Fuzzy soft sets

FUZZY SOFT SETS
S. Anita Shanthi
Department of Mathematics,
Annamalai University,
Annamalainagar-608002,
Tamilnadu, India.
E-mail : shanthi.anita@yahoo.com
S. Anita Shanthi (Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@FUZZY SOFT SETS 1 / 1

Introduction
1.Introduction
There are theories viz. theory of probability, theory of evidence,
theory of fuzzy sets, theory of intuitionistic fuzzy sets, theory of
vague sets, theory of interval mathematics, theory of rough sets
which can be considered as mathematical tools for dealing with
uncertainties. But these theories have their inherent diﬃculties as
pointed out by Molodstov. The reason for these diﬃculties is possibly
the inadequacy of the parametrization tool of the theories.

Introduction
Introduction
Consequently Molodtsov initiated the concept of soft theory as new
mathematical tool for dealing with uncertainties which is free from
the above diﬃculties. Soft set theory has rich potential for
applications in several directions, few of which have been discussed.

2. Theory of Soft Sets
Deﬁnition 2.1.
Let X be a non-empty set. A fuzzy set A is characterized by its
membership function µA : X → [0, 1] and µA(x) is interpreted as the
degree of membership of element x in X. A is completely determined
by the set of tuples, A = {(x, µA(x)) : x ∈ X}.
Let U be an initial universe set and E be a set of parameters. Let
P(U) denote the power set of U and A ⊂ E.

Deﬁnition 2.2.
The pair (F, A) is called a soft set over U, where F is a mapping
given by F : A → P(U). In other words, a soft set over U is
parametrized family of subsets of the universe U. For ǫ ∈ A, F(ǫ)
may be considered as the set of ǫ-approximate elements of the soft
set (F, A).

Example 2.3.
Suppose that, U is the set of houses under consideration, E is the set
of parameters. Each parameter is a linguistic expression.
E = { expensive; beautiful; wooden; cheap; in the green
surroundings; modern;in good condition; in bad condition}. In this
case, to deﬁne a soft set means to point out expensive houses,
beautiful houses and so on. The soft set (F, E) describes the
”attractiveness of the houses” which Mr. X (say) is planning to buy.

Suppose that, there are six houses in the universe U given by
U = {h1, h2, h3, h4, h5, h6}, and A = {e1, e2, e3, e4, e5} where e1
stands for the parameter ’expensive’, e2 stands for the ’beautiful’, e3
stands for the parameter ’wooden’, e4 stands for the parameter
’cheap’, e5 stands for the parameter ’in the green surroundings’.
Suppose that, F(e1) = {h2, h4}, F(e2) = {h1, h3}, F(e3) =
{h3, h4, h5}, F(e4) = {h1, h3, h5}, F(e5) = {h1}. The soft set (F, A)
is a parametrized family {F(ei ), i = 1, 2, ..., 5} of subsets of the set U
and gives us a collection of approximate description of the object.

Consider the mapping F which is ”houses (•) ” where dot (•)is to
ﬁlled up by the parameter e ∈ A. Therefore F(e1) means
”houses(expensive)” whose functional-value is the set {h2, h4}. Thus,
we can view the soft-set (F, A) as a collection of approximations as
below:
(F, A) = { expensive houses ={h2, h4}, beautiful houses ={h1, h3},
wooden houses ={h3, h4, h5}, cheap houses ={h1, h3, h5}, houses in
the green surroundings ={h1}, where each approximation has two
parts (i) a predicate p and
(ii) an approximate value-set γ ( or simply to be called value-set γ ).

Tabular representation of a soft set
U expensive beautiful wooden cheap in green surroundings
h1 0 1 0 1 1
h2 1 0 0 0 0
h3 0 1 1 1 0
h4 1 0 1 0 0
h5 0 0 1 1 0
h6 0 0 0 0 0

Deﬁnition 2.4.
For two soft sets (F, A) and (G, B) over a common universe U, we
say that (F, A) is a soft subset of (G, B) if
(i) A ⊂ B,
(ii) For all ǫ ∈ A, F(ǫ) and G(ǫ) are identical approximations.
We write (F, A) ⊂ (G, B) . (F, A) is said to be a soft super set of
(G, B), if (G, B) is a soft subset of (F, A). We denote it by
(F, A) ⊃ (G, B).

Deﬁnition 2.5.( Equality of two soft sets)
Two soft sets (F, A) and (G, B) over a common universe U are said
to be soft equal if (F, A) is a soft subset of (G, B) and (G, B) is a
soft subset of (F, A).

Example 2.6.
Let A = {e1, e3, e5} ⊂ E and B = {e1, e2, e3, e5} ⊂ E. Clearly A ⊂ B.
Let (F, A) and (G, B) be two soft sets over the same universe
U = {h1, h2, h3, h4, h5, h6} such that
G(e1) = {h2, h4}, G(e2) = {h1, h3},
G(e3) = {h3, h4, h5}, G(e5) = {h1} and F(e1) = {h2, h4},
F(e3) = {h3, h4, h5}, F(e5) = {h1}. Therefore, (F, A) ⊂ (G, B).

Deﬁnition 2.7.(NOT set of parameters)
Let E = {e1, e2, ..., en} be a set of parameters. The NOT of set E
denoted by ¬E is deﬁned by ¬E = {e1, e2, ..., en} where ∨ei = notei .

Proposition 2.8.
1.¬(¬A) = A
2.¬(A ∪ B) = (¬A ∪ ¬B)
3.¬(A ∩ B) = (¬A ∩ ¬B).
Example 2.9.
Consider the example as presented in Example 2.3. Here ¬E = {not
expensive; not beautiful; not in the green surroundings; ... ;not in
bad condition }.

Deﬁnition 2.10. (Complement of a soft set)
The complement of a soft set (F, A) is denoted by (F, A)c
and is
deﬁned by (F, A)c
= (Fc
, ¬A), where Fc
: ¬A → P(U) is a mapping
given by Fc
(α) = U − F(¬α) for all α ∈ ¬A. Let us call Fc
to be
the soft complement function of F. Clearly (Fc
)c
is same as F and
((F, A)c
)c
= (F, A).
Example 2.11. Consider Example 2.3. Here (F, A)c
= {not expensive
houses= {h1, h3, h5, h6}, not beautiful houses= {h2, h4, h5, h6}, not
wooden houses = {h1, h2, h6}, not cheap houses = {h2, h4}, not in
the green surroundings houses = {h2, h3, h4, h5, h6}}.

Deﬁnition 2.12. (Null soft set)
A soft set (F, A) over U is said to be a Null soft set denoted by φ, if
for all ǫ ∈ A F(ǫ) = φ(null-set).
Deﬁnition 2.13.(Absolute soft set). A soft set (F, A) over U is said
to be absolute soft set denoted by A, if for all ǫ ∈ A, F(ǫ) = U.
Clearly, A
c
= φ and φc
= A.

Deﬁnition 2.14.(AND operation on two soft sets)
. If (F, A) and (G, B) be two soft sets then” (F, A) AND (G, B)”
denoted by (F, A) ∧ (G, B) is deﬁned by (F, A) ∧ (G, B)=(H, A × B),
Where H(α, β) = F(α) ∩ G(β) for all (α, β) ∈ A × B.

Example 2.15.
Consider the soft set (F, A) which describe the ”cost of the houses”
and the soft set (G, B) which describes the ”attractiveness of the
houses”. Suppose that U = {h1, h2, h3, h4, h5, h6, h7, h8, h9, h10},
A = { very costly; costly; cheap } and B = { beautiful; in the green
surroundings; cheap }. Let F( very costly)={h2, h4, h7, h8},
F(costly)= {h1, h3, h5}, F( cheap)= {h6, h9, h10} and

G( beautiful)={h2, h3, h7}, G(in the green surroundings)
= {h5, h6, h8}, G( cheap)= {h6, h9, h10}. Then
(F, A) ∧ (G, B) = (H, A × B), where H( very costly,
beautiful)={h2, h7}, H(very costly, in the green surroundings)= {h8},
H(very costly, cheap)= φ, H( costly, beautiful)={h3}, H( costly, in
the green surroundings)= {h5}, H( costly, cheap)= φ, H( cheap,
beautiful)=φ, H( cheap, in the green surroundings)= {h6}, H(cheap,
cheap)= {h6, h9, h10}.

Deﬁnition 2.16.(OR operation on two soft sets)
If (F, A) and (G, B) are two soft sets, then (F, A) OR (G, B) denoted
by (F, A) ∨ (G, B) is deﬁned by (F, A) ∨ (G, B) = (O, A × B),where
O(α, β) = F(α) ∪ G(β), for all (α, β) ∈ A × B.

Example 2.17.
Consider the Example 2.15 above. We see that
(F, A) ∨ (G, B) = (O, A × B), where O( very costly)={h2, h4, h7, h8},
O( very costly, in the green surroundings)={h2, h4, h5, h6, h7, h8},
O( very costly, cheap)={h2, h4, h5, h6, h7, h8, h9, h10},
O( costly, beautiful)={h1, h2, h3, h5, h7},
O( costly, in the green surroundings)={h1, h3, h5, h6, h8},
O( costly, cheap)={h1, h3, h5, h6, h9, h10}
O( cheap, beautiful)={h2, h3, h6, h7, h9, h10},
O( cheap, in the green surroundings)={h5, h6, h8, h9, h10},
O( cheap, cheap)={h6, h9, h10}.

The following De Morgan’s types of results are true.
Proposition 2.18.
(i) ((F, A) ∨ (G, B))c
= (F, A)c
∧ (G, B)c
.
(ii) ((F, A) ∧ (G, B))c
= (F, A)c
∨ (G, B)c
.

Deﬁnition 2.19.
Union of two soft sets (F, A) and (G, B) over the common universe
U is the soft set (H, C), where C = A ∪ B, and for all e ∈ C,
H(e) =



F(e), e ∈ A − B
G(e), e ∈ B − A
F(e) ∪ G(e), e ∈ A ∩ B.

We write (F, A) ∪ (G, B) = (H, C).
In the above example, (F, A) ∪ (G, B) = (H, C), where
H(very costly)={h2, h4, h7, h8},
H( costly)={h1, h3, h5},
H(cheap)={h6, h9, h10},
H(beautiful)={h2, h3, h7} and
H(in the green surroundings)={h5, h6, h8}.

Deﬁnition 2.20.
Intersection of two soft sets (F, A) and (G, B) over the common
universe U is the soft set(H, C), where C = A ∩ B, and for all
e ∈ C, H(e) = F(e) or G(e), (as both are same set).
We write (F, A) ∩ (G, B) = (H, C).
In the above example intersection of two soft sets (F, A) and (G, B)
is the soft set (H, C), where C = {cheap} and
H(cheap) = {h6, h9, h10}.

Proposition 2.21.
(i) (F, A) ∪ (F, A) = (F, A).
(ii) (F, A) ∩ (F, A) = (F, A).
(iii) (F, A) ∪ φ = φ, where φ is the null soft set.
(iv) (F, A) ∩ φ = φ.
(v) (F, A) ∪ A = A, where A is the absolute soft set.
(vi) (F, A) ∩ A = (F, A).

Proposition 2.22.
(i) ((F, A) ∪ (G, B))c
= (F, A)c
∪ (G, B)c
.
(ii) ((F, A) ∩ (G, B))c
= (F, A)c
∩ (G, B)c
.

3. Fuzzy Soft Sets
3. Fuzzy Soft Sets
In this section we define what are called fuzzy-soft sets. In case of
soft sets, we observe that in most cases the elements of the
parameter sets are linguistic expressions involving fuzzy words. This
motivates the definition of fuzzy soft sets.
Definition 3.1.
Let U be an initial universe set and E be a set of parameters. Let
P(U) denote the set of all fuzzy sets of U. Let A ⊂ E. A pair (F, A)
is called a fuzzy soft set over U, where F is a mapping given by
F : A → P(U).

3. Fuzzy Soft Sets
We give an example of a fuzzy soft set.
Example 3.2.
Suppose that, U is the set of houses under consideration, E is the set
of parameters. Each parameter is a fuzzy word or a sentence
involving fuzzy words.
E = { expensive; beautiful; wooden; cheap; in green surroundings }.
In this case, to deﬁne a fuzzy soft set means to point out expensive
houses, beautiful houses and so on. The fuzzy soft set (F, E)
describes the ”attractiveness of the houses” which Mr. X (say) is
planning to buy.

3. Fuzzy Soft Sets
Suppose that, there are six houses in the universe U given by
U = {h1, h2, h3, h4, h5, h6}, and E = {e1, e2, e3, e4, e5} where e1
stands for the parameter ’expensive’, e2 stands for the ’beautiful’, e3
stands for the parameter ’wooden’, e4 stands for the parameter
’cheap’, e5 stands for the parameter ’in the green surroundings’.
Suppose that, F(e1) = {h1/.5, h2/1, h3/.4, h4/1, h5/.3, h6/0},
F(e2) = {h1/1, h2/.4, h3/1, h4/.4, h5/.6, h6/.8},
F(e3) = {h1/.2, h2/.3, h3/1, h4/1, h6/0},
F(e4) = {h1/1, h2/0, h3/1, h4/.2, h5/1, h6/.2},
F(e5) = {h1/1, h2/.1, h3/.5, h4/.3, h5/.2, h6/.3}.

3. Fuzzy Soft Sets
The fuzzy soft set (F, E) is a parametrized family
{F(ei ), i = 1, 2, ..., 5} of all fuzzy subsets of the set U and gives us a
collection of approximate description of the object. Consider the
mapping F which is ”houses (•) ” where dot (•)is to ﬁlled up by the
parameter e ∈ E. Therefore F(e1) means ”houses(expensive)” whose
functional-value is the fuzzy set
{h1/.5, h2/1, h3/.4, h4/1, h5/.3, h6/0}.

3. Fuzzy Soft Sets
Thus, we can view the fuzzy soft-set (F, E) as a collection of fuzzy
approximations as below:
(F, E) = expensive houses = {h1/.5, h2/1, h3/.4, h4/1, h5/.3, h6/0},
beautiful houses ={h1/1, h2/.4, h3/1, h4/.4, h5/.6, h6/.8},
wooden houses ={h1/.2, h2/.3, h3/1, h4/1, h6/0},
cheap houses ={h1/1, h2/0, h3/1, h4/.2, h5/1, h6/.2},
houses in the green surroundings
={h1/1, h2/.1, h3/.5, h4/.3, h5/.2, h6/.3}.

3. Fuzzy Soft Sets
where each approximation has two parts
(i) a predicate p
(ii) an approximate value- fuzzy set γ ( or simply to be called
value-set γ ).
For example, for the approximation
”expensive houses”= {h1/.5, h2/1, h3/.4, h4/1, h5/.3, h6/0} we have
(i) the predicate name is expensive houses,
(ii) the approximate value-set is
{h1/.5, h2/1, h3/.4, h4/1, h5/.3, h6/0}.

3. Fuzzy Soft Sets
For the purpose of storing a soft set in a computer, we could
represent a soft set in the form of a table, as shown below
(corresponding to the soft set in the above example).
In this table, the entries are hij corresponding to the house hi and
parameter ej , where hij = membership value of hi in F(ej ).

3. Fuzzy Soft Sets
Tabular representation of a fuzzy soft set
U expensive beautiful wooden cheap in green surroundings
h1 0.5 1.0 0.2 1.0 1.0
h2 1.0 0.4 0.3 0.0 0.1
h3 0.4 1.0 1.0 1.0 0.5
h4 1.0 0.4 1.0 0.2 0.3
h5 0.3 0.6 1.0 1.0 0.2
h6 0.0 0.8 0.0 0.2 0.3

3. Fuzzy Soft Sets
Deﬁnition 3.3.
For two fuzzy soft sets (F, A) and (G, B) over a common
universe U, we say that (F, A) is a soft subset of (G, B) if
(i) A ⊂ B,
(ii) For all ǫ ∈ A, F(ǫ) and G(ǫ) are identical approximations.
We write (F, A)⊂(G, B) .
(F, A) is said to be a fuzzy soft super set of (G, B), if (G, B) is a
fuzzy soft subset of (F, A). We denote it by (F, A)⊃(G, B).

3. Fuzzy Soft Sets
Example 3.4.
Consider two fuzzy soft sets (F, A) and (G, B), where
A={ cheap, in the green surroundings}
B={ cheap, in the green surroundings, in good condition} and
F(cheap)={h1/1, h2/.5, /h3/.5, /h4/1, h5/.7}
F(in the green surroundings)={h1/.5, h2/.6, h3/.3, /h4/1, h5/1}
G(cheap)={h1/1, h2/.6, /h3/.5, /h4/1, h5/1}
G(in the green surroundings)={h1/.6, h2/.6, h3/.5, /h4/1, h5/1}
G(in good repair)={h1/.4, h2/.6, /h3/.5, /h4/.8, h5/1}.
Clearly (F, A)⊂(G, B) .

3. Fuzzy Soft Sets
Deﬁnition 3.4.(Equality of two fuzzy soft sets)
Two fuzzy soft sets (F, A) and (G, B) over a common universe U are
said to be fuzzy soft equal if (F, A) is a fuzzy soft subset of (G, B)
and (G, B) is a fuzzy soft subset of (F, A).

3. Fuzzy Soft Sets
Deﬁnition 3.5(Complement of a fuzzy soft set)
The complement of a fuzzy soft set (F, A) is denoted by (F, A)c
and
is deﬁned by (F, A)c
= (Fc
, ¬A), where Fc
: ¬A → P(U) is a
mapping given by F(α) = fuzzy complement of F(¬α} for all
α ∈ ¬A. Let us call Fc
to be the fuzzy soft complement function of
F. Clearly (Fc
)c
is same as F and ((F, A)c
)c
= (F, A).

3. Fuzzy Soft Sets
Example 3.6.
Consider the Example 3.1.
Here (F, E)c
=
{not expensive houses} = {h1/.5, h2/0, h3/.6, h4/0, h5/.7, h6/1},
not beautiful houses= {h1/0, h2/.6, h3/0, h4/.6, h5/.4, h6/.2},
not wooden houses = {h1/.8, h2/.1, h3/0, h4/.8, h5/0, h6/1},
not cheap houses = {h1/0, h2/1, h3/0, h4/.8, h5/0, h6/.8},
not in the green surroundings houses
= {h1/0, h2/.9, h3/.5, h4/.7, h.5, h6/.7}}.

3. Fuzzy Soft Sets
Deﬁnition 3.7. (Null fuzzy soft set)
A fuzzy soft set (F, A) over U is said to be a Null fuzzy soft set
denoted by φ, if for all ǫ ∈ A F(ǫ) = φ (null fuzzy set of U).

3. Fuzzy Soft Sets
Deﬁnition 3.8. (Absolute fuzzy soft set)
A fuzzy soft set (F, A) over U is said to be absolute fuzzy soft set
denoted by A,if for all ǫ ∈ A, F(ǫ) = U. Clearly, A
c
= φ and φc
= A.

3. Fuzzy Soft Sets
Deﬁnition 3.9. (AND operation on two fuzzy soft sets)
If (F, A) and (G, B) be two fuzzy soft sets then” (F, A) AND
(G, B)” denoted by (F, A) ∧ (G, B) is deﬁned by
(F, A) ∧ (G, B)=(H, A × B), Where H(α, β) = F(α)∩G(β), where
∩ is the operation’ fuzzy intersection’ of two fuzzy soft sets.

3. Fuzzy Soft Sets
Example 3.10.
Consider the fuzzy soft set (F, A) which describe the ”‘cost of the
houses” and the fuzzy soft set (G, B) which describes the
”attractiveness of the houses”. Suppose that
U = {h1, h2, h3, h4, h5, h6, h7, h8}, A = { very costly; costly; cheap }
and B = { beautiful; in the green surroundings; cheap }.

3. Fuzzy Soft Sets
Let
F( very costly)={h1/.7, h2/.1, h3/.8, h4/1, h5/.9, h6/.3, h7/1, h8/1},
F(costly)= {h1/1, h2/1, h3/.8, h4/1, h5/.9, h6/.3, h7/1, h8/1},
F(cheap)= {h1/.4, h2/.2, h3/.5, h4/.4, h5/.3, h6/1, h7/.1, h8/.6} and
G(beautiful)={h1/.6, h2/1, h3/1, h4/.8, h5/.6, h6/.8, h7/1, h8/.8},
G(in the green surroundings)
= {h1/.5, h2/.7, h3/.8, h4/.6, h5/1, h6/1, h7/.9, h8/1},
G(cheap)= {h1/.4, h2/.2, h3/.5, h4/.4, h5/.3, h6/1, h7/1, h8/.6}.

3. Fuzzy Soft Sets
Then (F, A) ∧ (G, B) = (H, A × B), where
H(very costly, beautiful)
= {h1/.6, h2/.1, h3/.8, h4/.8, h5/.6, h6/.3, h7/1, h8/.8},
H(very costly, in the green surroundings)
= {h1/.5, h2/.7, h3/.8, h4/.6, h5/.9, h6/.3, h7/.9, h8/1},
H(very costly, cheap)
= {h1/.4, h2/.1, h3/.5, h4/.4, h5/.3, h6/.3, h7/1, h8/.6},
H(costly, beautiful)
= {h1/.6, h2/1, h3/.8, h4/.8, h5/.6, h6/.3, h7/1, h8/.8},

3. Fuzzy Soft Sets
H(costly, in the green surroundings)
= {h1/.5, h2/.7, h3/.8, h4/.6, h5/.1, h6/.4, h7/.9, h8/1},
H(costly, cheap)
= {h1/.4, h2/.2, h3/.5, h4/.4, h5/.3, h6/.4, h7/.1, h8/.6},
H(cheap, beautiful)
= {h1/.4, h2/.2, h3/.5, h4/.4, h5/.3, h6/.8, h7/1, h8/.6},
H(cheap, in the green surroundings)
= {h1/.4, h2/.2, h3/.5, h4/.4, h5/.3, h6/1, h7/.1, h8/.6},
H(cheap, cheap)
= {h1/.4, h2/.2, h3/.5, h4/.4, h5/.3, h6/1, h7/.1, h8/.6}.

3. Fuzzy Soft Sets
Deﬁnition 3.11 (OR operation on two soft sets)
If (F, A) and (G, B) be two fuzzy soft sets then (F, A) OR (G, B)
denoted by (F, A) ∨ (G, B) and is deﬁned by
(F, A) ∨ (G, B) = (O, A × B),Where O(α, β) = F(α)∪G(β), where
∪ is the operation ’fuzzy union’of two fuzzy sets.

3. Fuzzy Soft Sets
Example 3.12.
Consider the Example 2.5 above. We see that
(F, A) ∨ (G, B) = (O, A × B),
O(very costly, beautiful)
={h1/.7, h2/1, h3/1, h4/1, h5/.9, h6/.8, h7/1, h8/1},
O(very costly, in the green surroundings)
= {h1/.7, h2/1, h3/.8, h4/1, h5/1, h6/1, h7/1, h8/1},
O(very costly, cheap)
= {h1/.7, h2/1, h3/.5, h4/.4, h5/.3, h6/.3, h7/.1, h8/.6} and
O(costly, beautiful)
={h1/1, h2/1, h3/1, h4/1, h5/1, h6/.8, h7/.1, h8/1},

3. Fuzzy Soft Sets
O(costly, in the green surroundings)
= {h1/1, h2/1, h3/1, h4/1, h5/1, h6/1, h7/1, h8/1},
O(costly, cheap)
= {h1/1, h2/1, h3/1, h4/1, h5/1, h6/1, h7/1, h8/1},
O(cheap, beautiful)
={h1/.6, h3/1, h4/.8, h5/.6, h6/.8, h7/1, h8/.8},
O(cheap, in the green surroundings)
={h1/.5, h2/.7, h3/.8, h4/.6, h5/1, h6/1, h7/.9, h8/1},
O(cheap, cheap)
= {h1/.4, h2/.2, h3/.5, h4/.4, h5/.3, h6/1, h7/.1, h8/.6}.

3. Fuzzy Soft Sets
The following De Morgan’s types of results are true.
Proposition 3.13.
(i) ((F, A) ∨ (G, B))c
= (F, A)c
∧ (G, B)c
.
(ii) ((F, A) ∧ (G, B))c
= (F, A)c
∨ (G, B)c
.

3. Fuzzy Soft Sets
Deﬁnition 3.14.
Union of two fuzzy soft sets (F, A) and (G, B) over the common
universe U is the fuzzy soft set (H, C), where C = A ∪ B, and for all
e ∈ C,
H(e) =



F(e), e ∈ A − B
G(e), e ∈ B − A
F(e) ∪ G(e), e ∈ A ∩ B.
We write (F, A)∪(G, B) = (H, C).

3. Fuzzy Soft Sets
Example 3.15.
Consider the Example 3.4. Here (F, A)∪(G, B) = (H, C), where
H(very costly)={h1/.7, h2/.1, h3/.8, h4/1, h5/.9, h6/.3, h7/1, h8/1},
H( costly)={h1/1, h2/1, h3/1, h4/1, h5/1, h6/1, h7/1, h8/1},
H(cheap)={h1/.4, h2/.2, h3/.5, h4/.4, h5/.3, h6/1, h7/1, h8/.6},
H(beautiful)={h1/.6, h2/1, h3/1, h4/.8, h5/.6, h6/.8, h7/1, h8/.8}
and H(in the green surroundings)
={h1/.5, h2/.7, h3/.8, h4/.6, h5/1, h6/1, h7/.9, h8/1}.

3. Fuzzy Soft Sets
Deﬁnition 3.16.
Intersection of two fuzzy soft sets (F, A) and (G, B) over the
common universe U is the soft set (H, C), where C = A ∩ B, and for
all e ∈ C, H(e) = F(e) or G(e), (as both are same set).
We write (F, A)∩(G, B) = (H, C).

3. Fuzzy Soft Sets
Example 3.17.
Consider the Example 3.4. Here (F, A)∩(G, B) = (H, C), where
C(cheap) and
H(cheap = {h1/.4, h2/.2, h3/.5, h4/.4, h5/.3, h6/1, h7/.9, h8/.6}.

3. Fuzzy Soft Sets
Proposition 3.18.
(i) (F, A)∪(F, A) = (F, A).
(ii) (F, A)∩(F, A) = (F, A).
(iii) (F, A)∪φ = (F, A).
(iv) (F, A)∩φ = φ, where φ is the null soft set.
(v) (F, A)∪A = A, where A is the absolute soft set.
(vi) (F, A)∩A = (F, A).

References
References
[1] K.Atannassov, Intuitionistic fuzzy sets, Fuzzy sets and systems,
20 (1986), 87-96.
[2] P.K.Maji, R.Biswas and A.R.Roy, On soft set theory, Computers
and Mathematics with Applications, 45 (2003), 555-562.
[3] P.K.Maji, R.Biswas and A.R.Roy, Fuzzy soft sets, Journal of
Fuzzy Mathematics, 9(3)(2001), 589-602.

References
[4] D.Molodtsov, Soft set theory-First results, Computers and
Mathematics with Applications, 37 (1999), 19-31.
[5] L.A.Zadeh, Fuzzy sets, Infor. and Control, 8 (1965), 338-353.
[6] H.J.Zimmerman, Fuzzy set theory and its applications, Kluwer
Academic Publishers, Boston, (1996).

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