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 Œ 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 Œ 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 {a‘A « Ea _ f, a Œ [0, 1]}

= sup {m ‘w Œ 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 = {x‘x ² 3} Poss = 1 .

Possibility  tells us about the possibility of "not A", hence the necessity of the occurrence of A,

Nec = 1 - Poss .