On some new operations in soft set theory

By Muhammad Irfan Ali, Feng Feng, Xiaoyan Liu, Won Keun Min, Muhammad Shabir. Computers and Mathematics with Applications, 2009

1,705 citations recorded by the source corpus.

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)-(vi), Proposition 2.4 and Proposition 2.6 (iii), (iv)) in a previous paper by Maji et al. [P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562] 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.

Subfields: Computational Theory and Mathematics, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/09c280e56ac20145b297c7b4eabc30fc61d7588b

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Citations in the corpus

On Soft Topological Spaces (2013, background): “The soft set theory has been applied to many different fields (see, for example, [ 1], [2], [4], [5], [7], [8], [11], [12], [13], [14], [15], [16], [22], [25], [27]).”
Sum of Soft Topological Spaces (2020, neutral): “[4] have made some amendments for some results obtained by [3] and have defined new types of soft union and intersection between two soft sets.”
FUZZY SOFT SET THEORY AND ITS APPLICATIONS (2011, background): “[1] pointed out that these works have some weak points.”
Soft matrix theory and its decision making (2010, neutral): “[28].”
Soft Multisets Theory (2011, neutral): “10 [2].”
Neutrosophic Soft Set (2013, background): “Corresponding Author: Pabitra Kumar Maji (pabitra maji@yahoo.com )”
On soft linear spaces and soft normed linear spaces (2013, background): “Ali et al. [2] and Shabir and Irfan Ali ([2],[18]) studied soft semigroups and soft ideals over a semi group which characterize generalized fuzzy ideals and fuzzy ideals with thresholds of a semigroup.”
Vague soft sets and their properties (2010, background): “and additional properties of soft sets can be found in [11,12].”
A NOTE ON SOFT NEAR-RINGS AND IDEALISTIC SOFT NEAR-RINGS (2011, methodology): “Based on analysis of several operations on soft sets introduced in [12], Ali et al. [2] introduced some new operations on soft sets which we apply to the concept of soft near-rings throughout this paper.”
A Novel Approach towards Bipolar Soft Sets and Their Applications (2020, background): “Later on, Ali et al. [5] piercing out some inadequacies in the operations deﬁned in [4] bequeathed some new operations to soft sets like extended union, restricted union, restricted intersection, and the restricted diﬀerence of two soft sets.”
Decompositions of Some Types of Supra Soft Sets and Soft Continuity (2014, background): “4 [3] The complement of a soft set ) , ( A F , denoted by ) , (  A F , is defined by ) , ( = ) , ( A F A F   , ) ( : X P A F   is mapping given by ) ( = ) ( e F X e F   , A e  and F  is called the soft complement function of F .”
Interval-valued intuitionistic fuzzy soft sets and their properties (2010, background): “[10] present some new algebraic operations for soft sets and prove that certain DeMorgan’s laws hold in soft set theorywith respect to these new definitions.”
Picture fuzzy sets (2015, result): “Now we present a new definition , an example and some results which are the generalization of some propositions given in [1, 12].”
Soft set theory and uni-int decision making (2010, background): “[1] introduced the analysis of several operations on soft sets.”
Soft sets combined with fuzzy sets and rough sets: a tentative approach (2010, background): “Definition 8 (Ali et al. 2009) Let (F, A) and (G, B) be two soft sets over a common universe U.”
Soft Semi Compactness via Soft Ideals (2014, background): “After presentation of the operations of soft sets [20], the properties and applications of soft set theory have been studied increasingly [5,16,23,26].”
On Continuity of Soft Mappings (2015, background): “Definition 3.[10] The complement of a soft set (F,A), denoted by(F,A)c, is defined by(F,A)c = (Fc,A).”