Upload
khangminh22
View
2
Download
0
Embed Size (px)
Citation preview
ΓerΓ§i & Soylu (2020) Kuratowski Theorems in Soft Topology, ADYU J SCI, 10(1), 273-284
* Corresponding Author DOI: 10.37094/adyujsci.597721
Kuratowski Theorems in Soft Topology
MΓΌge ΓERΓΔ°1, GΓΌltekin SOYLU2, *
1Haliç University, Department of Mathematics, İstanbul, Turkey [email protected], ORCID: 0000-0001-9018-463X
2Akdeniz University, Department of Mathematics, Antalya, Turkey [email protected], ORCID: 0000-0003-2640-5690
Received: 28.07.2019 Accepted: 03.04.2020 Published: 25.06.2020
Abstract
This paper deals with the soft topological counterparts of concepts introduced by
Kuratowski. First the closure operator is investigated in the soft topological setting and afterwards
the Kuratowski Closure-Complement Theorem is stated and proved.
Keywords: Soft topology; Kuratowski closure operator; Kuratowski Closure-Complement
Theorem.
Soft Topolojideki Kuratowski Teoremleri
Γz
Bu Γ§alΔ±Εmada Kuratowski tarafΔ±ndan ortaya konan bazΔ± topolojik kavramlarΔ±n Soft
topolojideki karΕΔ±lΔ±klarΔ± ele alΔ±nmΔ±ΕtΔ±r. Γncelikle kapanΔ±Ε operatΓΆrΓΌ soft topolojide tanΔ±mlanmΔ±Ε
ve incelemesi yapΔ±lmΔ±Ε daha sonra Kuratowski KapanΔ±Ε-TΓΌmleyen Teoremi ifade edilmiΕ ve
kanΔ±tlanmΔ±ΕtΔ±r.
Anahtar Kelimeler: Soft topoloji; Kuratowski kapanΔ±Ε operatΓΆrΓΌ; Kuratowski KapanΔ±Ε-
TΓΌmleyen Teoremi.
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
274
1. Introduction
Various theories have been proposed with the purpose of dealing with different types of
uncertainties. Besides to probability theory the most known ones are the theory of fuzzy sets [1],
the theory of vague sets [2], the theory of rough sets [3]. Nevertheless not suprisingly all these
theories have their own drawbacks. In 1999, Molodtsov [4] introduced the notion of soft set theory
claiming to overcome the drawbacks of the theories mentioned above. Molodtsov proposed
applications of this new tool in several directions, such as smoothness of functions, game theory,
operations research, Riemann integration, Perron integration, probability, theory of measurement.
After the introduction of soft sets, several researchers started to extend the theory in different
paths. In 2003, Maji et al [5] defined and studied several fundamental notions of soft set theory.
The outcome of soft set theory in algebraic structures was introduced by AktaΕ and ΓaΔman [6].
They not only defined the notion of soft groups but also obtained most of their basic properties.
In 2011, Shabir and Naz [7] brought to light the idea of soft topological spaces. They interestingly
observed that a soft topological space is actually a parameterized family of topological spaces.
This paper participates to all those discussions in the direction of soft topological spaces.
First the concept of Kuratowski closure operator is introduced in the soft topological setting. The
correspondence of a closure operator with a soft topology is given. In addition the well-known
Kuratowski Closure-Complement Theorem is stated and proved for soft topological spaces.
2. Preliminaries
In this section, the fundamental definitions and results of soft set theory and its topology
are presented. They may be found in earlier studies [4 , 5, 7-10].
Definition 1. Let π denote the universe of discourse and πΈ be a set of parameters. Let β(π)
denote the power set of π and π΄ be a non-empty subset of πΈ. A pair (πΉ, π΄) is called a soft set over
π, where F is a mapping given by πΉ: π΄ βΆ β(π).
Basically, a soft set over π is a parametrized family of subsets of the universe π. In the
sequel π will allways be the universe of discourse and πΈ the set of parameters unless stated
otherwise.
For the sake of simplicity we hereafter will suppose any given soft set (πΉ, π΄) over π with
parameters set πΈ is extended as following:
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
275
πΉ!: πΈ β β(π), πΉ!(π) = /πΉ(π) , πππ β π΄β , πππ β πΈ β π΄. By this extension we will denote any given
soft set (πΉ, π΄) by (πΉ, πΈ) just by replacing πΉ! with πΉ.
Definition 2. Let π be an initial universe set and E be an universe set of parameters. Let
(πΉ, πΈ) and (πΊ, πΈ) be soft sets over a common universe set π. Then (πΉ, πΈ) is a soft subset of (πΊ, πΈ),
denoted by (πΉ, πΈ) ββΌ(πΊ, πΈ), if for all π β πΈ, πΉ(π) β πΊ(π).
(πΉ, πΈ) is called a soft super set of (πΊ, πΈ), if (πΊ, πΈ)is a soft subset of (πΉ, πΈ). We denote it
by (πΉ, πΈ) ββΌ(πΊ, πΈ).
Definition 3. Two soft set (πΉ, πΈ) and (πΊ, πΈ) over a common universe π are said to be soft
equal if, (πΉ, πΈ) is a soft subset of (πΊ, πΈ) and (πΊ, πΈ) is a soft subset of (πΉ, πΈ).
Definition 4. A soft set (πΉ, πΈ) over π is said to be the empty soft set denoted by π·# if for
all π β πΈ, πΉ(π) = β .
Definition 5. A soft set (πΉ, πΈ) over π is said to be an absolute soft set denoted by π# if for
all π β πΈ, πΉ(π) = π.
Clearly π#$ = Ξ¦# and Ξ¦#$ = π#.
Definition 6. The union (π», πΈ) of two soft sets (πΉ, πΈ) and (πΊ, πΈ) over the common universe
π, denoted (πΉ, πΈ) βͺβΌ(πΊ, πΈ), is defined as π»(π) = πΉ(π) βͺ πΊ(π), for all π β πΈ.
Definition 7. The intersection (π», πΈ) of two soft sets (πΉ, πΈ) and (πΊ, πΈ) over the common
universe π, denoted (πΉ, πΈ) β©βΌ(πΊ, πΈ), is defined as π»(π) = πΉ(π) β© πΊ(π), for all π β πΈ.
Definition 8. The complement of a soft set (πΉ, πΈ) is denoted by (πΉ, πΈ)$ and is defined by
(πΉ, πΈ)$: = (πΉ$ , πΈ), where πΉ$: πΈ βΆ β(π) is a mapping given by πΉ$(π) = π β πΉ(π), for all π β
πΈ.
Proposition 9. Let (πΉ, πΈ) and (πΊ, πΈ) be the soft sets over π. Then
i) A(πΉ, πΈ) βͺβΌ(πΊ, πΈ)B
$= (πΉ, πΈ)$ β©
βΌ(πΊ, πΈ)$,
ii) A(πΉ, πΈ) β©βΌ(πΊ, πΈ)B
$= (πΉ, πΈ)$ βͺ
βΌ(πΊ, πΈ)$.
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
276
Definition 10. Let π be an initial universe and πΈ be the non-empty set of parameters. The
difference (π», πΈ) of two soft sets (πΉ, πΈ) and (πΊ, πΈ) over π, denoted by (πΉ, πΈ)\βΌ(πΊ, πΈ), is defined
as π»(π) = πΉ(π)\πΊ(π), for all π β πΈ.
Definition 11. Let (πΉ, πΈ) be a soft set over π and π’ β π. We say that π’ ββΌ(πΉ, πΈ) read as π’
belongs to the soft set (πΉ, πΈ) whenever π’ β πΉ(π), for all π’ β πΈ.
Note that for any π’ β π, π’ ββΌ(πΉ, πΈ) if π’ β πΉ(π), for some π β πΈ.
Definition 12. Let π be a non-empty subset of π, then π# denotes the soft set (π, πΈ) over
π for which π(π) = π, for all π β πΈ.
Definition 13. Let π’ β π, then (π’, πΈ) denotes the soft set over π for which π’(π) = {π’}, for
all π β πΈ.
Definition 14. Let π be a collection of soft sets over π, then π is said to be a soft topology
on π if
T1) Ξ¦#,π# belong to π,
T2) The union of any number of soft sets in π belongs to π,
T3) The intersection of any two soft sets in π belongs to π.
The triplet (π, π, πΈ) is called a soft topological space over π.
Definition 15. Let (π, π, πΈ) be a soft topological space over π, then the members of π are
said to be soft open sets in π.
Definition 16. Let (π, π, πΈ) be a soft topological space over π. A soft set (πΉ, πΈ) over U is
said to be a soft closed set in π, if its complement (πΉ, πΈ)$ belongs to π.
Proposition 17. Let (π, π, πΈ) be a soft topological space over π and πΈ be the non-empty
set of parameters. Then
i) Ξ¦#,π# are closed soft sets over π,
ii) The intersection of any number of soft closed sets is a soft closed set over π,
iii) The union of any two soft closed sets is a soft closed set over π.
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
277
Definition 18. Let (π, π, πΈ) be a soft space over π and (πΉ, πΈ) be a soft set over π. Then
the soft closure of (πΉ, πΈ), denoted by (πΉ, πΈ) is the intersection of all soft closed super sets of
(πΉ, πΈ).
Clearly (πΉ, πΈ) is the smallest soft closed set over π which contains (πΉ, πΈ).
Theorem 19. Let (π, π, πΈ) be soft topological space over π and (πΉ, πΈ), (πΊ, πΈ) are soft sets
over π. Then the following hold:
i) Ξ¦# = Ξ¦# and π# = π#,
ii) (πΉ, πΈ) βK (πΉ, πΈ),
iii) (πΉ, πΈ) is a soft closed set if and only if (πΉ, πΈ) = (πΉ, πΈ),
iv) (πΉ, πΈ) = (πΉ, πΈ),
v) (πΉ, πΈ) βK (πΊ, πΈ) implies (πΉ, πΈ) βK (πΊ, πΈ),
vi) (πΉ, πΈ) βͺK (πΊ, πΈ) = (πΉ, πΈ) βͺK (πΊ, πΈ),
vii) (πΉ, πΈ) β©K (πΊ, πΈ) βK (πΉ, πΈ) β©K (πΊ, πΈ).
Definition 20. Let (π, π, πΈ) be a soft topological space over U then soft interior of soft set
(πΉ, πΈ) over U is denoted by (πΉ, πΈ)β and is defined as the union of all soft open sets contained in
(πΉ, πΈ).
Thus (πΉ, πΈ)β is the largest soft open set contained in (πΉ, πΈ).
Theorem 21. Let (π, π, πΈ) be a soft topological space over π and (πΉ, πΈ), (πΊ, πΈ) are soft
sets over π. Then the followings hold:
i) Ξ¦#β = Ξ¦# and π#β = π#,
ii) (πΉ, πΈ)β βK (πΉ, πΈ),
iii) ((πΉ, πΈ)β)β = (πΉ, πΈ)β,
iv) (πΉ, πΈ) is a soft open set if and only if (πΉ, πΈ)β = (πΉ, πΈ),
v) (πΉ, πΈ) βK (πΊ, πΈ) implies (πΉ, πΈ)β βK (πΊ, πΈ)β,
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
278
vi) ((πΉ, πΈ) β©K (πΊ, πΈ))β = (πΉ, πΈ)β β©K (πΊ, πΈ)β,
vii) (πΉ, πΈ)β βͺK (πΊ, πΈ)β βK ((πΉ, πΈ) βͺK (πΊ, πΈ))β.
Theorem 22. Let (πΉ, πΈ) be a soft set of soft topological space over π. Then
i) ((πΉ, πΈ)$)β = L(πΉ, πΈ)M$,
ii) ((πΉ, πΈ)$) = ((πΉ, πΈ)β)$,
iii) (πΉ, πΈ)β = A((πΉ, πΈ)$)B$.
Definition 23. Let (π, π, πΈ) be a soft topological space over π then the soft boundary of
soft set (πΉ, πΈ) over π is denoted by (πΉ, πΈ) and defined as (πΉ, πΈ) = (πΉ, πΈ) β©βΌ((πΉ, πΈ)$).
Remark 24. From the above definition it follows directly that the soft sets (πΉ, πΈ) and
(πΉ, πΈ)$ have same soft boundary.
Theorem 25. Let (πΉ, πΈ) be a soft set of soft topological space over π. Then the followings
hold:
(1) ((πΉ, πΈ))$ = (πΉ, πΈ)β βͺβΌ((πΉ, πΈ)$)β = (πΉ, πΈ)β βͺ
βΌ(πΉ, πΈ)β
(2) (πΉ, πΈ) = (πΉ, πΈ)β βͺβΌ(πΉ, πΈ)
(3) (πΉ, πΈ) = (πΉ, πΈ) β©βΌ((πΉ, πΈ)$) = (πΉ, πΈ)\
βΌ(πΉ, πΈ)β
(4) (πΉ, πΈ)β = (πΉ, πΈ)\βΌ(πΉ, πΈ).
3. Soft Kuratowski Closure Operator
Theorem 26. Let (π, π, πΈ) be a soft topological space. The operator π:β(π#) β β(π#),
defined by πO((πΉ, πΈ)) = (πΉ, πΈ) satisfies following properties:
K1) (πΉ, πΈ) βK πO((πΉ, πΈ)),
K2) πO(πO((πΉ, πΈ))) = πO((πΉ, πΈ)),
K3) πO((πΉ, πΈ) βͺK (πΊ, πΈ)) = πO((πΉ, πΈ)) βͺK πO((πΊ, πΈ)),
K4) πO(Ξ¦#) = Ξ¦#.
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
279
Conversely for any operator πO:β(π#) β β(π#) satisfying these four conditions there
exists a unique soft topology π&' on π such that for all (πΉ, πΈ) βK π# the soft closure of (πΉ, πΈ)is
just πO((πΉ, πΈ)).
Proof. (K1)-(K4) are obvious from Theorem 19.
For the second part of the theorem let us define π&' as following:
π&' = {(πΉ, πΈ) βK π#: πO((πΉ, πΈ)$) = (πΉ, πΈ)$}
We will show that the family π&' is a soft topology on π# .
T1) πO(π#$) = πO(Ξ¦#) and by (K4) πO(Ξ¦#) = Ξ¦# therefore π# β π&' . πO(Ξ¦#$ ) = πO(π#) and
since by (K1) π# β πO(π#) we obeserve that πO(π#) = π# and Ξ¦# β π&' .
T2) Let (πΉ, πΈ), (πΊ, πΈ) β π&' then we have πO((πΉ, πΈ)$) = (πΉ, πΈ)$ and πO((πΊ, πΈ)$) = (πΊ, πΈ)$.
Now by the Morganβs πO(((πΉ, πΈ) β©K (πΊ, πΈ))$) = πO((πΉ, πΈ)$ βͺK (πΊ, πΈ)$) and by (K3)
πO((πΉ, πΈ)$ βͺK (πΊ, πΈ)$) = πO((πΉ, πΈ)$) βͺK πO((πΊ, πΈ)$) = (πΉ, πΈ)$ βͺK (πΊ, πΈ)$ = ((πΉ, πΈ) β©K (πΊ, πΈ))$
which means that (πΉ, πΈ) β©K (πΊ, πΈ) β π&' .
T3) Let (πΉ( , πΈ) β π&' for βπ β πΌ. For βπ β πΌ we have (πΉ( , πΈ) βK βͺK(β* (πΉ( , πΈ) therefore,
π#\R βͺK(β* (πΉ( , πΈ) βK π#\R(πΉ( , πΈ) and with help of (K3) it can be seen that for βπ β πΌ,
πO Aπ#\R βͺK(β* (πΉ( , πΈ)B βK πO(π#\R(πΉ( , πΈ)). Since each (πΉ( , πΈ) is a member of π&' we have
πO(π#\R(πΉ( , πΈ)) = π#\R(πΉ( , πΈ)thus, πO Aπ#\R βͺK(β* (πΉ( , πΈ)B βKβ©K Lπ#\R(πΉ( , πΈ)M and finally by De
Morganβs
πO Aπ#\R βT(β*(πΉ( , πΈ)B βK Aπ#\R βT
(β*(πΉ( , πΈ)B. (1)
On the other hand by (K1),
π#\R βT(β*(πΉ( , πΈ) βK πO Aπ#\R βT
(β*(πΉ( , πΈ)B. (2)
Combining Eqn. (1) and Eqn. (2) we get the equality πO Aπ#\R βͺK(β* (πΉ( , πΈ)B = π#\R βͺK(β* (πΉ( , πΈ) hence
βͺK(β*(πΉ( , πΈ) β π&' .
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
280
Once we have seen that π&' is a soft topolgy the property (K2) will help us showing that
πO((πΉ, πΈ)) = (πΉ, πΈ):
Since by (K2) πO(πO((πΉ, πΈ))) = πO((πΉ, πΈ)) we have by definition of π&' , πO((πΉ, πΈ))$ β π&'
and therefore πO((πΉ, πΈ)) is a soft closed set which means,
(πΉ, πΈ) βK πO((πΉ, πΈ)). (3)
For the reverse inclusion we first observe that since (πΉ, πΈ) is a closed set, by definition of
π&' , πO(((πΉ, πΈ))) = (πΉ, πΈ). Additionally we have (πΉ, πΈ) βK (πΉ, πΈ) and by (K1)
πO((πΉ, πΈ)) βK πO((πΉ, πΈ)) = (πΉ, πΈ) which is the required inclusion,
πO((πΉ, πΈ)) βK (πΉ, πΈ). (4)
Thus by Eqn. (3) and Eqn. (4) πO((πΉ, πΈ)) = (πΉ, πΈ). The operator πO in this theorem is called
to be a Kuratowski soft closure operator.
Example 27. Let π = β, πΈ = {π+, π,, . . . , π-} and πO:β(β#) β β(β#) be defined as
following:
πO((πΉ, πΈ)) = VΞ¦# , if(πΉ, πΈ) = Ξ¦#;(πΉ, πΈ) βͺK Lβ2, πΈM, if(πΉ, πΈ) β Ξ¦#;
πO is a soft Kuratowski closure operator.
It can be easily verified that πO satisfies (K1)-(K4). The topology generated by πO is
π&' = ](πΉ, πΈ): (β2, πΈ) βK (πΉ, πΈ)$^ βͺK π# .
For sure the duallity between closednes and opennes of soft sets in soft topological spaces
reflects a dual concept to Soft Closure Operators: In the sequel we introduce the Soft Interior
Operator.
Theorem 28. In a soft topological space (π, π, πΈ) the operator πΉT:β(π#) β β(π#),
πΉT((πΉ, πΈ)) = (πΉ, πΈ)β satisfies following properties:
I1) Ξ¨T((πΉ, πΈ)) βK (πΉ, πΈ),
I2) Ξ¨TLΞ¨T((πΉ, πΈ))M = Ξ¨T((πΉ, πΈ)),
I3) Ξ¨T((πΉ, πΈ) β©K (πΊ, πΈ)) = Ξ¨TL(πΉ, πΈ)) β©K Ξ¨T((πΊ, πΈ)M,
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
281
I4) Ξ¨T(π#) = π#.
Conversely, suppose that the operator Ξ¨T:β(π#) β β(π#) satisfies the 4 conditions given
above then there exists a soft topology π./ on π such that for each (πΉ, πΈ) βK π# the soft interior
of (πΉ, πΈ) is just Ξ¨T((πΉ, πΈ)).
Proof. The fΔ±rst part of the theorem is obvious from Theorem 21. For the converse part it
can be first varified that πO((πΉ, πΈ)) = π#\RπΉT((πΉ, πΈ)$) is a soft Kuratowski closure. Afterwards it
can be seen that for the topology π0/ = ](πΉ, πΈ) βK π#: πΉT((πΉ, πΈ)) = (πΉ, πΈ)^ we have the equality
(πΉ, πΈ)β = πΉT((πΉ, πΈ)).
Example 29. Let π = β, πΈ = {π+, π,, . . . , π-} and πΉT:β(β#) β β(β#) be defined as
following:
Ξ¨T((πΉ, πΈ)) = Vβ# , if(πΉ, πΈ) = β;(πΉ, πΈ)\RLβ2, πΈM, if(πΉ, πΈ) β β.
Ξ¨T is a soft interior operator and the corresponding soft topology is the family
π./ = ](πΉ, πΈ) β β(β#): (πΉ, πΈ)) β©K (β2, πΈ) = Ξ¦#^.
4. Soft Kuratowski Closure-Complement Theorem
The Kuratowski Closure-Complement Theorem which was first proved by the Polish
mathematician Kazimierz Kuratowski in 1922 can be given by using soft sets as follows:
Theorem 30. Let (π, π, πΈ) be a soft topological space and (πΉ, πΈ) be a soft set over π. The
number of different soft sets obtained by soft complementing and soft closing the set (πΉ, πΈ) can
not exceed 14. Moreover, this number can be attained for a soft set in the soft standard topology.
Several proofs of the classical version of this theorem are demonstrated by different
mathematicians. The way we have choosen is analoguous to the proof of Strabel [11] but before
going to the proof of the theorem, we will review some algebraic notions that are in connection
with the operators used in the proof. Given a soft topological space (π, π, πΈ) define the soft
complement operator π and the soft closure operator π on soft subsets (πΉ, πΈ) βK π# by π(πΉ, πΈ) =
π#\R(πΉ, πΈ) and π(πΉ, πΈ) = (πΉ, πΈ), respectively. Obviously ππ(πΉ, πΈ) = (πΉ, πΈ). Starting with any
soft topological space (π, π, πΈ), possible distinct operators on (π, π, πΈ) that can be obtained by
composing the elements of the set {π, π} yield to a monoid with the identity element ππ. This
monoid is called the Kuratowski monoid on (π, π, πΈ).
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
282
For any soft topological space (π, π, πΈ), a natural partial order on the Kuratowski monoid
exists.(π, π, πΈ) If β+ and β, are elements of the Kuratowski monoid on (π, π, πΈ), we define the
partial order β€ as β+β€β, if for every (πΉ, πΈ) βK π#, β+ (πΉ, πΈ) β€β, (πΉ, πΈ).
After this short introduction we are ready for the proof of the theorem.
Proof. Let (π, π, πΈ) be a soft topological space. We mentioned above that ππ = ππ. At the
same time ππ = π. Therefore it can be easily observed that a member of the Kuratowski monoid
on (π, π, πΈ) has to be equivalent to one of the operators: ππ, π, π, ππ, ππ, πππ, πππ, ππππ, ππππ,
πππππ, πππππ, ππππβ¦ππ, ππππβ¦ππ, ππππβ¦πππ or ππππβ¦πππ.
We will now obtain that πππ = πππππππ. Firstly,
πππL(πΉ, πΈ)M = ππLπ#\R(πΉ, πΈ)M
= π ALπ#\R(πΉ, πΈ)Mβ βͺK (πΉ, πΈ)B
= (πΉ, πΈ)β
Now ππππππ β€ πππ since ππππππ(πΉ, πΈ) is the interior of πππ(πΉ, πΈ). By the fact that
ππ = π, it follows that πππππππ β€ ππππ = πππ. Also, ππππ β€ π since ππππ(πΉ, πΈ) is the
interior of π(πΉ, πΈ). Hence, πππππ β€ ππ = π. Then ππππππ β₯ ππ, and therefore πππππππ β₯
πππ. We conclude that πππ = πππππππ. From this it can be deduced that any word on {π, π} that
is longer than 7 places has to be equivalent to a word of length at most 7. Thus, for any soft
topological space (π, π, πΈ), each operator in the Kuratowski monoid on (π, π, πΈ) is equivalent to
at least one of the following:
ππ, π, π, ππ, ππ, πππ, πππ, ππππ, ππππ, πππππ, πππππ, ππππππ, ππππππ, πππππππ
Therefore, for a soft topological space (π, π, πΈ), the Kuratowski monoid on (π, π, πΈ) can
have order at most 14 and hence for any (πΉ, πΈ) βK π#, there are at most 14 distinct soft sets that
can be obtained via soft closures and soft complements of (πΉ, πΈ).
To complete the proof of the theorem we need to show that this bound of 14 can be attained
for a soft set. The following set will serve to this objective. (πΉ, πΈ) βK β# given by
(πΉ, πΈ) = ]Lπ+, (0,1)M, Lπ,, (1,2)M, (π1, {3}), (π2, [4,5] β© β)^
attains the bound of 14; that is, we can produce 14 distinct soft sets from (πΉ, πΈ) by taking soft
complements and soft closures. These soft sets are:
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
283
ππ(πΉ, πΈ) = ]Lπ+, (0,1)M, Lπ,, (1,2)M, (π1, {3}), (π2, [4,5] β© β)^, (i)
π(πΉ, πΈ) = VLπ+, (ββ, 0] βͺ [1, +β)M, Lπ,, (ββ, 1] βͺ [2, +β)M,(π1, (ββ, 3] βͺ (3,+β)), Lπ2, (ββ, 4) βͺ (5,+β) βͺ ([4,5] β© πΌ)M
t, (ii)
π(πΉ, πΈ) = {(π+, [0,1]), (π,, [1,2]), (π1, {3}), (π2, [4,5])}, (iii)
ππ(πΉ, πΈ) = VLπ+, (ββ, 0) βͺ (1,+β)M, Lπ,, (ββ, 1) βͺ (2,+β)M,(π1, (ββ, 3) βͺ (3,+β)), Lπ2, (ββ, 4) βͺ (5,+β)M
t, (iv)
ππ(πΉ, πΈ) = VLπ+,(ββ, 0] βͺ [1, +β)M, Lπ,, (ββ, 1] βͺ [2, +β)M,
(π1, β), (π2, β)t, (v)
πππ(πΉ, πΈ) = ]Lπ+, (0,1)M, Lπ,, (1,2)M^, (vi)
πππ(πΉ, πΈ) = VLπ+, (ββ, 0] βͺ [1, +β)M, Lπ,, (ββ, 1] βͺ [2, +β)M,(π1, β), Lπ2, (ββ, 4] βͺ [5, +β)M
t, (vii)
ππππ(πΉ, πΈ) = ]Lπ+, (0,1)M, Lπ,, (1,2)M, Lπ2, (4,5)M^, (viii)
ππππ(πΉ, πΈ) = {(π+, [0,1]), (π,, [1,2])}, (ix)
πππππ(πΉ, πΈ) = VLπ+,(ββ, 0) βͺ (1,+β)M, Lπ,, (ββ, 1) βͺ (2,+β)M,
(π1, β), (π2, β)t, (x)
πππππ(πΉ, πΈ) = {(π+, [0,1]), (π,, [1,2]), (π2, [4,5])}, (xi)
ππππππ(πΉ, πΈ) = VLπ+, (ββ, 0) βͺ (1,+β)M, Lπ,, (ββ, 1) βͺ (2,+β)M,(π1, β), Lπ2, (ββ, 4) βͺ (5,+β)M
t, (xii)
ππππππ(πΉ, πΈ) = VLπ+,(ββ, 0] βͺ [1, +β)M, Lπ,, (ββ, 1] βͺ [2, +β)M,
(π1, β), (π2, β)t, (xiii)
πππππππ(πΉ, πΈ) = ]Lπ+, (0,1)M, Lπ,, (1,2)M^. (xiv)
References
[1] Zadeh, L.A., Fuzzy sets, Information and Control, 8(3), 338β353, 1965.
[2] Gau, W.-L., Buehrer, D.J., Vague sets, IEEE Transactions on Systems, Man, and Cybernetics, 23(2), 610β614, 1993.
[3] Pawlak, Z., Rough sets, International Journal of Computer & Information Sciences, 11(5), 341β356, 1982.
ΓerΓ§i & Soylu (2020) ADYU J SCI, 10(1), 273-284
284
[4] Molodtsov, D., Soft set theory-first results, Computers & Mathematics with Applications, 37(4-5), 19β31, 1999.
[5] Maji, P.K., Biswas, R., Roy, A.R., Soft set theory, Computers & Mathematics with Applications, 45(4-5), 555β562, 2003.
[6] AktaΕ, H., ΓaΔman, N., Soft sets and soft groups, Information Sciences, 177(13), 2726β2735, 2007.
[7] Shabir, M., Naz, M., On soft topological spaces, Computers & Mathematics with Applications, 61(7), 1786β1799, 2011.
[8] ΓaΔman, N., KarataΕ, S., EnginoΔlu, S., Soft topology, Computers & Mathematics with Applications, 62(1), 351β358, 2011.
[9] Feng, F., Jun, Y:B., Zhao, X., Soft semirings, Computers & Mathematics with Applications, 56(10), 2621β2628, 2008.
[10] Hussain, S., Ahmad, B., Some properties of soft topological spaces, Computers & Mathematics with Applications, 62(11), 4058β4067, 2011.
[11] Strabel, G., The Kuratowski Closure-Complement Theorem, Accessed on 15.02.2019. www.mathtransit.com/2009 strabel.pdf, 2009.