Finding a Link between Randomness and Fuzziness

If sample realizations are intervals, if the upper and the lower boundaries of such intervals are realizations of two independently distributed random variables, the two probability laws together lead to some interesting assertions. In this article, we shall attempt to remove certain confusions regarding the relationship between probability theory and fuzzy mathematics.


Introduction
Let X and Y be two random variables independently distributed real intervals [ ]  α β a two- population event (t.p.e.), we are interested to calculate the probability associated with such a t.p.e., and thereafter to explain its physical significance.

Simple and Compound Two-Population Events
In other words, in terms of intervals The identities (1) ( ) In the same way, [ ] ( ) Identities ( 3) and (4) give us the probabilities of union and intersection respectively of two simple two population events.
Our discussions on probability of a t.p.e.ends here.In what follows, we shall explain the physical significance of identity (2).It would thereafter be shown that a t.p.e. is indeed a possibilistic event.

Physical Significance of a Two-Population Event
Zadeh ( [1]) introduced the mathematics of fuzziness long forty nine years ago.It has made inroads into almost all applied branches of knowledge.However, most of the mathematicians shy away from this branch of mathematics for various reasons.One such is the belief that randomness and fuzziness are two independent concepts.
In this section we shall show the following: a) Randomness can actually be at the root of fuzziness.b) Two independent probability laws can define a fuzzy number.First, let us review certain necessary definitions.

Some Basic Definitions Related to Fuzziness
Let E be the universe of discourse.A fuzzy set A in E is characterized by a membership function

A x υ
for x E ∈ represents the grade of membership of x in A. Thus a fuzzy set A is defined as An α-cut of a fuzzy set is an ordinary set of elements with membership grade greater than or equal to a threshold α, 0 1 α ≤ ≤ .Thus an α-cut A α of a fuzzy set A is characterized by [2] ( ) { } A fuzzy set is said to be convex if all its α-cuts are convex sets [2].
A fuzzy number is a convex normalized fuzzy set A defined on the real line E such that i) there exists an 0 x E ∈ such that ( ) x a b ∈ is known as left reference function and x b c ∈ is known as the right reference function of the fuzzy number [3].For such an L-R (left-right) fuzzy number, the left reference function is non-decreasing and the right reference function is non-increasing.Finally, the membership function of a fuzzy set is viewed as its possibility distribution [4].

Set Superimposition
When we overwrite, the overwritten portion looks darker.Indeed, the doubly represented portion looks doubly dark.To explain this, we would need a set operation (S) of superimposition of two sets A and B defined as: where ( ) ( ) are two real intervals such that ( ) , we would get from ( 5) where ( ) ( ) max , a a a = , ( ) ( ) , and ( ) ( ) . Identity (6) explains why-if two line segments are superimposed, the common portion looks doubly dark [5].Mazarbhuiya et al. used this operation to define fuzzy arithmetic operations [6] and to solve a fuzzy equation [7].Set superimposition is also used in mining temporal data ([8] [9]).In ( [10] [11]), Baruah tries to establish link between probability law and possibility law.

Let now, [ ] ( )
, a b be two fuzzy sets with constant membership 1/2 everywhere.Following (6) we can write: An obvious extension of ( 7) is: if At this point, we would need a classical result from order statistics to proceed further.

The Glivenko-Cantelli Theorem
Define ( ) where 1 2 , , , n x x x  are realizations of a random variable X, and ( ) ( ) , , , n x x x  are 1 2 , , , n x x x  arranged in ascending order.
F(x) is random here with where ( ) F x is the underlying theoretical distribution function.
( ) n F x being an empirical distribution func- tion.
In such a case, the Glivenko-Cantelli Theorem on order statistics [12] states that ( ) ( ) uniformly in x with probability 1.

The Glivenko-Cantelli Theorem Applied to Superimposed Equally Fuzzy Sets
Consider the identity (8) once again.Let X and Y be random and independently distributed in two real intervals Application of Glivenko-Cantelli Theorem on identity (8) now after the imposition of randomness, as n → ∞ , on n superimposed equally fuzzy sets  , gives us the following: i) ( ) In other words, a probability distribution function We can therefore conclude that randomness can actually be at the root of fuzziness, and that two independent probability laws can define a fuzzy number.

Discussions
The possibility distribution function in Ω = Ω Ω  .The fuzzy mathematicians have all along been trying to impose a PΩ on the same Ω on which FΩ is defined.For this mistake, they have all along been saying that fuzziness and randomness are two independent concepts.
In fact, every sample realization , 1, 2, , , , a b c one can define possibilistic events, which are indeed t.p.e.'s defined in Section 1 of this article.Probabilities of such t.p.e.'s or possibilistic events, simple or compound, can be found following (2), ( 3) and (4).As long as one continues to impose a single probability law in a Ω where a possibility distribution is defined, the question of involving randomness to define fuzziness cannot arise.Independent probability laws on 1 Ω and 2 Ω , can finally define a possibility law on 1 2 Ω Ω  .In other words, possibility can indeed be a measure, and indeed it is a product measure.
It is to be noted that identities (2) and ( 10) are directly related.In fact, (10) explains the physical significance of (2).We would like to conclude our discussions with an important comment.The fuzzy mathematicians invariably use the triangular fuzzy number in computations.No fuzzy mathematician has as yet supplied a logical answer as to why they use the triangular fuzzy number only.An obvious answer hidden in escapism is that it is simple to use.Simplicity of the triangular number in fact lies elsewhere.If .

G
are uniform probability distribution functions, we simply get the triangular number.What we mean just as the uniform law is the simplest in field of probability, so is the triangular number the simplest in fuzzy mathematics.
In other words, the simplicity of the triangular fuzzy number is rooted at the simplicity of the uniform probability law.This is the actual reason why the triangular number is simple.
To illustrate the concept, let us take a simple example.Let illustrates the fact that randomness can actually be at the root of fuzziness and two probability laws are sufficient to define a fuzzy number.
distribution functions of X and Y respectively.Let simple probabilistic events, [ ] , α β is totally reference as an event because its boundaries are from two probabilistic populations.For simplicity let us call [ ]

=
be a t.p.e.It is easy to see that distribution functions of X and Y respectively.

,
a b can indeed be the left reference function, and a complementary probability distribution function or survival function ( ) 1 G y − in [ ] , b c can similarly be the right reference function of a fuzzy number.
is the membership function of a fuzzy number [ ] , , a b c , can be viewed as two different functions, one in the interval

1 , 1 ,
of as values of unit fuzzy membership, around which intervals expressed as [ ] would give us fuzzy numbers.Accordingly, in [ ] 1 a b there can be a probability law, while in [ ] 1 b c there can be another probability law.Within the fuzzy number [ ]

1 f 1 g
-Prade right reference function.If we observe clearly, then we find that ( ) F x is actually a proba- bility distribution function whose probability density function is ( ) or survival function whose probability density function is ( ) and (2) are rather elementary.But (2) in particular can have far reaching consequences, which we are going to assert later.The t.p.e.[ ]