Some formal relationships among soft sets, fuzzy sets, and their extensions

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. (C) 2015 Elsevier Inc. All rights reserved.