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Probability and Fuzziness: A Case Study

Probability and Fuzziness: A Case Study

Mamoni Dhar (Science College, Kokrajhar, India)
Copyright: © 2022 |Pages: 15
DOI: 10.4018/978-1-7998-7979-4.ch020
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Abstract

From the beginning of the theory of fuzzy sets, it has been a matter of concern for many researchers in the field to find an appropriate connection between probability and possibility. One such effort can be found in the randomness-fuzziness consistency principle where a fuzzy number of the type [α,β,γ] is expressed as a combination of two probability laws. The main intention of this work is to show that a fuzzy number of the type [α,β,β] or [β,β,γ] can be expressed by one probability law with the help of superimposition of uniformly fuzzy intervals.
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2. Definitions And Preliminaries

In this section some basic definitions and results are pt forward for better understanding of the underlying concept.

2.1 Crisp Set

A classical or conventional or crisp set contains objects that satisfy some precise properties. For example

  • A= numbers between 3 and 5

  • = {x∈R / 3 ≤ x ≤ 5}

    978-1-7998-7979-4.ch020.m01

2.2 Fuzzy Set

Fuzzy sets are sets whose elements have degrees of membership. Unlike classical sets where an element either belongs or does not belong or does not belong to the set, in a fuzzy set there is a gradual assessment of the membership of elements in a set. In other words, fuzzy set contains objects having imprecise properties with varying degree. To be more precise if X is an universe of discourse and x is any element of X, then the fuzzy set, A defined on X may be defined as an ordered pairA = {𝜇A(x), x; ∀x∈X}where 𝜇A: X→[0,1], 𝜇A(x) is the membership function or degree of membership of x in A or degree of belongingness of x in A or degree of possessing some imprecise property represented by A.

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