Relationship between sigma-entropy and sigma-similarity measure of fuzzy sets
Purpose - To provide a simplified proof for Liu's results concerning the relationship between sigma-entropy and sigma-similarity measure. Design/methodology/approach - First obtains Liu's clear and simple functional expression of sigma-similarity measures based on the structure of sigma-entropy by using expressions of equivalent functions of sigma-entropy. Then, by employing the general expression of equivalent functions of sigma-similarity measure of fuzzy sets, provides a different proof of Liu's results. Findings - Here, the proof is much simpler than the original one provided by Liu. And, on the basis of Shannon's probability entropy as introduced by De Luca and Termini, a sigma-similarity measure is constructed, on which a No answer to an open question as proposed by Liu is provided. Originality/value - Constructs a new approach to some known results and provides a definite answer to an open question.
fuzzy logic