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On some new operations in soft set theory. (English) Zbl 1186.03068
Summary: Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. In this paper, we first point out that several assertions (Proposition 2.3 (iv)-$\left(vi\right)$, Proposition 2.4 and Proposition 2.6 (iii), (iv)) in a previous paper by P.K. Maji, R. Biswas and A.R. Roy [Comput. Math. Appl. 45, No. 4–5, 555–562 (2003; Zbl 1032.03525)] are not true in general, by counterexamples. Furthermore, based on the analysis of several operations on soft sets introduced in the same paper, we give some new notions such as the restricted intersection, the restricted union, the restricted difference and the extended intersection of two soft sets. Moreover, we improve the notion of complement of a soft set, and prove that certain De Morgan’s laws hold in soft set theory with respect to these new definitions.

##### MSC:
 03E72 Fuzzy set theory 68T37 Reasoning under uncertainty
##### Keywords:
soft sets; union; intersection; complement; difference
##### References:
 [1] Molodtsov, D.: Soft set theory–first results, Comput. math. Appl. 37, 19-31 (1999) · Zbl 0936.03049 · doi:10.1016/S0898-1221(99)00056-5 [2] Maji, P. K.; Biswas, R.; Roy, A. R.: Soft set theory, Comput. math. Appl. 45, 555-562 (2003) [3] Aktaş, H.; Çağman, N.: Soft sets and soft groups, Inform. sci. 177, 2726-2735 (2007) · Zbl 1119.03050 · doi:10.1016/j.ins.2006.12.008 [4] Yang, C. F.: A note on soft set theory, Comput. math. Appl. 56, 1899-1900 (2008) · Zbl 1152.68647 · doi:10.1016/j.camwa.2008.03.019 [5] Feng, F.; Jun, Y. B.; Zhao, X. Z.: Soft semirings, Comput. math. Appl. 56, 2621-2628 (2008)