Algebraic Properties of \( Z \)-Numbers Under Multiplicative Arithmetic Operations

  • R. A. AlievEmail author
  • A. V. Alizadeh
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 896)


Prof. L.A. Zadeh introduced the concept of a Z-number for description of real-world information. A Z-number is an ordered pair \( Z = (A,B) \) of fuzzy numbers \( A \) and \( B \) used to describe a value of a random variable \( X \). \( A \) is an imprecise estimation of a value of \( X \) and \( B \) is an imprecise estimation of reliability of \( A \). A series of important works on computations with Z-numbers and applications were published. However, no study exists on properties of operations of Z-numbers. Such theoretical study is necessary to formulate the basics of the theory of Z-numbers. In this paper we prove that Z-numbers exhibit fundamental properties under multiplicative arithmetic operations.


Fuzzy arithmetic Probabilistic arithmetic Associativity law Distributivity law Z-number 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Joint MBA ProgramAzerbaijan State Oil and Industry UniversityBakuAzerbaijan
  2. 2.Department of Control and Systems EngineeringAzerbaijan State Oil and Industry UniversityBakuAzerbaijan

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