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Some formal relationships among soft sets, fuzzy sets, and their extensions. (English) Zbl 1346.03046
Summary: We prove that every hesitant fuzzy set on a set $$E$$ can be considered either a soft set over the universe $$[0, 1]$$ or a soft set over the universe $$E$$. Concerning converse relationships, for denumerable universes we prove that any soft set can be considered even a fuzzy set. Relatedly, we demonstrate that every hesitant fuzzy soft set can be identified with a soft set, thus a formal coincidence of both notions is brought to light. Coupled with known relationships, our results prove that interval type-2 fuzzy sets and interval-valued fuzzy sets can be considered as soft sets over the universe $$[0, 1]$$. Altogether we contribute to a more complete understanding of the relationships among various theories that capture vagueness and imprecision.

##### MSC:
 3e+72 Theory of fuzzy sets, etc.
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