Rough Soft Sets: A novel Approach

Transcription

1 International Journal of Computational pplied Mathematics. ISSN Volume 12, Number 2 (2017), pp Research India Publications Rough Soft Sets: novel pproach Dr. Khaja Moinuddin ssistant Professor, Department of Mathematics Maulana zad National Urdu University,achibowli,Hyderabad-32 bstract In this paper lower soft set, upper soft set are introduced then the notion of Rough soft set is given. Here few interesting properties of these sets are established some results are discussed in this contest. MS Subject Classification: 1157, 20-XX, 20K27, 03xx Keywords: Rough set, Soft set, lower Soft set, Upper soft set, Rough soft set. 1. INTRODUCTION: The problem of imperfect knowledge has been tackled for a long time by philosophers, logicians mathematicians. There are many theories like uzzy set theory, Rough set theory Soft set theory to tackle the problems involving uncertainty. In order to study the control problems of complicate systems dealing with fuzzy information, merican cyberneticist L.. Zadeh introduced fuzzy set theory in his classical paper [8] of polish applied mathematician computer scientist Zdzislaw Pawlak introduced rough set theory in his classical paper [5] of Rough set theory is a new mathematical approach to imperfect knowledge. This theory presents still another attempt to deal with the uncertainty or vagueness. The rough set theory has attracted the attention of many researchers practitioners who contributed essentially to its development application. In 1999, Molodtsov [4] introduced the soft set theory as a general mathematical tool for dealing with uncertainty or vagueness. Soft set theory is still a better approach to

2 538 Dr. Khaja Moinuddin deal with problems involving uncertainty. Molodtsov recognized the importance of the role of parameters introduced the theory of Soft sets. He has shown several applications of this theory in many fields like economics, engineering, medical sciences, etc. Later, this theory became a very good source of research for many mathematicians computer scientists of recent years because of its wide range of applicability. The concepts of Rough Soft sets Soft Rough sets were also emerged. He has shown several applications of this theory in many fields like economics, engineering, medical sciences, etc. Later, this theory became a very good source of research for many mathematicians computer scientists of recent years because of its wide range of applicability. The concepts of Rough Soft sets Soft Rough sets were also emerged. In the present work, the lower upper soft sets are introduced in a different approach. The notion of Rough soft sets is also introduced few results are investigated in this context. Let U be an initial universe set (or simply ) be a collection of all possible parameters with respect to U, where parameters are the characteristics or properties of objects in U. Let PU ( ) be the collection of all subsets of U let st for the empty set. irstly some basic concepts of Soft sets Rough sets are presented in the following consecutive sections. U 2. SOT STS: In this section, some basic definitions of soft sets that are needed in further study of this paper are presented. 2.1 Definition: pair (, ) is called a soft set over U, if : P( U). We write for (, ). 2.2 Definition: Let,. Then we say that be soft sets over a common universe set U (a) is a soft subset of, denoted by, if (i) (ii) ( e) ( e) e.

3 Rough Soft Sets: novel pproach 539 (b) equals, denoted by, if. 2.3 Definition: soft set over U is called a null soft set, denoted by, if e, e (). 2.4 Definition: soft set over U is called an absolute soft set, denoted by, if e, () e U. 2.5 Definition: The union of two soft sets is the soft set H, where C, for all e C, C We write HC. () e if e H ( e) ( e) if e ( e) ( e) if e over a common universe U 2.6 Definition: The intersection of two soft sets universe U is the soft set H( e) ( e) ( e). We write HC. C over a common H, where C, for all e C, 2.7 Definition: or a soft set over U, the relative complement of by c is defined by, where c 1 ( e) U ( e) for all e. 2.8 Definition: The difference of two soft sets 1 c is defined by /. a b a is denoted 1 : P( U) is a mapping given by, denoted by \ it a b 3. ROUH STS In this section, some basic definitions of Rough sets that are necessary for further study of this work are presented.

4 540 Dr. Khaja Moinuddin 3.1 Definition: relation R on a non-empty set S is said to be an equivalence relation on S if (a) xrx for all x S (reflexivity) (b) xry yrx (symmetry) (c) xry yrz xrz (transitivity) We denote the equivalence class of an element x S with respect to the equivalence relation R by the symbol [ ] R x R [ x] y S : yrx. 3.2 Definition: Let X U define the following.. Let R be an equivalence relation on U. Then we (a) The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X using R. That is the set R ( X ) x : R [ x] X. * (b) The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X using R. That is the set R * ( X ) x : R [ x] X. (c) The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor as not X using R. That is * the set ( X ) R R ( X ) R *( X ). It is clear that R X X R X. ( ) * ( ) * 3.3 Definition: set X U is said to be a Rough set with respect to an equivalence relation R on U, if the boundary region. * ( X ) R R ( X ) R *( X ) is non-empty.

5 Rough Soft Sets: novel pproach ROUH SOT STS inally in this section, the lower soft set,upper soft set Rough soft sets are introduced in a different approach few interesting results are established in this context. 4.1 Definition: Let U be the set of parameters the universe set respectively. Let R be an equivalence relation on U if is a soft set then we define two soft sets : PU : PU e e R e as follows. R e for every e. We call the soft sets, the upper soft set the lower soft set respectively. 4.2 Definition: We say that a soft set, a Rough soft set if the difference \ is a non-null soft set. 4.3 Definition: y a Crisp soft set, We mean a soft set such that. 4.4 Proposition: If is a soft set over the universe set U then Proof: Let be a soft set over the universe set U R be an equivalence relation on U. Let Hence If Hence Let x R e R x e for e Rx e Rx x R e R e e for every e (1) then Rx e so x e x R e x R e e for every e (2) e, Then xrx e

6 542 Dr. Khaja Moinuddin R x e x R e e R e for every e (3) y (1), (2), (3) we have R e e R e for every e (4) rom (4), it follows that 4.5 Proposition: If is the null soft set defined by e X e (a) (b) X X X U for every e, then X is the absolute soft set, 4.6 Proposition: If are any two soft sets such that then (a) (b) 4.7 Proposition: If then (a) (b) (c) (d)

7 Rough Soft Sets: novel pproach Proposition: If is any soft set then (a) (b) (c) (d) RRNCS [1] Herstein I.N., Topics in lgebra,. 2 nd edition, Wiley &Sons, New York, [2] Li.., Notes on the Soft Operations, rpn Journal of System Software, Vol-66, pp , [3] Maji. P. K., iswas. R., Roy.. R., Soft set theory, Comp. Math. ppl., Vol-45, pp , [4] Molodtsov. D., Soft Set Theory- irst Results, Comp. Math. ppl., Vol-37, pp , [5] Pawlak, Z., Rough Sets, International Journal of Computer Information Sciences, Vol-11, pp , [6] Pawlak, Z., Rough Sets Theoretical spects of Reasoning about Data, Kluwer cademic Publishers, Dordrecht, [7] Polkowski, L., Rough Sets, Mathematical oundations, Springer Verlag, erlin,2002. [8] Zadeh, L.., uzzy sets, Inform. Control., vol-8,pp ,1965.

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