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

• Every hesitant fuzzy set can be considered a soft set on a common universe.
• Every soft set on a denumerable universe can be considered a fuzzy set.
• Interval type-2 fuzzy sets, interval-valued fuzzy sets can be considered as soft sets.
• We prove coincidence of notions of hesitant fuzzy soft set, fuzzy soft set, soft set.

We prove that every hesitant fuzzy set on a set E   can be considered either a soft set over the universe [0,1][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][0,1]. Altogether we contribute to a more complete understanding of the relationships among various theories that capture vagueness and imprecision.