.
J. Pitman, {it Probability}, Springer-Verlag, New York, 1993.
B. Riev can, {it A descriptive definition of the probability on intuitionistic fuzzy sets},
In: Proc. EUSFLAT'2003 (Wagenknecht, M. and Hampel, R., eds.), Zittau-Goerlitz Univ. Appl. Sci., Dordrecht, (2003),  263-266.
B. Riev can, {it Representation of probabilities on IFS events}, Advanced in Soft Computing, Soft Methodology and Random Information Systems (M. L$acute{rm o}$pez et. al., eds) Springer, Berlin, (2004), 243-246.
B. Riev can, {it On a problem of Radko Mesiar: general form of IF probabilities}, Fuzzy Sets and Systems,
{bf 152}textbf{(11)} (2006), 1485-1490.
B. Riev can, {it M-probability theory on IF-events}, In: Proc. EUSFLAT'2007 (M. v Stv epniv cka,
V. Nov$acute{rm a}$k, and U. Bodenhofer eds.), Universitas Ostraviensis, (2007), 227-230.
L. A. Zadeh, {it Probability measures of fuzzy events}, Journal of Mathematical Analysis and Applications,
{bf 23}textbf{(2)} (1968), 421-427.
L. A. Zadeh, {it Toward a generalized theory of uncertainty (GTU) - an outline}, Information Sciences,
{bf 172}textbf{(1-2)} (2005), 1-40.
L. A. Zadeh, {it Is there a need for fuzzy logic?}, Information Sciences, {bf 178}textbf{(13)} (2008), 2751-2779.