U E is uniquely defined as

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1 VOL 2, NO 9, Oct 212 ISSN RPN Journal of Science Technology ll rights reserved Notes on Soft Matrices Operations 1 D Singh, 2 Onyeozili, I (corresponding author) 1 Former Professor, Indian Institute of Technology, Bombay, Mathematics Department, hmadu Bello University, Zaria,Nigeria 2 Department of Mathematics, University of buja, Nigeria, (PhD Research Scholar) { 1 mathdss@yahoocom, 2 eozili@gmailcom} BSTRCT In this paper, we describe soft sets, soft matrices, soft sub matrices, soft matrix operations their basic properties We also recall the definition of operations on soft sets show that they are equivalent to the corresponding operations on their soft matrices Keywords :Soft sets, soft matrices, soft matrix operations 1INTRODUCTION In 1999, soft sets were introduced by Molodtsov [1] with an objective to help solving problems involving uncertainties which other extant modern theories dealing with uncertainties such as probability theory, fuzzy set theory, rough set theory, etc, cannot successfully solve mainly due to their inherent parameterization inadequacies In the past ten years or so, progress towards systematization of foundation for soft set theory its applications has been perceptible Maji et al,[2,3] developed many new operations on soft sets, established their properties presented an application to a decision making problem Cagman Enginoglu [4,5] developed a theory of soft matrices successfully applied it to a decision-making problem They also constructed a uni-int decision making method In this paper, we will be largely drawing definitions from [5] In sections 2 3, we describe soft sets, soft matrices, soft submatrices operations on them In section 4, we recall the definition of operations on soft sets show that they are equivalent to the corresponding operations on soft matrices representing them We have included only those references which pertain to the aim of this paper However, as the subject is still in its infancy stage, inclusion of a plethora of definitions examples may be unavoidable 2SOFT SETS, SOFT MTRICES ND SOFT SUBMTRICES We describe representations of soft sets by matrices, called soft matrices, which are found useful as they can be efficiently stored in a computer Definition 21 [1] Let U be an initial universe, P(U) be the power set of U, E be the set of all possible parameters with respect to U, E pair F, E on the universe U, also denoted (F,), is called a soft set defined as the set of ordered pairs viz, F, E e, f ( e) : ee, f ( e) P( U) where f : E P( U ) such that f() e (the empty set) if e Thus a soft set on the universe U is a parameterized family of subsets of U Here f is called an approximate function of the soft set f ( ), e e E, f E Every set may be understood as the set of e-elements or e- approximate elements of the soft set or e-approximate value set or simply e-approximate set In other words, a soft set may be considered as a collection of approximations or a value- class Henceforth, we denote subsets of E by,b,c, Example 21 Let U u1, u2, u3, u4, u5, u6 E e1, e2, e3, e4, e5 Let 1, 3, 4, 1 2, 4, 3 f e4 u1, u3, u5 f e e e f e u u f e U we can view the soft set, E as consisting of the following collection of approximations: f, E e, 1 u, 2 u4, e, 3 U, e, 4 u,, 1 u3 u5 lso, f e f e since e2, e5 2 5 Definition 22 [4] Let f, E be a soft set over U a subset R of U E is uniquely defined as 861

2 VOL 2, NO 9, Oct 212 ISSN RPN Journal of Science Technology ll rights reserved R ( u, e): e, u f ( e), called a relation form of the soft set, f E The characteristic function of where R is defined as R : UE {,1}, 1, ( u, e) R; R ( ue, ), ( u, e) R Let U u, u,, u, E e, e,, e 1 2 m 1 2 n E R can be represented by a matrix as follows: Let a R ui, ej We can represent a matrix a11 a12 a1 n a21 a22 2n a mn am1 am2 amn n soft matrix of the soft set f call it an m, E over U In other words, a soft set is uniquely represented by its corresponding soft matrix For convenience, while writing soft matrices, we may suppress their dimensions unless necessary also represent them by capital letters M, N, P, etc Let the set of all m n soft matrices over U be denoted SM ( U ) m n or just SM(U), henceforth considered to represent the universe of soft matrices It may be noted that, similar to matrix representations of relations in set theory, the tabular form of representing a soft set was already depicted in [3], which is exactly being represented as a matrix Example 22 Let U u1, u2, u3, u4, u5 E e, e, e, e be a universe set, be a set of all parameters with respect to U Let e1, e3, e4, fe1 u3, u4, f e3 nd f e u, u, u the soft set f by,a E is given R of E f,,,,,,, E e u u e u u u The relation form f, 3, 1, 4, 1, 1, 4, 3, 4, 5, 4 a of the soft set f, R u e u e u e u e u e Hence the soft matrix by E is given 1 a 1 1, i 1,2,,5; j 1,2,,4 1 1 s noted earlier, f e, since there is no element in U 3 e related to the parameter 3, it does not appear in the aforesaid description of the soft set f, E Definition 23 [4] Let a SM ( U) a is called (a) zero soft matrix, denoted, if a i j; a, if a 1 j I j : e i (b) n -universal soft matrix, denoted j (Note that it is so called, since a 1 only for the parameters appearing in the set E); (c) universal soft matrix denoted a 1 i j Example 23 I, if Let U u, u, u, u, E e, e, e, e a, b, c SM ( U) 44 If e1, e2, e3, f e1 f e2 f e3, then a is a zero soft matrix given by 862

3 VOL 2, NO 9, Oct 212 ISSN RPN Journal of Science Technology ll rights reserved 2, 4, B 2 B 4, then b If B e e f e U f e B-universal soft matrix given by If b is a C E, fe ei U for each i, then c I is a universal soft matrix given by Definition 24 [4] I Let M a, N b SM ( U) following: we define the, if (i) M is a soft submatrix of N, denoted M N a b for each i j In this case, we also say that M is dominated by N or N dominates M Note that, similar to k R ( k 1), the k-dimensional real space, holds without the holding of either < or = We define M N comparable, denoted M N, iff M N or N M; (ii) M is a proper soft submatrix of N denoted M N, if a b a b for at least one term for all i j In this case, we say that M is properly dominated by Ns (iii) M is a strictly proper soft submatrix of N, denoted M N, if M N a b for each i j In this case we say that M is strictly dominated by N (iv) M N are soft equal matrices, denoted N, if a for each i j Equivalently, if b M M N N M, then M N It is immediate to see that is a partial ordering (reflexive, antisymmetric transitive) on the class of soft matrices 3OPERTIONS ON SOFT MTRICES We discuss the operations of union, intersection complement, difference products of soft matrices their basic properties Definition 31 [4] Let M a, N b SM ( U) P c SM ( U) is called the a soft matrix (i) union of M N, denoted M N, if c max a, b for all i j; (ii) intersection of M N, denoted M N c min a, b for all i j; (iii) complement of M, denoted M, if c 1 a for all i j; (iv) difference of N from M, also called the relative complement of N in M, denoted M N or M\N,if P M N In view of the (ii) above, M N are said to be disjoint if M N Example 31 Let M 1 1 N 1 1 1, if 863

4 VOL 2, NO 9, Oct 212 ISSN RPN Journal of Science Technology ll rights reserved, (iii) (iv) (v) (i) (ii) M N M N 1 1 1; 1 1 M N that M N are disjoint; ; which implies M N 1 1 M N ; 1 (i) (ii) (iii) (iv) (v) M M M ; M M M (Idempotent laws) M M ; M I M (Identity laws) M I I ; M (Domination laws) I ; I (De Morgan's laws) M M I M M ; (De Morgan's laws) M N M N M N M N ; (vi) (vii) (viii) (ix) (De Morgan's laws) M M for all M (Involution law/double complement) M N N M ; M N N M (Commutative laws) ; (ssociative laws) M N P M N P M N P M N P (x) M N P M NM P; M N P M NM P (Distributive laws) Proof: Most of the proofs follow from definitions Let us, for example, prove the first parts of (vi), (ix) (x) (vi) For each i j, Proposition 31: Properties of Soft Matrix Operations Let M a, N b, P c SM ( U) 864

5 VOL 2, NO 9, Oct 212 ISSN RPN Journal of Science Technology ll rights reserved M N a b max a, b 1max a, b min 1 a, 1 b a b M N a b c ik ip where c min a, b P n( j 1) k ip ik N : SM ( U) SM ( U) (ii) OR-product of M N, denoted M mn SM ( U ) mn 2 such that, a b c mn ik ip where c, is defined max a, b P n( j 1) k ip ik (ix) (x) max a,max b, c maxmax a, b, c a b c M N P a b c M N P max a, min b, c minmax a, b, maxa, c a b a c M N P a b c M N M P Remark: Proof of (ix) for example can also be seen by using two-valued truth table of a well-formed formula containing three atomic propositions Definition 32 [4]: Product of Soft Matrix Let M a, N b SM ( U) ik m n (iii) ND-NOT-product of M N, denoted M defined : SM ( U) SM ( U) mn SM ( U ) mn 2 such that, a b c mn ik ip where c min a,1b ip ik P n( j 1) k (iv) OR-NOT-product of M N, denoted M defined Proposition 32 [4] : SM ( U) SM ( U) mn SM ( U ) mn 2 such that, a b c mn ik ip where c max a,1b ip ik P n( j 1) k Let hold: M a, N bik SM ( U) (i) (ii) N N, is, is the following ( M N) M N ; ( M N) M N (De Morgan s laws) ( M N) M N ; ( M N) M N (De Morgan s laws) (i) ND-product of M N, denoted M N defined : SM ( U) SM ( U) mn SM ( U ) mn 2 such that mn, is 865

6 VOL 2, NO 9, Oct 212 ISSN RPN Journal of Science Technology ll rights reserved Proof: The proofs follow from definitions Example 32 M a, N b SM ( U) Let ik 4 4 given by M N M N a bik c ip Similarly, the other products M N, M N M N can be found lso, Thus ( M N), M N ( M N) M N Note that the commutative laws are not valid for products of soft matrices 4COMPRING OPERTIONS ON SOFT SETS WITH OPERTIONS ON THEIR CORRESPONDING SOFT MTRICES We recall from section 2 that a soft set can be uniquely characterized by a soft matrix vice versa Thus a soft set f, E over U is formally equal to its soft matrix a where m U mn E respectively n E, the cardinality of U In order to show that the operations on soft sets that on their corresponding soft matrices are equivalent, we first recall the following definitions on soft sets Definition 41 [5] Let, the f E f, B E be two soft sets over U We define (i) union of f, E f, B f, E f, E B, as a soft set f, C f ( e) f ( e) f ( e) e E (ii) intersection of E,defined by the approximate function C B f, E f, B f, E f, E B,as a soft set 866

7 VOL 2, NO 9, Oct 212 ISSN RPN Journal of Science Technology ll rights reserved f, D f ( e) f ( e) f ( e) e E (iii) complement of E,defined by the approximate function D B f, E denoted, C C f E f, E or f C, set f C, E,defined by the approximate E,as a soft function f ( e) U f ( e) e E (iv) difference of C f, E f, B, \ B, f E f E,as a soft set f, K E,defined by the approximate function f ( e) f ( e) f ( e) f ( e) f ( e) e E (v) ND-product of K B B C f, E f, B f, E fb, E,as a soft set f E E G,,defined by the function f ( x, y) f ( x) f ( y) ( x, y) E E G B, their soft matrices are respectively given by f, E f, B f, E fb, E,as a soft set f E E (vi) OR-product of H, We illustrate them as follows: Example 41,defined by the function f ( x, y) f ( x) f ( y) ( x, y) E E H B Let U u1, u2, u3, u4, u5 E e1, e2, e3, e4 Let e1, e2 B e2, e3, e4 sets f E f E are given by be the universe set be the set of parameters, B, Suppose that the soft f,,,,,, E e u u e u u f,,,,,,,, B E e u u e u u e U respectively, From Definition 41: a 1 b f, E fb, E fc, E, where c 1 1 a b fc ( e) f ( e) fb( e), e E f, C E e, 1 u, 2 u4, e, 2 u,, 1 u2 u3, e3 u, 1 u4, e, 1 4 U its soft matrix c (ii) f, E fb, E fd, E, where f ( e) f ( e) f ( e), e E (i) D B 867

8 VOL 2, NO 9, Oct 212 ISSN RPN Journal of Science Technology ll rights reserved f E e u D,, 2 1 its soft matrix d 1 d a b C (iii) f, E f C, E, where f ( e) U f ( e) e E C f C, E e1, u1, u3, e2, u2, u4, u5, e3, U, e4, U its soft matrix a C (v) f, E f, E f, E E, where B G a 1 1 1a C (iv) f, E \ f, E f, E B K, where f ( e) f ( e) f ( e) f ( e) f ( e) e E K B B C 1 k 1 a b 1 f ( x, y) f ( x) f ( y) ( x, y) E E G B its soft matrix given by g ip f,,,,, K E e u u e u its soft matrix k e2, e2, u1, e2, e3, u1, e2, e4, u1, u3 f, E E e, e, u, e, e, u, e, e, u, u G is g 1 a b 1 1 ip ik (vi) f, E f, E f, E E, where f ( x, y) f ( x) f ( y) ( x, y) E E B H H B 868

9 VOL 2, NO 9, Oct 212 ISSN RPN Journal of Science Technology ll rights reserved e1, e3, u1, u2, u4, e1, e4, U, e2, e1, u1, u3, e2, e2, u1, u2, u3, e2, e3, u1, u3, u4, e2, e4, U, e3, e2, u1, u2, e3, e3, u1, u4, e3, e4, U e4, e2, u1, u2, e4, e3, u1, u4, e4, e4, U f, E E e, e, u, u, e, e, u, u, u, H its soft matrix h ip Similarly,the other products f, E fb, E a bik f, E f, E a b h a b ip ik B ik can also be found 5CONCLUSION fter defining soft sets soft matrices, we describe operations on both of them show that they are equivalent Despite having found some sporadic applications, one of the important areas of future research is concerned with constructing an efficient algorithm for solving a complex problem using soft matrices REFERENCES [1] D Molodtsov, Soft Set Theory First Results, Computers Mathematics with pplications, 37 (1999), [2] PK Maji, RBiswas, R Roy, Soft Set Theory, Computers Mathematics with pplications, 45 (23) [3] PK Maji, R Biswas, R Roy, n pplication of Soft Set in a Decision-making Problem, Computers Mathematics with pplications, 44 (22) [4] N Cagman S Enginoglu, Soft Set Theory Uni-int decision making, European Journal of Operational Research, 27 (21) [5] N Cagman S Enginoglu, Soft Matrix Theory its Decisionmaking, Computers Mathematics with pplications, 59 (21)