Construction of the membership surface of imprecise vector

In this article, a method has been developed to construct the membership surface of imprecise vector based on Randomness-Impreciseness Consistency Principle. The Randomness-Impreciseness Consistency Principle leads to define a normal law of impreciseness using two different laws of randomness. The Dubois-Prade left and right reference functions of an imprecise number are distribution function and complementary distribution function respectively. In this article, based on the Randomness-Impreciseness Consistency Principle we have successfully obtained the membership surface of imprecise vector and demonstrated with the help of numerical examples.


Introduction
L(x) being a continuous non-decreasing function in the interval [a, b], and R(x) being a continuous non-increasing function in the interval [b, c], with L(a) = R(c) = 0 and L (b) = R(b) = 1. Dubois and Prade named L(x) as left reference function and R(x) as right reference function of the concerned fuzzy number. A continuous non-decreasing function of this type is also called a distribution function with reference to a Lebesgue-Stieltjes measure (De Barra 1987).
In this article, on the simple assumption that the Dubois-Prade left reference function is a distribution function, and similarly the Dubois-Prade right reference function is a complementary distribution function, we are going to demonstrate the method of obtaining the membership surface of an imprecise vector. Here the term imprecise is used instead of fuzzy because, in the Zadehian theory of fuzzy sets there are two flaws (Baruah 2011a, Baruah 2012). First, it had been accepted that the fuzzy sets do not in any way conform to the classical measure theoretic formalisms. Secondly, it had been agreed upon that given a fuzzy set neither its intersection with its complement is the null set, nor its union with the complement is the universal set. The Zadehian definition of complement of a fuzzy set is defective (Baruah 1999). In the Zadehian definition of complementation, fuzzy membership function and fuzzy membership value have been taken to be the same, and that is where the defect lies. Indeed fuzzy membership function and fuzzy membership value are two different things for the complement of a normal fuzzy set (Baruah 2011a). Instead of saying (Baruah 2011b;Baruah 2011c) that the mathematics of fuzziness has been incorrectly explained, Baruah has started the whole process anew, introducing the theory of imprecise sets, which might initially look similar to the theory of fuzzy sets. Baruah (2011b) has successfully shown the construction of a normal imprecise number with the help of the operation of superimposition of real intervals. Das et al. (2013) has shown the construction of normal imprecise number using data from earthquake waveform and has studied the pattern of the membership curve of the waveform. In this article, instead of superimposing real intervals, it will be discussed about how to obtain the membership surface of a two dimensional imprecise vector if we superimpose some plates in the two dimensional plane. In Figure 1 two plates are superimposed restricting the condition that the intersection of the two plates is not void.

The mathematical explanation of imprecise vector
We can easily visualize in Figure 1, that the probability of the shaded area is 1 and the probability for the unshaded area of the plates will be ½. But if the number of superimposition is large then it will be very difficult to obtain the probabilities by simply observing the imposition of the plates. So, in that situation a different technique can be used to obtain the probabilities when the number of operation of superimposition is very large. At first, it is discussed, about the operation of superimposition in the two dimensional case when the variable X is imprecise but Y is not imprecise.
Similarly, we can also show that, the membership surface of a normal imprecise vector N = [(0, α), (0, β), (0, γ)] can be expressed as Here, we have also considered the value of x is zero. But instead of zero if we consider any precise value of x, then in the above membership surface only the value of x will be changed. Now, we are going to discuss about the method how to obtain the membership surface of the vector (X, Y), where the variables x and y both are imprecise.
Consider an imprecise vector (X, Y), where X and Y are imprecise represented by X = [a, b, c] and Y = [p, q,  r] respectively. Assume that X and Y are independently distributed. Let the membership function of X and Y be μ X|Y (x, y) and μ Y|X (x, y) as mentioned below Then the membership surface of the imprecise vector (X, Y) can be obtained as follows For a three dimensional imprecise vector (X, Y, Z), where the membership functions of X, Y and Z are as mentioned below: Then the membership surface of the imprecise vector (X, Y, Z) will be as shown below: otherwise: