Conditional logical entropy of fuzzy σ-algebras

Article type: Research Article

Authors: Ebrahimzadeh, Abolfazl^{a; *} | Jamalzadeh, Javad^b

Affiliations: [a] Young Researchers and Elite Club, Zahedan Branch, Islamic Azad University, Zahedan, Iran | [b] Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

Correspondence: [*] Corresponding author. Abolfazl Ebrahimzadeh, Young Researchers and Elite Club, Zahedan Branch, Islamic Azad University, Zahedan, Iran. Tel.: +98 9159619270; Fax: +98 5431135244; E-mail: a.ebrahimzade@iauzah.ac.ir.

Abstract: In the papers of Mona Khare the notion of Shannon entropy of fuzzy σ-algebras on an F-probability measure space has been defined and some results concerning this measure have been presented. The purpose of the present paper is to provide analogues of these results for the case of the logical entropy. We define the logical entropy and conditional logical entropy of fuzzy σ-algebras on the F-probability measure space and prove the basic properties of these measures. We also define the concept of logical mutual information on the F-probability measure space and prove some properties concerning this measure. Shannon entropy of fuzzy σ-algebras can be replaced by logical entropy of fuzzy σ-algebras as a measure of information of experiments whose outcomes are fuzzy events.

Keywords: Fuzzy σ-algebra, logical entropy, conditional logical entropy, logical mutual information

DOI: 10.3233/JIFS-162303

Journal: Journal of Intelligent & Fuzzy Systems, vol. 33, no. 2, pp. 1019-1026, 2017