On Fuzzy Soft Multi Set and Its Application in Information Systems

International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
DOI:10.5121/ijcax.2015.2303 29
On Fuzzy Soft Multi Set and Its Application in
Information Systems
Anjan Mukherjee1
and Ajoy Kanti Das2
1
Department of Mathematics,Tripura University, Agartala-799022,Tripura, India
2
Department of Mathematics, ICV-College, Belonia -799155, Tripura, India
ABSTRACT
Research on information and communication technologies have been developed rapidly since it can be
applied easily to several areas like computer science, medical science, economics, environments,
engineering, among other. Applications of soft set theory, especially in information systems have been
found paramount importance. Recently, Mukherjee and Das defined some new operations in fuzzy soft
multi set theory and show that the De-Morgan’s type of results hold in fuzzy soft multi set theory with
respect to these newly defined operations. In this paper, we extend their work and study some more basic
properties of their defined operations. Also, we define some basic supporting tools in information system
also application of fuzzy soft multi sets in information system are presented and discussed. Here we define
the notion of fuzzy multi-valued information system in fuzzy soft multi set theory and show that every fuzzy
soft multi set is a fuzzy multi valued information system.
KEYWORDS
Soft set, fuzzy set, soft multi set, fuzzy soft multi set, information system.
1. INTRODUCTION
In recent years vague concepts have been used in different areas such as information and
communication technologies, medical applications, pharmacology, economics and engineering
since; these kinds of problems have their own uncertainties. There are many mathematical tools
for dealing with uncertainties; some of them are fuzzy set theory [18] and soft set theory [12]. In
soft set theory there is no limited condition to the description of objects; so researchers can
choose the form of parameters they need, which greatly simplifies the decision making process
and make the process more efficient in the absence of partial information. Although many
mathematical tools are available for modelling uncertainties such as probability theory, fuzzy set
theory, rough set theory, interval valued mathematics etc, but there are inherent difficulties
associated with each of these techniques. Soft set theory is standing in a unique way in the sense
that it is free from the above difficulties. Soft set theory has a rich potential for application in
many directions, some of which are reported by Molodtsov [12] in his work. Later on Maji et
al.[10, 11] presented some new definitions on soft sets and discussed in details the application of
soft set in decision making problem. Based on the analysis of several operations on soft sets
introduced in [12], Ali et al. [2] presented some new algebraic operations for soft sets and proved
that certain De Morgan’s law holds in soft set theory with respect to these new definitions.
Combining soft sets [12] with fuzzy sets [18], Maji et al. [9] defined fuzzy soft sets, which are
rich potential for solving decision making problems. Alkhazaleh and others [[1], [4], [5], [7],
[16], [17]] as a generalization of Molodtsov’s soft set, presented the definition of a soft multi set

30
and its basic operations such as complement, union, and intersection etc. In 2012 Alkhazaleh and
Salleh [3] introduced the concept of fuzzy soft multi set theory and studied the application of
these sets and recently, Mukherjee and Das[14] defined some new algebraic operations for fuzzy
soft multi sets and proved that certain De Morgan's law holds in fuzzy soft multi set theory with
respect to these new definitions. They also presented an application of fuzzy soft multi set based
decision making problems in [13].
Molodtsov [12] presented some applications of soft set theory in several directions, which
includes: the study of smoothness of functions, game theory, operations research, Riemann-
integration, probability, theory of measurement, etc. It has been shown that there is compact
connection between soft sets and information system. From the concept and the example of fuzzy
soft multisets given in the section 4.3, it can be seen that a fuzzy soft multi set is a multi-valued
information system. In this paper we define the notion of fuzzy multi-valued information system
in fuzzy soft multi set theory and application of fuzzy soft multi sets in information system were
presented and discussed.
2. PRELIMINARY NOTES
In this section, we recall some basic notions in soft set theory and fuzzy soft multi set theory. Let
U be an initial universe and E be a set of parameters. Let P(U) denotes the power set of U and
A⊆E.
A pair (F, A) is called a soft set over U, where F is a mapping given by F: A→ P(U). In other
words, soft set over U is a parameterized family of subsets of the universe U.
2.1. Definition [12]
Let { }:iU i I∈ be a collection of universes such that i I iU φ∈ =I and let { }:iUE i I∈ be a
collection of sets of parameters. Let ( )i I iU FS U∈= ∏ where ( )iFS U denotes the set of all fuzzy
subsets of iU , ii I UE E∈= ∏ and A E⊆ . A pair (F, A) is called a fuzzy soft multi set over U,
where :F A U→ is a mapping given by ,e A∀ ∈
( )
( ) : :
( )
i
F e
u
F e u U i I
uµ
   
= ∈ ∈      
For illustration, we consider the following example.
2.2. Example
Let us consider three universes { }1 1 2 3 4 5, , , ,U h h h h h= , { }2 1 2 3 4, , ,U c c c c= and { }3 1 2 3, ,U v v v= be the
sets of “houses,” “cars,” and “hotels”, respectively. Suppose Mr. X has a budget to buy a house, a
car and rent a venue to hold a wedding celebration. Let us consider a intuitionistic fuzzy soft
multi set (F, A) which describes “houses,” “cars,” and “hotels” that Mr. X is considering for
accommodation purchase, transportation purchase, and a venue to hold a wedding celebration,
respectively. Let { }1 2 3
, ,U U UE E E be a collection of sets of decision parameters related to the above
universes, where
{ }1 1 1 1,1 ,2 ,3expensive, cheap, woodenU U U UE e e e= = = =

31
{ }2 2 2 2,1 ,2 ,3beautiful, cheap, sporty ,U U U UE e e e= = = =
{ }3 3 3 3,1 ,2 ,3expensive, model, beautiful .U U U UE e e e= = = =
Let ( )3
1i iU FS U== ∏ , 3
1 ii UE E== ∏ and A E⊆ , such that
{ }1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 31 ,1 ,1 ,1 2 ,1 ,2 ,1 3 ,2 ,3 ,1 4 ,3 ,3 ,1 5 ,1 ,1 ,2 6 ,1 ,2 ,2( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , ) .U U U U U U U U U U U U U U U U U UA a e e e a e e e a e e e a e e e a e e e a e e e= = = = = = =
Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of
choice parameters. Then fuzzy soft multi set (F, A) can be represent as in Table 1.
Table 1. The tabular representation of the fuzzy soft multi-set (F, A)
Ui a1 a2 a3 a4 a5 a6
U1
h1
h2
h3
h4
h5
0.8
0.4
0.9
0.4
0.1
0.6
0.5
0
0.4
1
0.5
0.7
1
0.4
0.8
0.4
0.5
0.1
0
0.8
0.7
0.5
1
0.4
1
0.6
0.4
0.8
1
0.1
U2
c1
c2
c3
c4
0.8
1
0.8
1
0.9
0.6
1
0
0.6
0.1
0
1
0.8
0
0.8
1
0
0.1
0.7
1
0.7
0.1
0.9
0.1
U3
v1
v2
v3
0.8
0.7
0.6
0.6
0.6
0
0
0.7
0.7
1
0.7
0
1
0.8
0
0.1
0.9
0.1
The restricted union of two fuzzy soft multi sets (F, A) and (G, B) over U is a fuzzy soft multi set
(H,C)whereC A B= ∩ and ,e C∀ ∈
( )( ) ( ), ( )H e F e G e= U
{ }( ) ( )
: :
max ( ), ( )
i
F e G e
u
u U i I
u uµ µ
    = ∈ ∈ 
    
and is written as ( , ) ( , ) ( , ).RF A G B H C=∪%
2.5 Definition [14]
The extended union of two fuzzy soft multi sets (F, A) and (G, B) over U is a fuzzy soft multi set
(H, D), where D A B= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
H e G e if e B A
F e G e if e A B
 ∈ −

= ∈ −

∈ ∩U
where ( )( ), ( )F e G eU
{ }( ) ( )
: :
max ( ), ( )
i
F e G e
u
u U i I
u uµ µ
    = ∈ ∈ 
    

32
and is written as ( , ) ( , ) ( , ).EF A G B H D=∪%
2.6 Definition [14]
The restricted intersection of two intuitionistic fuzzy soft multi sets (F, A) and (G, B) over U is a
fuzzy soft multi set (H, D) where D A B= ∩ and ,e D∀ ∈
( )( ) ( ), ( )H e F e G e= I
{ }( ) ( )
: :
min ( ), ( )
i
F e G e
u
u U i I
u uµ µ
    = ∈ ∈ 
    
and is written as ( , ) ( , ) ( , ).RF A G B H D=∩%
2.7 Definition [14]
The extended intersection of two fuzzy soft multi sets (F, A) and (G, B) over U is a fuzzy soft
multi set (H, D), where D A B= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) ,
F e if e A B
H e G e if e B A
F e G e if e A B
 ∈ −

= ∈ −

∈ ∩I
Where
( )( ), ( )F e G eI
{ }( ) ( )
: :
min ( ), ( )
i
F e G e
u
u U i I
u uµ µ
    = ∈ ∈ 
    
and is written as ( , ) ( , ) ( , ).EF A G B H D=∩%
If (F, A) and (G, B) be two fuzzy soft multi sets over U, then ( ) ( )" , AND , "F A G B is a fuzzy soft
multi set denoted by ( ) ( ), ,F A G B∧ and is defined by ( ) ( ) ( ), , , ,F A G B H A B∧ = × where H is
mapping given by H: A×B→U and
( )
{ }( ) ( )
, , ( , ) : : .
min ( ), ( )
i
F a G b
u
a b A B H a b u U i I
u uµ µ
    ∀ ∈ × = ∈ ∈ 
    
If (F, A) and (G, B) be two fuzzy soft multi sets over U, then ( ) ( )" , OR , "F A G B is a fuzzy soft
multi set denoted by ( ) ( ), ,F A G B∨ and is defined by ( ) ( ) ( ), , , ,F A G B K A B∨ = × where K is
mapping given by K: A×B→U and
( )
{ }( ) ( )
, , ( , ) : :
max ( ), ( )
i
F a G b
u
a b A B K a b u U i I
u uµ µ
    ∀ ∈ × = ∈ ∈ 
    

33
An information system is a 4-tuple S = (U, A, V, f ), where U = {u1, u2, u3, …., un} is a non-empty
finite set of objects, A = {a1, a2, a3,…., am} is a non-empty finite set of attributes, V =∪a∈A Va, Va
is the domain of attribute a, f : U × A→V is an information function, such that f (u, a) ∈ Va for
every (u, a) ∈ U × A, called information(knowledge) function.
3. MAIN RESULTS
Mukherjee and Das [14] defined some new operations in fuzzy soft multi set theory and show that
the De Morgan’s types of results hold in fuzzy soft multi set theory with respect to these newly
defined operations. In this section, we extend their work and study some more basic properties of
their defined operations.
3.1. Proposition (Associative Laws)
Let (F, A), (G, B) and (H, C) are three fuzzy soft multi sets over U, then we have the following
properties:
1. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R RF A G B H C F A G B H C=∩ ∩ ∩ ∩% % % %
2. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R RF A G B H C F A G B H C=∪ ∪ ∪ ∪% % % %
Proof. 1. Assume that ( , ) ( , ) ( , )RG B H C I D=∩% , where D B C= ∩ and ,e D∀ ∈
{ }( ) ( )min
( ) ( ) ( ) : :
( ), ( )
i
G e H e
u
I e G e H e u U i I
u uµ µ
    = ∩ = ∈ ∈ 
    
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A I D=∩ ∩ ∩% % % , we suppose that ( )( , ) ( , ) ,RF A I D K M=∩% where
M A D A B C= ∩ = ∩ ∩ and ,e M∀ ∈
( )
( ) ( ) ( ) : :
( )
i
K e
u
K e F e I e u U i I
uµ
   
= ∩ = ∈ ∈      
{ } { }{ } { }( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )min min min min( ) ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( )K e F e I e F e G e H e F e G e H e
where
u u u u u u u u uµ µ µ µ µ µ µ µ µ= = =
Suppose that ( , ) ( , ) ( , ),RF A G B J N=∩% where N A B= ∩ and ,e N∀ ∈
{ }( ) ( )min
( ) ( ) ( ) : :
( ), ( )
i
F e G e
u
J e F e G e u U i I
u uµ µ
    = ∩ = ∈ ∈ 
    
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C J N H C=∩ ∩ ∩% % % , we suppose that ( , ) ( , ) ( , )RJ N H C O N C= ∩∩% where

34
( )
, ( ) ( ) ( ) : :
( )
i
O e
u
e N C A B C O e J e H e u U i I
uµ
   
∀ ∈ ∩ = ∩ ∩ = ∩ = ∈ ∈      
{ } { }{ }
{ }
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
min min min
min
( ) ( ), ( ) ( ), ( ) , ( )
( ), ( ), ( ) ( )
O e J e H e F e G e H e
F e G e H e K e
u u u u u u
u u u u
µ µ µ µ µ µ
µ µ µ µ
= =
= =
Consequently, K and O are the same operators. Thus
( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ).R R R RF A G B H C F A G B H C=∩ ∩ ∩ ∩% % % %
2. Assume that ( , ) ( , ) ( , )RG B H C I D=∪% , where D B C= ∩ and ,e D∀ ∈
{ }( ) ( )max
( ) ( ) ( ) : :
( ), ( )
i
G e H e
u
I e G e H e u U i I
u uµ µ
    = ∪ = ∈ ∈ 
    
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A I D=∪ ∪ ∪% % % , we suppose that ( )( , ) ( , ) ,RF A I D K M=∪% where
M A D A B C= ∩ = ∩ ∩ and ,e M∀ ∈
( )
( ) ( ) ( ) : :
( )
i
K e
u
K e F e I e u U i I
uµ
   
= ∪ = ∈ ∈      
{ } { }{ } { }( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )max max max max( ) ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( )K e F e I e F e G e H e F e G e H e
where
u u u u u u u u uµ µ µ µ µ µ µ µ µ= = =
Suppose that ( , ) ( , ) ( , ),RF A G B J N=∪% where N A B= ∩ and ,e N∀ ∈
{ }( ) ( )max
( ) ( ) ( ) : :
( ), ( )
i
F e G e
u
J e F e G e u U i I
u uµ µ
    = ∪ = ∈ ∈ 
    
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C J N H C=∪ ∪ ∪% % % , we suppose that ( , ) ( , ) ( , )RJ N H C O N C= ∩∪%
where
( )
, ( ) ( ) ( ) : :
( )
i
O e
u
e N C A B C O e J e H e u U i I
uµ
   
∀ ∈ ∩ = ∩ ∩ = ∪ = ∈ ∈      
{ } { }{ }
{ }
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
max max max
max
( ) ( ), ( ) ( ), ( ) , ( )
( ), ( ), ( ) ( )
O e J e H e F e G e H e
F e G e H e K e
u u u u u u
u u u u
µ µ µ µ µ µ
µ µ µ µ
= =
= =
( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ).R R R RF A G B H C F A G B H C=∪ ∪ ∪ ∪% % % %

35
3.2. Proposition (Distributive Laws)
properties:
( ) ( ) ( )(1). ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R R RF A G B H C F A G B F A H C=∪ ∩ ∪ ∩ ∪% % % % %
( ) ( ) ( )(2). ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R R RF A G B H C F A G B F A H C=∩ ∪ ∩ ∪ ∩% % % % %
Proof. (1). Assume that ( , ) ( , ) ( , )RG B H C I D=∩% , where D B C= ∩ and ,e D∀ ∈
( ) ( ) ( )I e G e H e= ∩ =
{ }( ) ( )min
: :
( ), ( )
i
G e H e
u
u U i I
u uµ µ
    ∈ ∈ 
    
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A I D=∪ ∩ ∪% % % , we suppose that ( )( , ) ( , ) ,RF A I D K M=∪% where
M A D A B C= ∩ = ∩ ∩ and ,e M∀ ∈
( )
( ) ( ) ( ) : :
( )
i
K e
u
K e F e I e u U i I
uµ
   
= ∪ = ∈ ∈      
{ }
{ }{ }
{ } { }{ }
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
max
max min
min
( ) ( ), ( )
( ), ( ), ( )
max ( ), ( ) ,max ( ), ( )
K e F e I e
F e G e H e
F e G e F e H e
where u u u
u u u
u u u u
µ µ µ
µ µ µ
µ µ µ µ
=
=
=
Suppose that ( , ) ( , ) ( , ),RF A G B J N=∪% where N A B= ∩ and ,e N∀ ∈
{ }( ) ( )max
( ) ( ) ( ) : :
( ), ( )
i
F e G e
u
J e F e G e u U i I
u uµ µ
    = ∪ = ∈ ∈ 
    
Again, let ( , ) ( , ) ( , ),RF A H C S T=∪% where T A C= ∩ and ,e T∀ ∈
{ }( ) ( )max
( ) ( ) ( ) : :
( ), ( )
i
F e H e
u
S e F e H e u U i I
u uµ µ
    = ∪ = ∈ ∈ 
    
Since ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R RF A G B F A H C J N S T=∪ ∩ ∪ ∩% % % % , we suppose that
( , ) ( , ) ( , )RJ N S T O N T= ∩∩% where ,e N T A B C∀ ∈ ∩ = ∩ ∩
( )
( ) ( ) ( ) : :
( )
i
O e
u
O e J e S e u U i I
uµ
   
= ∩ = ∈ ∈      

36
{ }
{ } { }{ }
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
min
min max max
( ) ( ), ( )
( ), ( ) , ( ), ( ) ( )
O e J e S e
F e G e F e H e K e
u u u
u u u u u
µ µ µ
µ µ µ µ µ
=
= =
( ) ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R R RF A G B H C F A G B F A H C=∪ ∩ ∪ ∩ ∪% % % % %
(2). Assume that ( , ) ( , ) ( , )RG B H C I D=∪% , where D B C= ∩ and ,e D∀ ∈ ( ) ( ) ( )I e G e H e= ∪
{ }( ) ( )max
: :
( ), ( )
i
G e H e
u
u U i I
u uµ µ
    = ∈ ∈ 
    
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A I D=∩ ∪ ∩% % % , we suppose that ( )( , ) ( , ) ,RF A I D K M=∩% where
M A D A B C= ∩ = ∩ ∩ and ,e M∀ ∈
( )
( ) ( ) ( ) : :
( )
i
K e
u
K e F e I e u U i I
uµ
   
= ∩ = ∈ ∈      
{ }
{ }{ }
{ } { }{ }
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
min
min max
max
( ) ( ), ( )
( ), ( ), ( )
min ( ), ( ) ,min ( ), ( )
K e F e I e
F e G e H e
F e G e F e H e
where u u u
u u u
u u u u
µ µ µ
µ µ µ
µ µ µ µ
=
=
=
Suppose that ( , ) ( , ) ( , ),RF A G B J N=∩% where N A B= ∩ and ,e N∀ ∈
{ }( ) ( )min
( ) ( ) ( ) : :
( ), ( )
i
F e G e
u
J e F e G e u U i I
u uµ µ
    = ∩ = ∈ ∈ 
    
Again, let ( , ) ( , ) ( , ),RF A H C S T=∩% where T A C= ∩ and ,e T∀ ∈
{ }( ) ( )min
( ) ( ) ( ) : :
( ), ( )
i
F e H e
u
S e F e H e u U i I
u uµ µ
    = ∩ = ∈ ∈ 
    
Since ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R RF A G B F A H C J N S T=∩ ∪ ∩ ∪% % % % , we suppose that
( , ) ( , ) ( , )RJ N S T O N T= ∩∪% where ,e N T A B C∀ ∈ ∩ = ∩ ∩
( )
( ) ( ) ( ) : :
( )
i
O e
u
O e J e S e u U i I
uµ
   
= ∪ = ∈ ∈      

37
{ }
{ } { }{ }
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
max
max min min
( ) ( ), ( )
( ), ( ) , ( ), ( ) ( )
O e J e S e
F e G e F e H e K e
u u u
u u u u u
µ µ µ
µ µ µ µ µ
=
= =
( ) ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R R RF A G B H C F A G B F A H C=∩ ∪ ∩ ∪ ∩% % % % %
properties:
1. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B H C F A G B H C=∪ ∪ ∪ ∪% % % %
2. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B H C F A G B H C=∩ ∩ ∩ ∩% % % %
Proof. 1. Suppose that ( , ) ( , ) ( , )EG B H C I D=∪% , where D B C= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
G e if e B C
I e H e if e C B
G e H e if e B C
 ∈ −

= ∈ −

∈ ∩U
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C F A I D=∪ ∪ ∪% % % , we suppose that ( )( , ) ( , ) ,EF A I D J M=∪% where
M A D A B C= ∪ = ∪ ∪ and ,e M∀ ∈
( )
( )
( )
( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
J e G e H e if e B C A
F e H e if e A C B
G e F e if e A B C
F e G e H e if e A B
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩ ∩
U
U
U
U ,C













Assume that ( , ) ( , ) ( , )EF A G B K S=∪% , where S A B= ∪ and ,e S∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
K e G e if e B A
F e G e if e A B
 ∈ −

= ∈ −

∈ ∩U
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C K S H C=∪ ∪ ∪% % % , we suppose that ( )( , ) ( , ) ,EK S H C L T=∪% where
T S C A B C= ∪ = ∪ ∪ and ,e T∀ ∈

38
( )
( )
( )
( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
L e G e H e if e B C A
F e H e if e A C B
G e F e if e A B C
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩ ∩
U
U
U
U ,C













Therefore it is clear that M T= and , ( ) ( )e M J e L e∀ ∈ = , that is J and L are the same operators.
Thus ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B H C F A G B H C=∪ ∪ ∪ ∪% % % % .
2. Suppose that ( , ) ( , ) ( , )EG B H C I D=∩% , where D B C= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
G e if e B C
I e H e if e C B
G e H e if e B C
 ∈ −

= ∈ −

∈ ∩I
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C F A I D=∩ ∩ ∩% % % , we suppose that ( )( , ) ( , ) ,EF A I D J M=∩% where
M A D A B C= ∪ = ∪ ∪ and ,e M∀ ∈
( )
( )
( )
( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
F e H e if e A C B
G e F e if e A B C
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩ ∩
I
I
I
I ,C













Assume that ( , ) ( , ) ( , )EF A G B K S=∩% , where S A B= ∪ and ,e S∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
K e G e if e B A
F e G e if e A B
 ∈ −

= ∈ −

∈ ∩I
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C K S H C=∩ ∩ ∩% % % , we suppose that ( )( , ) ( , ) ,EK S H C L T=∩% where
T S C A B C= ∪ = ∪ ∪ and ,e T∀ ∈

39
( )
( )
( )
( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
L e G e H e if e B C A
F e H e if e A C B
G e F e if e A B C
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩ ∩
I
I
I
I ,C













Therefore it is clear that M T= and , ( ) ( )e M J e L e∀ ∈ = , that is J and L are the same operators.
Thus ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B H C F A G B H C=∩ ∩ ∩ ∩% % % % .
3.4. Proposition (Distributive Laws)
properties:
( ) ( ) ( )(1). ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E E EF A G B H C F A G B F A H C=∩ ∪ ∩ ∪ ∩% % % % %
( ) ( ) ( )(2). ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E E EF A G B H C F A G B F A H C=∪ ∩ ∪ ∩ ∪% % % % %
Proof. (1). Suppose that ( , ) ( , ) ( , )EG B H C I D=∪% , where D B C= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
G e if e B C
I e H e if e C B
G e H e if e B C
 ∈ −

= ∈ −

∈ ∩U
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C F A I D=∩ ∪ ∩% % % , we suppose that ( )( , ) ( , ) ,EF A I D J M=∩% where
M A D A B C= ∪ = ∪ ∪ and ,e M∀ ∈
( )
( )
( )
( )( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
F e H e if e A C B
F e G e if e A B C
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩
U
I
I
I U ,C











 ∩
Assume that ( , ) ( , ) ( , )EF A G B K S=∩% , where S A B= ∪ and ,e S∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
K e G e if e B A
F e G e if e A B
 ∈ −

= ∈ −

∈ ∩I
and ( , ) ( , ) ( , )EF A H C N T=∩% , where T A C= ∪ and ,e T∀ ∈

40
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A C
N e H e if e C A
F e H e if e A C
 ∈ −

= ∈ −

∈ ∩I
Since ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B F A H C K S N T=∩ ∪ ∩ ∪% % % % , we suppose that ( )( , ) ( , ) ,EK S N T O P=∪%
where ( ) ( )P S T A B A C A B C= ∪ = ∪ ∪ ∪ = ∪ ∪ and ,e P∀ ∈
( )
( )
( )
( )( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
O e G e H e if e B C A
F e H e if e A C B
F e G e if e A B C
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩
U
I
I
I U ,C











 ∩
Therefore it is clear that M P= and , ( ) ( )e M J e O e∀ ∈ = , that is “J” and “O” are the same
operators. Thus ( ) ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E E EF A G B H C F A G B F A H C=∩ ∪ ∩ ∪ ∩% % % % % .
(2). Suppose that ( , ) ( , ) ( , )EG B H C I D=∩% , where D B C= ∪ and ,e D∀ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C F A I D=∪ ∩ ∪% % % , we suppose that ( )( , ) ( , ) ,EF A I D J M=∪% where
M A D A B C= ∪ = ∪ ∪ and ,e M∀ ∈
( )
( )
( )
( )( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
F e H e if e A C B
F e G e if e A B C
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩
I
U
U
U I ,C











 ∩
Assume that ( , ) ( , ) ( , )EF A G B K S=∪% , where S A B= ∪ and ,e S∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
K e G e if e B A
F e G e if e A B
 ∈ −

= ∈ −

∈ ∩U
and ( , ) ( , ) ( , )EF A H C N T=∪% , where T A C= ∪ and ,e T∀ ∈

41
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A C
N e H e if e C A
F e H e if e A C
 ∈ −

= ∈ −

∈ ∩U
Since ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B F A H C K S N T=∪ ∩ ∪ ∩% % % % , we suppose that ( )( , ) ( , ) ,EK S N T O P=∩%
where ( ) ( )P S T A B A C A B C= ∪ = ∪ ∪ ∪ = ∪ ∪ and ,e P∀ ∈
( )
( )
( )
( )( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
O e G e H e if e B C A
F e H e if e A C B
F e G e if e A B C
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩
I
U
U
U I ,C











 ∩
Therefore it is clear that M P= and , ( ) ( )e M J e O e∀ ∈ = , that is “J” and “O” are the same
operators. Thus ( ) ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E E EF A G B H C F A G B F A H C=∪ ∩ ∪ ∩ ∪% % % % % .
properties:
1. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )F A G B H C F A G B H C∧ ∧ = ∧ ∧
2. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )F A G B H C F A G B H C∨ ∨ = ∨ ∨
Proof. 1. Assume that ( , ) ( , ) ( , )G B H C I B C∧ = × , where ( ), ,b c B C∀ ∈ ×
{ }( ) ( )min
( , ) ( ) ( ) : :
( ), ( )
i
G b H c
u
I b c G b H c u U i I
u uµ µ
    = ∩ = ∈ ∈ 
    
Since ( )( , ) ( , ) ( , ) ( , ) ( , )F A G B H C F A I B C∧ ∧ = ∧ × , we suppose that ( )( , ) ( , ) , ( )F A I B C K A B C∧ × = × ×
where ( ) ( ), , ,a b c A B C A B C∀ ∈ × × = × ×
( , , )
( , , ) ( ) ( , ) : :
( )
i
K a b c
u
K a b c F a I b c u U i I
uµ
   
= ∩ = ∈ ∈      
{ } { }{ }
{ }
( , , ) ( ) ( , ) ( ) ( ) ( )
( ) ( ) ( )
min min min
min
( ) ( ), ( ) ( ), ( ), ( )
( ), ( ), ( )
K a b c F e I b c F e G b H c
F e G b H c
where u u u u u u
u u u
µ µ µ µ µ µ
µ µ µ
= =
=
We take ( , ) .a b A B∈ × Suppose that ( , ) ( , ) ( , )F A G B J A B∧ = × where ( ), ,a b A B∀ ∈ ×
{ }( ) ( )min
( , ) ( ) ( ) : :
( ), ( )
i
F a G b
u
J a b F a G b u U i I
u uµ µ
    = ∩ = ∈ ∈ 
    

42
Since ( )( , ) ( , ) ( , ) ( , ) ( , )F A G B H C J A B H C∧ ∧ = × ∧ , we suppose that ( , ) ( , ) ( ,( ) )J A B H C O A B C× ∧ = × ×
where ( ) ( ), , ,a b c A B C A B C∀ ∈ × × = × ×
( , , )
( , , ) ( , ) ( ) : :
( )
i
O a b c
u
O a b c J a b H c u U i I
uµ
   
= ∩ = ∈ ∈      
{ } { }{ }
{ }
( , , ) ( , ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( , , )
min min min
min
( ) ( ), ( ) ( ), ( ) , ( )
( ), ( ), ( ) ( )
O a b c J a b H c F e G b H c
F e G b H c K a b c
u u u u u u
u u u u
µ µ µ µ µ µ
µ µ µ µ
= =
= =
( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ).F A G B H C F A G B H C∧ ∧ = ∧ ∧
2. Assume that ( , ) ( , ) ( , )G B H C I B C∨ = × , where ( ), ,b c B C∀ ∈ ×
{ }( ) ( )max
( , ) ( ) ( ) : :
( ), ( )
i
G b H c
u
I b c G b H c u U i I
u uµ µ
    = ∪ = ∈ ∈ 
    
Since ( )( , ) ( , ) ( , ) ( , ) ( , )F A G B H C F A I B C∨ ∨ = ∨ × , we suppose that ( )( , ) ( , ) , ( )F A I B C K A B C∨ × = × ×
( ) ( )
( , , )
where , , , ( , , ) ( ) ( , ) : :
( )
i
K a b c
u
a b c A B C A B C K a b c F a I b c u U i I
uµ
   
∀ ∈ × × = × × = ∪ = ∈ ∈      
{ } { }{ }
{ }
( , , ) ( ) ( , ) ( ) ( ) ( )
( ) ( ) ( )
max max max
max
( ) ( ), ( ) ( ), ( ), ( )
( ), ( ), ( )
K a b c F e I b c F e G b H c
F e G b H c
where u u u u u u
u u u
µ µ µ µ µ µ
µ µ µ
= =
=
We take ( , ) .a b A B∈ × Suppose that ( , ) ( , ) ( , )F A G B J A B∨ = × where ( ), ,a b A B∀ ∈ ×
{ }( ) ( )max
( , ) ( ) ( ) : :
( ), ( )
i
F a G b
u
J a b F a G b u U i I
u uµ µ
    = ∪ = ∈ ∈ 
    
Since ( )( , ) ( , ) ( , ) ( , ) ( , )F A G B H C J A B H C∨ ∨ = × ∨ , we suppose that ( , ) ( , ) ( ,( ) )J A B H C O A B C× ∨ = × ×
where
( ) ( )
( , , )
, , , ( , , ) ( , ) ( ) : :
( )
i
O a b c
u
a b c A B C A B C O a b c J a b H c u U i I
uµ
   
∀ ∈ × × = × × = ∪ = ∈ ∈      
{ } { }{ }
{ }
( , , ) ( , ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( , , )
max max max
max
( ) ( ), ( ) ( ), ( ) , ( )
( ), ( ), ( ) ( )
O a b c J a b H c F e G b H c
F e G b H c K a b c
where u u u u u u
u u u u
µ µ µ µ µ µ
µ µ µ µ
= =
= =

43
( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ).F A G B H C F A G B H C∨ ∨ = ∨ ∨
4. An Application Of Fuzzy Soft Multi Set Theory In Information
System
In this section we define some basic supporting tools in information system and also application
of fuzzy soft multi sets in information system are presented and discussed.
4.1. Definition
A fuzzy multi-valued information system is a quadruple ( ), , ,systemInf X A f V= where X is
a non empty finite set of objects, A is a non empty finite set of attribute, a
a A
V V
∈
= ∪ , where V is
the domain (a fuzzy set,) set of attribute, which has multi value and :f X A V× → is a total
function such that ( ), af x a V∈ for every ( ), .x a X A∈ ×
4.1. Proposition
If ( ),F A is a fuzzy soft multi set over universe U, then ( ),F A is a fuzzy multi-valued
information system.
Proof. Let { }:iU i I∈ be a collection of universes such that i I iU φ∈ =I and let { }:iE i I∈
be a collection of sets of parameters. Let ( )i I iU IFS U∈= ∏ where ( )iIFS U denotes the set of
all fuzzy subsets of iU , ii I UE E∈= ∏ and A E⊆ . Let ( ),F A be an fuzzy soft multi set over
U and i
i I
X U
∈
= ∪ . We define a mapping f where :f X A V× → , defined as
( ) ( ) ( ), / F a
f x a x xµ= .
Hence a
a A
V V
∈
= ∪ where aV is the set of all counts of in ( )F a and ∪ represent the classical set
union. Then the fuzzy multi-valued information system ( ), , ,X A f V represents the fuzzy soft
multi set ( ),F A .
4.1. Application in information system
Let us consider three universes { }1 1 2 3 4 5, , , ,U h h h h h= , { }2 1 2 3 4, , ,U c c c c= and
{ }3 1 2 3, ,U v v v= be the sets of “houses,” “cars,” and “hotels”, respectively. Suppose Mr. X has a
budget to buy a house, a car and rent a venue to hold a wedding celebration. Let us consider a
intuitionistic fuzzy soft multi set (F, A) which describes “houses,” “cars,” and “hotels” that Mr. X
is considering for accommodation purchase, transportation purchase, and a venue to hold a

44
wedding celebration, respectively. Let { }1 2 3
, ,U U UE E E be a collection of sets of decision
parameters related to the above universes, where
{ }
{ }
{ }
1 1 1 1
2 2 2 2
3 3 3 3
,1 ,2 ,3
,1 ,2 ,3
,1 ,2 ,3
expensive, cheap, wooden
beautiful, cheap, sporty ,
expensive, model, beautiful .
U U U U
U U U U
U U U U
E e e e
E e e e
E e e e
= = = =
= = = =
= = = =
Let ( )3
1i iU FS U== ∏ , 3
1 ii UE E== ∏ and A E⊆ , such that
{
}
1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3
1 ,1 ,1 ,1 2 ,1 ,2 ,1 3 ,2 ,3 ,1
4 ,3 ,3 ,1 5 ,1 ,1 ,2 6 ,1 ,2 ,2
( , , ), ( , , ), ( , , ),
( , , ), ( , , ), ( , , ) .
U U U U U U U U U
U U U U U U U U U
A a e e e a e e e a e e e
a e e e a e e e a e e e
= = = =
= = =
Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of
choice parameters. Let
( ) { } { } { }( )
( ) { } { } { }( )
( ) { }
1 1 2 3 4 5 1 2 3 4 1 2 3
2 1 2 3 4 5 1 2 3 4 1 2 3
3 1 2 3 4 5 1 2
/ 0.2, / 0.4, / 0, / 0.4, / 0 , / 0.8, / 0.1, / 0, /1 , / 0.8, / 0.7, / 0 ,
/ 0.3, / 0.5, / 0, / 0.4, /1 , / 0.9, / 0.6, /1, / 0 , / 0.6, / 0.6, / 0 ,
/ 0.5, / 0.7, /1, / 0.4, / 0 , / 0.6,
F a h h h h h c c c c v v v
F a h h h h h c c
=
=
= { } { }( )
( ) { } { } { }( )
( ) { } { } { }( )
( )
3 4 1 2 3
4 1 2 3 4 5 1 2 3 4 1 2 3
5 1 2 3 4 5 1 2 3 4 1 2 3
6 1 2
/ 0.1, / 0, /1 , / 0, / 0.7, / 0.7 ,
/ 0.4, / 0.5, / 0.1, / 0, /1 , / 0.8, / 0, / 0.8, /1 , /1, / 0.7, / 0 ,
/ 0.7, / 0.5, /1, / 0.4, /1 , / 0, / 0.1, / 0.7, /1 , /1, / 0.8, / 0 ,
/ 0.6, /
c c v v v
F a h h
=
=
= { } { } { }( )3 4 5 1 2 3 4 1 2 30.4, / 0.8, /1, / 0.1 , / 0, / 0.1, /1, /1 , /1, / 0, /1 ,h h h c c c c v v v
Then the fuzzy soft multi set ( ),F A defined above describes the conditions of some “house”,
“car” and “hotel” in a state. Then the quadruple ( ), , ,X A f V corresponding to the fuzzy soft
multi set given above is a fuzzy multi-valued information system.
Where
3
1
i
i
X U
=
= ∪ and A is the set of parameters in the fuzzy soft multi set and
{ }
{ }
1
2
3
1 2 3 4 5 1 2 3 4 1 2 3
1 2 3 4 5 1 2 3 4 1 2 3
1 2 3 4 5 1 2
/ 0.2, / 0.4, / 0, / 0.4, / 0, / 0.8, / 0.1, / 0, /1, / 0.8, / 0.7, / 0 ,
/ 0.3, / 0.5, / 0, / 0.4, /1, / 0.9, / 0.6, /1, / 0, / 0.6, / 0.6, / 0 ,
/ 0.5, / 0.7, /1, / 0.4, / 0, / 0.6,
a
a
a
V h h h h h c c c c v v v
V h h h h h c c
=
=
= { }
{ }
{ }
4
5
6
3 4 1 2 3
1 2 3 4 5 1 2 3 4 1 2 3
1 2 3 4 5 1 2 3 4 1 2 3
1 2
/ 0.1, / 0, /1, / 0, / 0.7, / 0.7 ,
/ 0.4, / 0.5, / 0.1, / 0, /1, / 0.8, / 0, / 0.8, /1, /1, / 0.7, / 0 ,
/ 0.7, / 0.5, /1, / 0.4, /1, / 0, / 0.1, / 0.7, /1, /1, / 0.8, / 0 ,
/ 0.6, /
a
a
a
c c v v v
V h h
=
=
= { }3 4 5 1 2 3 4 1 2 30.4, / 0.8, /1, / 0.1, / 0, / 0.1, /1, /1, /1, / 0, /1h h h c c c c v v v
,

45
For the pair ( )1 1,h a we have ( )1 1,f h a = 0.2, for ( )2 1,h a , we have ( )2 1,f h a = 0.4. Continuing
in this way we obtain the values of other pairs. Therefore, according to the result above, it is seen
that fuzzy soft multi sets are fuzzy soft multi-valued information systems. Nevertheless, it is
obvious that fuzzy soft multi-valued information systems are not necessarily fuzzy soft multi sets.
We can construct an information table representing fuzzy soft multi set ( ),F A defined above as
follows.
4.1. An Information Table
Table 2. An information table
X a1 a2 a3 a4 a5 a6
h1
h2
h3
h4
h5
0.2
0.4
0
0.4
0
0.3
0.5
0
0.4
1
0.5
0.7
1
0.4
0
0.4
0.5
0.1
0
1
0.7
0.5
1
0.4
1
0.6
0.4
0.8
1
0.1
c1
c2
c3
c4
0.8
0.1
0
1
0.9
0.6
1
0
0.6
0.1
0
1
0.8
0
0.8
1
0
0.1
0.7
1
0
0.1
1
1
v1
v2
v3
0.8
0.7
0
0.6
0.6
0
0
0.7
0.7
1
0.7
0
1
0.8
0
1
0
1
5. Conclusion
The theory of soft set, fuzzy soft set and multi set are vital mathematical tools used in handling
uncertainties about vague concepts. In this paper, we have introduced the notion of fuzzy multi-
valued information system in fuzzy soft multi set theory. Here we shall present the application of
fuzzy soft multi set in information system and show that every fuzzy soft multi set is a fuzzy
multi valued information system.
References
[1] K.Alhazaymeh & N. Hassan, (2014) “Vague Soft Multiset Theory”, Int. J. Pure and Applied Math.,
Vol. 93, pp511-523.
[2] M.I.Ali, F. Feng, X. Liu, W.K. Minc & M. Shabir, (2009) “On some new operations in soft set
theory”, Comp. Math. Appl., vol. 57, pp1547-1553.
[3] S.Alkhazaleh & A.R. Salleh, (2012) “Fuzzy Soft Multi sets Theory”, Hindawi Publishing
Corporation, Abstract and Applied Analysis, Article ID 350603, 20 pages, doi: 10.1155/2012/350603.
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pp3561– 3573.
[5] K.V Babitha & S.J. John, (2013) “On Soft Multi sets”, Ann. Fuzzy Math. Inform., Vol. 5 pp35-44.
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46
[10] P.K. Maji, R. Biswas & A.R. Roy, (2003) “Soft set theory”, Comp. Math. Appl., Vol. 45 pp555-562.
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