Fuzzy soft sets

Soft sets, fuzzy soft sets, operations on fuzzy soft sets

1. FUZZY SOFT SETS
S. Anita Shanthi
Department of Mathematics,
Annamalai University,
Annamalainagar-608002,
Tamilnadu, India.
E-mail : [email protected]
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 1 / 1

2. 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 roughsets
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.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 2 / 1

3. 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.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 3 / 1

4. 2. Theory of Soft Sets
2. Theory of Soft Sets
Definition 2.1.
LetXbe a non-empty set. Afuzzy setAis characterized by its
membership functionA:X→[0,1] andA(x) is interpreted as the
degree of membership of elementxinX.Ais completely determined
by the set of tuples,A={(x, A(x)) :x∈X}.
LetUbe an initial universe set andEbe a set of parameters. Let
P(U) denote the power set ofUandA⊂E.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 4 / 1

5. 2. Theory of Soft Sets
Definition 2.2.
The pair (F,A) is called a soft set overU, whereFis a mapping
given byF:A→P(U). In other words, a soft set overUis
parametrized family of subsets of the universeU. Forǫ∈A,F(ǫ)
may be considered as the set ofǫ-approximate elements of the soft
set (F,A).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 5 / 1

6. 2. Theory of Soft Sets
Example 2.3.
Suppose that,Uis the set of houses under consideration,Eis 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.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 6 / 1

7. 2. Theory of Soft Sets
Suppose that, there are six houses in the universeUgiven by
U={h1,h2,h3,h4,h5,h6}, andA={e1,e2,e3,e4,e5}wheree1
stands for the parameter ’expensive’,e2stands for the ’beautiful’,e3
stands for the parameter ’wooden’,e4stands for the parameter
’cheap’,e5stands 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 setU
and gives us a collection of approximate description of the object.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 7 / 1

8. 2. Theory of Soft Sets
Consider the mappingFwhich is ”houses (•) ” where dot (•)is to
filled up by the parametere∈A. ThereforeF(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 predicatepand
(ii) an approximate value-setγ( or simply to be called value-setγ).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 8 / 1

9. 2. Theory of Soft Sets
Tabular representation of a soft set
Uexpensivebeautifulwoodencheapin 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
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 9 / 1

10. 2. Theory of Soft Sets
Definition 2.4.
For two soft sets (F,A) and (G,B) over a common universeU, we
say that (F,A) is a soft subset of (G,B) if
(i)A⊂B,
(ii) For allǫ∈A,F(ǫ) andG(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).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 10 / 1

11. 2. Theory of Soft Sets
Definition 2.5.( Equality of two soft sets)
Two soft sets (F,A) and (G,B) over a common universeUare 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).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 11 / 1

12. 2. Theory of Soft Sets
Example 2.6.
LetA={e1,e3,e5} ⊂EandB={e1,e2,e3,e5} ⊂E.ClearlyA⊂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}andF(e1) ={h2,h4},
F(e3) ={h3,h4,h5},F(e5) ={h1}.Therefore, (F,A)⊂(G,B).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 12 / 1

13. 2. Theory of Soft Sets
Definition 2.7.(NOT set of parameters)
LetE={e1,e2, ...,en}be a set of parameters. The NOT of set E
denoted by¬Eis defined by¬E={e1,e2, ...,en}where∨ei= notei.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 13 / 1

14. 2. Theory of Soft Sets
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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 14 / 1

15. 2. Theory of Soft Sets
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
= (F
c
,¬A), whereF
c
:¬A→P(U) is a mapping
given byF
c
(α) =U−F(¬α) for allα∈ ¬A. Let us callF
c
to be
the soft complement function of F. Clearly (F
c
)
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}}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 15 / 1

16. 2. Theory of Soft Sets
Definition 2.12. (Null soft set)
A soft set (F,A) overUis said to be a Null soft set denoted byφ, if
for allǫ∈A F(ǫ) =φ(null-set).
Definition 2.13.(Absolute soft set). A soft set (F,A) overUis said
to be absolute soft set denoted byA, if for allǫ∈A,F(ǫ) =U.
Clearly,A
c
=φandφ
c
=A.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 16 / 1

17. 2. Theory of Soft Sets
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),
WhereH(α, β) =F(α)∩G(β) for all (α, β)∈A×B.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 17 / 1

18. 2. Theory of Soft Sets
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 thatU={h1,h2,h3,h4,h5,h6,h7,h8,h9,h10},
A={very costly; costly; cheap}andB={beautiful; in the green
surroundings; cheap}. LetF( very costly)={h2,h4,h7,h8},
F(costly)={h1,h3,h5},F( cheap)={h6,h9,h10}and
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 18 / 1

19. 2. Theory of Soft Sets
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), whereH( 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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 19 / 1

20. 2. Theory of Soft Sets
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.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 20 / 1

21. 2. Theory of Soft Sets
Example 2.17.
Consider the Example 2.15 above. We see that
(F,A)∨(G,B) = (O,A×B), whereO( 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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 21 / 1

22. 2. Theory of Soft Sets
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
.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 22 / 1

23. 2. Theory of Soft Sets
Definition 2.19.
Union of two soft sets (F,A) and (G,B) over the common universe
Uis the soft set (H,C), whereC=A∪B, and for alle∈C,
H(e) =













F(e), e∈A−B
G(e), e∈B−A
F(e)∪G(e),e∈A∩B.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 23 / 1

24. 2. Theory of Soft Sets
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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 24 / 1

25. 2. Theory of Soft Sets
Definition 2.20.
Intersection of two soft sets (F,A) and (G,B) over the common
universeUis the soft set(H,C), whereC=A∩B, and for all
e∈C,H(e) =F(e) orG(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), whereC={cheap}and
H(cheap) ={h6,h9,h10}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 25 / 1

26. 2. Theory of Soft Sets
Proposition 2.21.
(i) (F,A)∪(F,A) = (F,A).
(ii) (F,A)∩(F,A) = (F,A).
(iii) (F,A)∪φ=φ, whereφis the null soft set.
(iv) (F,A)∩φ=φ.
(v) (F,A)∪A=A, whereAis the absolute soft set.
(vi) (F,A)∩A= (F,A).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 26 / 1

27. 2. Theory of Soft Sets
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
.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 27 / 1

28. 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 fuzzywords. This
motivates the definition of fuzzy soft sets.
Definition 3.1.
LetUbe an initial universe set andEbe a set of parameters. Let
P(U) denote the set of all fuzzy sets ofU. LetA⊂E. A pair (F,A)
is called afuzzy soft setoverU, whereFis a mapping given by
F:A→P(U).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 28 / 1

29. 3. Fuzzy Soft Sets
We give an example of a fuzzy soft set.
Example 3.2.
Suppose that,Uis the set of houses under consideration,Eis 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.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 29 / 1

30. 3. Fuzzy Soft Sets
Suppose that, there are six houses in the universeUgiven by
U={h1,h2,h3,h4,h5,h6}, andE={e1,e2,e3,e4,e5}wheree1
stands for the parameter ’expensive’,e2stands for the ’beautiful’,e3
stands for the parameter ’wooden’,e4stands for the parameter
’cheap’,e5stands 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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 30 / 1

31. 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 setUand gives us a
collection of approximate description of the object. Consider the
mappingFwhich is ”houses (•) ” where dot (•)is to filled up by the
parametere∈E. ThereforeF(e1) means ”houses(expensive)” whose
functional-value is the fuzzy set
{h1/.5,h2/1,h3/.4,h4/1,h5/.3,h6/0}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 31 / 1

32. 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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 32 / 1

33. 3. Fuzzy Soft Sets
where each approximation has two parts
(i) a predicatep
(ii) an approximate value- fuzzy setγ( or simply to be called
value-setγ).
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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 33 / 1

34. 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 arehijcorresponding to the househiand
parameterej, wherehij= membership value ofhiinF(ej).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 34 / 1

35. 3. Fuzzy Soft Sets
Tabular representation of a fuzzy soft set
Uexpensivebeautifulwoodencheapin 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
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 35 / 1

36. 3. Fuzzy Soft Sets
Definition 3.3.
For two fuzzy soft sets (F,A) and (G,B) over a common
universeU, we say that (F,A) is a soft subset of (G,B) if
(i)A⊂B,
(ii) For allǫ∈A,F(ǫ) andG(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).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 36 / 1

37. 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) .
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 37 / 1

38. 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 universeUare
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).
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39. 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
= (F
c
,¬A), whereF
c
:¬A→P(U) is a
mapping given byF(α) = fuzzy complement ofF(¬α}for all
α∈ ¬A. Let us callF
c
to be the fuzzy soft complement function of
F. Clearly (F
c
)
c
is same asFand ((F,A)
c
)
c
= (F,A).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 39 / 1

40. 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}}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 40 / 1

41. 3. Fuzzy Soft Sets
Definition 3.7. (Null fuzzy soft set)
A fuzzy soft set (F,A) overUis said to be a Null fuzzy soft set
denoted byφ, if for allǫ∈A F(ǫ) =φ(null fuzzy set ofU).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 41 / 1

42. 3. Fuzzy Soft Sets
Definition 3.8. (Absolute fuzzy soft set)
A fuzzy soft set (F,A) overUis said to be absolute fuzzy soft set
denoted byA,if for allǫ∈A,F(ǫ) =U. Clearly,A
c
=φandφ
c
=A.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 42 / 1

43. 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), WhereH(α, β) =F(α)∩G(β), where
∩is the operation’ fuzzy intersection’ of two fuzzy soft sets.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 43 / 1

44. 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}
andB={beautiful; in the green surroundings; cheap}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 44 / 1

45. 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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 45 / 1

46. 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},
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 46 / 1

47. 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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 47 / 1

48. 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),WhereO(α, β) =F(α)∪G(β), where
∪is the operation ’fuzzy union’of two fuzzy sets.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 48 / 1

49. 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},
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 49 / 1

50. 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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 50 / 1

51. 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
.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 51 / 1

52. 3. Fuzzy Soft Sets
Definition 3.14.
Union of two fuzzy soft sets (F,A) and (G,B) over the common
universeUis the fuzzy soft set (H,C), whereC=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).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 52 / 1

53. 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}
andH(in the green surroundings)
={h1/.5,h2/.7,h3/.8,h4/.6,h5/1,h6/1,h7/.9,h8/1}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 53 / 1

54. 3. Fuzzy Soft Sets
Definition 3.16.
Intersection of two fuzzy soft sets (F,A) and (G,B) over the
common universeUis the soft set (H,C), whereC=A∩B, and for
alle∈C,H(e) =F(e) orG(e), (as both are same set).
We write (F,A)∩(G,B) = (H,C).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 54 / 1

55. 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}.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 55 / 1

56. 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,whereφis the null soft set.
(v) (F,A)∪A=A, whereAis the absolute soft set.
(vi) (F,A)∩A= (F,A).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 56 / 1

57. 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.
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 57 / 1

58. 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).
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 58 / 1

59. Thank You
Thank You
S. Anita Shanthi(Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@yFUZZY SOFT SETS 59 / 1