8.6.7. Possibility Distribution
Let wo be an unknown value ranging over a set W, and let the piece of imprecise information be given as a set E, i.e., wo E is known for sure and E ³ 2. If we ask whether another set A contains wo, there can be two possible answers:
if A « E = f then it is impossible that wo A
if A « E _ f then it is possible
Formally, we obtain a mapping
PossE : 2W Æ [0, 1] , PossE(A) =
where 1 indicate "possible" and 0 "impossible".
When E becomes a fuzzy set , we define
Poss : 2W Æ [0, 1]
Poss = sup {aA « Ea _ f, a [0, 1]}
= sup {m w A}
Hence given a fuzzy set the small positive integer
= (1, 1, 0.8, 0.6, 0.4, 0.2) .
Given A = {3} , the possibility is 0.8
A = {xx ² 3} Poss = 1 .
Possibility tells us about the possibility of "not A", hence the necessity of the occurrence of A,
Nec = 1 - Poss .