Editing Possibility theory (section)

==Necessity==

Whereas [[probability theory]] uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the ''possibility'' and the ''necessity ''of the event. For any set <math>U</math>, the necessity measure is defined by

:<math>N(U) = 1 - \Pi(\overline U)</math>.

In the above formula, <math>\overline U</math> denotes the complement of <math>U</math>, that is the elements of <math>\Omega</math> that do not belong to <math>U</math>. It is straightforward to show that:

:<math>N(U) \leq \Pi(U)</math> for any <math>U</math>

and that:

:<math>N(U \cap V) = \min ( N(U), N(V))</math>.

Note that contrary to probability theory, possibility is not self-dual. That is, for any event <math>U</math>, we only have the inequality:

:<math>\Pi(U) + \Pi(\overline U) \geq 1</math>

However, the following duality rule holds:

:For any event <math>U</math>, either <math>\Pi(U) = 1</math>, or <math>N(U) = 0</math>

Accordingly, beliefs about an event can be represented by a number and a bit.