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To provide a simplified proof for Liu's results concerning the relationship between σ‐entropy and σ‐similarity measure.

First obtains Liu's clear and simple functional expression of σ‐similarity measures based on the structure of σ‐entropy by using expressions of equivalent functions of σ‐entropy. Then, by employing the general expression of equivalent functions of σ‐similarity measure of fuzzy sets, provides a different proof of Liu's results.

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 σ‐similarity measure is constructed, on which a No answer to an open question as proposed by Liu is provided.

Constructs a new approach to some known results and provides a definite answer to an open question.