Results from set-operations on Fuzzy soft sets
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The paper introduces and defines seven set operations on fuzzy soft sets, which extend classical set theory to handle imprecise phenomena. These operations are represented in matrix form and various relationships among them are established. The research emphasizes the significance of fuzzy soft sets in mathematical applications and decision-making processes.
![International
OPEN ACCESS Journal
Of Modern Engineering Research (IJMER)
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.8| Aug. 2014 | 53|
Results from set-operations on Fuzzy soft sets
D. R Jain1, Bhu Dev Sharma2
1Department of Mathematics, Jaypee Institute of Information Technology, Noida, UP, India
2Formerly at the Department of Mathematics, Jaypee Institute of Information Technology, Noida, UP, India
I. Introduction
Recent advances present phenomena, in many areas including engineering, social and medical sciences
that are neither deterministic nor stochastic in nature. These cannot be characterized in terms of classical set
theory. As such fundamental extensions and generalizations of sets in mathematics have been proposed.
Zadeh [1], in 1965, introduced the theory of fuzzy sets for dealing with imprecise phenomena. These
were further generalized by Atanassov [2,3], to what has come to be known as ‘Intuitionistic fuzzy sets’, to
characterize a broader class of vague phenomena. Molodstov [4], in 1999, on the other hand, introduced the
concept of ‘Soft set’ associating characteristics or parameters in considering subsets of a set.
Maji, et. al [5], inducing the concept of fuzzyness on soft-sets, introduced the concept of Fuzzy Soft
Sets. The hybrid ‘Fuzzy Soft Set theory’ has attracted the attention of researchers for its further study and
applications. Yong Yang and Chenli Ji [6], using matrix representation of Fuzzy Soft Sets considered
applications. The notion of Fuzzy Soft matrices has been further extended in [7] and applied in certain decision
making problems.
While set-operations, refer Verma & Sharma [8], on intitutionistic fuzzy sets have been studied, for
mathematical viability and usefulness, there is a need to examine and to study these over fuzzy-soft-sets. In this
paper we define seven operations analogous to [8] on fuzzy soft sets in terms of their matrices and prove various
different relations amongst these operations.
II. Preliminaries
In this section we give definitions and notions, refer [7], used in following work.
Definition 1: Fuzzy Soft Set - Let X be an initial universal set and E be a set of parameters. Let P
~
(x) denotes
the power set of all Fuzzy Subsets Sets of X. Let A E. A pair (F, A) is called Fuzzy Soft Set over X. where F
is a mapping given by F : A ( ).
~
P X
Definition 2: - Fuzzy Soft Class - The pair (X, E) denotes the collection of all Fuzzy Soft Sets on X with
attributes from E and is called Fuzzy Soft Class.
Definition 3: Fuzzy Soft Matrices
Let X = {x1, x2, ...... xm} be the universal set and E = {e1, e2, ......en} be the set of parameters. Let A E and (F,
A) be a Fuzzy Soft Set in the Fuzzy Soft Class (X, E). Then we represent the Fuzzy Soft Set (F, A) in the matrix
form as:
Amn = ij a mxn or simply by A = ij a
where
if e A
x if e A
a
j
j i j
ij 0
( )
.
Here ( ) j i x represent the membership of xi in the Fuzzy Set F(ej). We would identify a Fuzzy Soft Set with
its Fuzzy Soft matrix and vice versa. The set of all mxn Fuzzy Soft Matrices will be denoted by FSMmxn over X.
Definition 4: Set of Operations on FSMmxn
Let A = ij a mxn and B = ij b mxn be two Fuzzy Soft matrices over the universal set X.
Some operations on FSMmxn are defined as follows:
Abstract: In this paper considering a class of Fuzzy-Soft Sets, seven set operations are defined and
several relations arising from these set-operations are established using matrix representation of fuzzy soft
sets.
Key words: Fuzzy Soft Sets, equality, operations, Fuzzy Soft matrix.](https://image.slidesharecdn.com/g040805-5357-141001052401-phpapp01/85/Results-from-set-operations-on-Fuzzy-soft-sets-1-320.jpg)
![Results from set-operations on Fuzzy soft sets
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.8| Aug. 2014 | 57|
(8) (A . B) (A # B) =
ij ij
ij ij
ij ij a b
a b
a b
2
=
ij ij
ij ij
ij ij a b
a b
a b
2
min ,
= ij ij a b on using (2.6)
= A . B
and
(A . B) (A # B) =
ij ij
ij ij
ij ij a b
a b
a b
2
max ,
=
ij ij
ij ij
a b
2a b
on using (2.6)
= A # B
IV. Concluding Remarks
The results obtained in terms of various operations must go a long way in applications of Fuzzy Soft
Sets.
REFERERENCES
[1] L.A. Zadeh, Fuzzy Sets, Information and Control,8, 1965, 338-353.
[2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems 20(1) 1986 ,87-96.
[3] K.T. Atanassov, Intuitionistic fuzzy sets, Springer Physica-Verlag, Heidelberg,1999.
[4] D. Molodstov, Soft Set Theory-first Results, Computers Math with Applications, 37, 1999, 19-31.
[5] P.K. Maji, R. Biswas and A.R. Roy, Fuzzy Soft Sets, J. Fuzzy Math, 9(3), 2001, 589-602.
[6] Yong Yang and Chenli Ji, Fuzzy Soft Matrices and their Applications, AICI 2011, Part I, LNAI 7002, 2011, 618-627.
[7] Manash Jyoti Borah, Tridiv Jyoti Neog, Dusmanta Kumar Sut, Fuzzy Soft Matrix Theory and its Decision Making,
International Journal of Modern Engineering Research (IJMER) Vol. 2 Issue 2, 2012, 121-127.
[8] Rajkumar Verma, Bhu Dev Sharma, some new equalities connected with Intuitionistic fuzzy sets, Notes on
Intuitionistic fuzzy sets Vol. 19 ,2,2013, 25-30.](https://image.slidesharecdn.com/g040805-5357-141001052401-phpapp01/85/Results-from-set-operations-on-Fuzzy-soft-sets-5-320.jpg)
Results from set-operations on Fuzzy soft sets
- 1. International OPEN ACCESS Journal Of Modern Engineering Research (IJMER) | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.8| Aug. 2014 | 53| Results from set-operations on Fuzzy soft sets D. R Jain1, Bhu Dev Sharma2 1Department of Mathematics, Jaypee Institute of Information Technology, Noida, UP, India 2Formerly at the Department of Mathematics, Jaypee Institute of Information Technology, Noida, UP, India I. Introduction Recent advances present phenomena, in many areas including engineering, social and medical sciences that are neither deterministic nor stochastic in nature. These cannot be characterized in terms of classical set theory. As such fundamental extensions and generalizations of sets in mathematics have been proposed. Zadeh [1], in 1965, introduced the theory of fuzzy sets for dealing with imprecise phenomena. These were further generalized by Atanassov [2,3], to what has come to be known as ‘Intuitionistic fuzzy sets’, to characterize a broader class of vague phenomena. Molodstov [4], in 1999, on the other hand, introduced the concept of ‘Soft set’ associating characteristics or parameters in considering subsets of a set. Maji, et. al [5], inducing the concept of fuzzyness on soft-sets, introduced the concept of Fuzzy Soft Sets. The hybrid ‘Fuzzy Soft Set theory’ has attracted the attention of researchers for its further study and applications. Yong Yang and Chenli Ji [6], using matrix representation of Fuzzy Soft Sets considered applications. The notion of Fuzzy Soft matrices has been further extended in [7] and applied in certain decision making problems. While set-operations, refer Verma & Sharma [8], on intitutionistic fuzzy sets have been studied, for mathematical viability and usefulness, there is a need to examine and to study these over fuzzy-soft-sets. In this paper we define seven operations analogous to [8] on fuzzy soft sets in terms of their matrices and prove various different relations amongst these operations. II. Preliminaries In this section we give definitions and notions, refer [7], used in following work. Definition 1: Fuzzy Soft Set - Let X be an initial universal set and E be a set of parameters. Let P ~ (x) denotes the power set of all Fuzzy Subsets Sets of X. Let A E. A pair (F, A) is called Fuzzy Soft Set over X. where F is a mapping given by F : A ( ). ~ P X Definition 2: - Fuzzy Soft Class - The pair (X, E) denotes the collection of all Fuzzy Soft Sets on X with attributes from E and is called Fuzzy Soft Class. Definition 3: Fuzzy Soft Matrices Let X = {x1, x2, ...... xm} be the universal set and E = {e1, e2, ......en} be the set of parameters. Let A E and (F, A) be a Fuzzy Soft Set in the Fuzzy Soft Class (X, E). Then we represent the Fuzzy Soft Set (F, A) in the matrix form as: Amn = ij a mxn or simply by A = ij a where if e A x if e A a j j i j ij 0 ( ) . Here ( ) j i x represent the membership of xi in the Fuzzy Set F(ej). We would identify a Fuzzy Soft Set with its Fuzzy Soft matrix and vice versa. The set of all mxn Fuzzy Soft Matrices will be denoted by FSMmxn over X. Definition 4: Set of Operations on FSMmxn Let A = ij a mxn and B = ij b mxn be two Fuzzy Soft matrices over the universal set X. Some operations on FSMmxn are defined as follows: Abstract: In this paper considering a class of Fuzzy-Soft Sets, seven set operations are defined and several relations arising from these set-operations are established using matrix representation of fuzzy soft sets. Key words: Fuzzy Soft Sets, equality, operations, Fuzzy Soft matrix.
- 2. Results from set-operations on Fuzzy soft sets | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.8| Aug. 2014 | 54| (1) A B = C = ij c mxn where cij = max ij ij a , b , for all i and j (2) A B = C = ij c mxn where cij = min ij ij a , b , for all i and j (3) A B = C = ij c mxn where ij ij ij ij ij c a b a b , for all i and j (4) A . B = C = ij c mxn where cij = ij ij a b , for all i and j (5) A @ B = C = ij c mxn where cij = 2 1 ij ij a b , for all i and j (6) A $ B = C = ij c mxn where cij = ij ij a b , for all i and j (7) A # B = C = ij c mxn where ij c = ij ij ij ij a b a b 2 , for all i and j, for which we will accept that if 0 ij ij a b then 0 ij ij ij ij a b a b . III. Main Results Before starting discussion of the main results we prove some rather simple inequalities to be used in the subsequent work. ij ij a b ij ij ij ij 2 a b 2a b (2.1) ij ij ij ij a b a b ij ij a b (2.2) ij ij ij ij a b 2a b 0 ij ij ij ij 2 a b a b ij ij a b ij ij ij ij a b a b ( ) 2 1 ij ij a b (2.3) Next ij ij ij ij a b a b ij ij a b ij ij ij ij ij ij 2 a b a b a b (on using 2.1) 0 ij ij ij ij a b a b ij ij ij ij a b a b ) ij ij a b (2.4) Also ij ij ij ij a b a b ij ij ij ij a b a b 2 0 ( ) (1 ) ( ) (1 ) 2 2 ij ij ij ij ij ij a b a b b a ij ij ij ij a b a b ij ij ij ij a b a b 2 . (2.5) Further ij ij ij ij a b a b 2 - ij ij a b = 0 (2 ) ij ij ij ij ij ij a b a b a b . Thus
- 3. Results from set-operations on Fuzzy soft sets | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.8| Aug. 2014 | 55| ij ij ij ij a b a b 2 ij ij a b . (2.6) Theorem If A = ij a and B = ij b are any two FSMmxn, then (1) (A @ B) $ (A # B) = A $ B (2) (A B) (A . B) = A . B , (A B) (A . B) = A B (3) (A B) (A @ B) = A @ B , (A B) (A @ B) = A B (4) (A . B) (A @ B) = A . B , (A . B) (A @ B) = A @ B (5) (A B) (A $ B) = A $ B, (A B) (A $ B) = (A B) (6) (A . B) (A $ B) = A . B , (A . B) (A $ B) = A $ B (7) (A B) (A # B) = A # B , (A B) (A # B) = (A B) (8) (A . B) (A # B) = A . B , (A . B) (A # B) = (A # B) Proof of the Theorem: Using definitions, we have: (1) (A @ B) $ (A # B) = ij ij ij ij ij ij a b a b 2a b $ 2 = ij ij ij ij ij ij a b a b 2a b . 2 = ij ij a b =A $ B (2) (A B) (A . B) = ij ij ij ij ij ij a b a b a b = ij ij ij ij ij ij min a b a b ,a b = ij ij a b (on using 2.2) = A . B and (A B) (A . B) = ij ij ij ij ij ij max a b a b ,a b = ij ij ij ij a b a b = A B (3) (A B) (A @ B) = ij ij ij ij b a b a 2 ij ij a b = 2 min , ij ij ij ij ij ij a b a b a b = 2 ij ij a b on using (2.3) = A @ B and (A B) (A @ B) = 2 max , ij ij ij ij ij ij a b a b a b = ij ij ij ij a b a b on using (2.3) = (A B) (4) (A . B) (A @ B) = 2 ij ij ij ij a b a b
- 4. Results from set-operations on Fuzzy soft sets | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.8| Aug. 2014 | 56| = 2 min , ij ij ij ij a b a b = ij ij a b on using (2.1) = A . B and (A . B) (A @ B) = 2 max , ij ij ij ij a b a b = 2 ij ij a b on using (2.1) = A @ B (5) (A B) (A$ B) = ij ij ij ij ij ij a b a b a b = ij ij ij ij ij ij min a b a b , a b = ij ij a b on using (2.4) = A $ B and (A B) (A $ B) = ij ij ij ij ij ij max a b a b , a b = ij ij ij ij a b a b on using (2.4) = A B (6) (A . B) (A $ B) = ij ij ij ij a b a b = ij ij ij ij min a b , a b = ij ij a b = A . B and (A . B) (A $ B) = ij ij ij ij max a b , a b = a b A B ij ij $ (7) (A B) (A # B) = ij ij ij ij ij ij ij ij a b a b a b a b 2 = ij ij ij ij ij ij ij ij a b a b a b a b 2 min , = ij ij ij ij a b 2a b on using (2.5) = A # B and (A B) (A # B) = ij ij ij ij ij ij ij ij a b a b a b a b 2 max , = ij ij ij ij a b a b on using (2.5) = A B
- 5. Results from set-operations on Fuzzy soft sets | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.8| Aug. 2014 | 57| (8) (A . B) (A # B) = ij ij ij ij ij ij a b a b a b 2 = ij ij ij ij ij ij a b a b a b 2 min , = ij ij a b on using (2.6) = A . B and (A . B) (A # B) = ij ij ij ij ij ij a b a b a b 2 max , = ij ij ij ij a b 2a b on using (2.6) = A # B IV. Concluding Remarks The results obtained in terms of various operations must go a long way in applications of Fuzzy Soft Sets. REFERERENCES [1] L.A. Zadeh, Fuzzy Sets, Information and Control,8, 1965, 338-353. [2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems 20(1) 1986 ,87-96. [3] K.T. Atanassov, Intuitionistic fuzzy sets, Springer Physica-Verlag, Heidelberg,1999. [4] D. Molodstov, Soft Set Theory-first Results, Computers Math with Applications, 37, 1999, 19-31. [5] P.K. Maji, R. Biswas and A.R. Roy, Fuzzy Soft Sets, J. Fuzzy Math, 9(3), 2001, 589-602. [6] Yong Yang and Chenli Ji, Fuzzy Soft Matrices and their Applications, AICI 2011, Part I, LNAI 7002, 2011, 618-627. [7] Manash Jyoti Borah, Tridiv Jyoti Neog, Dusmanta Kumar Sut, Fuzzy Soft Matrix Theory and its Decision Making, International Journal of Modern Engineering Research (IJMER) Vol. 2 Issue 2, 2012, 121-127. [8] Rajkumar Verma, Bhu Dev Sharma, some new equalities connected with Intuitionistic fuzzy sets, Notes on Intuitionistic fuzzy sets Vol. 19 ,2,2013, 25-30.