Correct structures and similarity measures of soft sets along with historic comments of Prof. D.A. Molodtsov

Abstract

After the paper of Molodtsov [Molodtsov D. Soft set theory — First results, Computers and Mathematics with Applications, 1999, vol. 37, no. 4-5, pp. 19-31.] first appeared, soft set theory grew at a breakneck pace. Several authors have introduced various operations, relations, results, etc. as well as other aspects in soft set theory and hybrid structures incorrectly, despite their widespread use in mathematics and allied areas. In his paper [Molodtsov D.A. Equivalence and correct operations for soft sets, International Robotics and Automation Journal, 2018, vol. 4, no. 1, pp. 18-21.], Molodtsov, the father of soft set theory, pointed out several wrong results and notions. Molodtsov [Molodtsov D.A. Structure of soft sets, Nechetkie Sistemy i Myagkie Vychisleniya, 2017, vol. 12, no. 1, pp. 5-18.] also stated that the concept of soft set had not been fully understood and used everywhere. As a result, it is important to revisit the quirks of those conceptions and provide a formal account of the notion of soft set equivalency. Molodtsov already explored many correct operations on soft sets. We use some notions and results of Molodtsov [Molodtsov D.A. Structure of soft sets, Nechetkie Sistemy i Myagkie Vychisleniya, 2017, vol. 12, no. 1, pp. 5-18.] to create matrix representations as well as related operations of soft sets, and to quantify the similarity between two soft sets.

Stability in social networks

“…equal to A × D, and the corresponding mappings are given by the formula Wða, dÞ ¼ SðaÞ > FðdÞ, (a, d) ∈ A × D. Definition 2.8. ( [22]) The binary operation product (S, A) × (F, D) = (X, A × D) for a pair of soft sets (S, A) and (F, D) given over a universal set X is defined as follows. The parameter set is chosen equal to the direct product of the parameter sets, i.e.…”

“…Consider X = {a, b, c, d, e}, A ¼ ftall, shortg, and B ¼ fgirl, boyg. For more on soft set theory, one may refer to [18,21,22]. The hierarchy of friendship, as shown in figure 1, depicts the friends of a person depending on emotional closeness.…”

“…Here, we should note that the sets of attributes A × B and B × A are indistinguishable while calculating the union and intersection of two soft sets having the parameter sets A and B , respectively [ 22 ].…”