Heuristic Approach to Establish New Operators via Nano Topology ()
1. Introduction
Thivagar and Richard [1] introduced a nano topological space with respect to a subset X of an universe, which is defined in terms of lower and upper approximations and boundary region of X. This paper introduces and defines a new type of class of all after-composed set and the class of all fore composed set denoted by
and
and used in nano topological space induced by any binary relation. Some of their properties are studied and investigated. We define new operators such as nano equality, nano inclusion and nano power set concerning any binary relation in nano topology. Then, we show the differences and relationships between the notions of ordinary set theory and nano topological space. Finally, we present the application of network topology [2] in nano topology.
2. Preliminaries
Definition 2.1. [3] [4] [5] [6] For the pair of approximation space
where
is the non-empty finite set of objects called the universe, R be an binary relation on
. Then the set xR is defined as
is called as the right neighborhood of an element
.
Definition 2.2. [5] [7] Let
be approximation space and
. The subset X is called an after (resp.,fore) composed set if X contains all after (resp.,fore) sets for all elements of its points that is for all
(resp.,
). The class of all after composed sets and fore composed set in
is
and
.
Definition 2.3. [1] [8] Let
be a non-empty finite set of objects called the universe R be an equivalence relation on
named as the indiscerniblity relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair
is said to be the approximation space. Let
.
(i) The Lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and it is denoted by
. That is,
, where R(x) denotes the equivalence class determined by x.
(ii) The Upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by
.
(iii) The Boundary region of X with respect to R is the set of all objects which can be classified neither as X nor as not X with respect to R and it is denoted by
Definition 2.4. [1] Let
be the universe, R be an indiscernibility relation on
and
where
, then
satisfies the following axioms:
(i)
and
.
(ii) The union of elements of any sub collection of
is in
.
(iii) The intersection of the elements of any finite sub collection of
is in
.
That is,
forms a topology on
called as the nano topology on
with respect to X. We call
as the nano topological space.
Throughout this paper a binary relation is called as a relation and also after composed set,fore composed set are called as relational topology.
3. Relational Topology Based on Nano Topological Space
In this section, we define the after and fore-composed sets based on nano topological space are investigated.
Definition 3.1. Let
be a non empty finite set of objects called the universe R be any binary relation on
. The class of all after- (resp., fore-) composed sets in
is an ordered pair of approximation space and
where
and if lower and upper approximations and boundary region of X is given by
(i)
.
(ii)
.
(iii)
.
where
forms a topology on
called as the nano topology induced by relational topology on
with respect to any relation. We call
as the nano topology induced by relational topology.
Example 3.2. Let
and
with
. Then we have
,
,
,
and
,
,
,
. Here
and
. Then
,
and
. Hence
.
Proposition 3.3. The class
in approximation space
in a topology on
.
Proof: We shall prove that
is a topology on
and similarly for
. Clearly
and
are after composed sets,then
. Let
, and let
.Then
and
, which implies that
and
.Thus
, and then
. Now,let
.Then
imply that there exist
such that
, and hence
, that is
. Thus
is a topology on
.
4. New Operators in Nano Topology
In this section we will give the basic deviations for equality
, inclusion
and power set is an ordinary set theoretical operation approach to approximation space in nano topology equipped with relation. The relation represents the basic and necessary concept to define the after and fore composed sets induced by relation over nano topological space.
Definition 4.1. Let
be nano topological space induced by relational topology. Then the two subsets
are called as follows:
(i) Nano Lower-equal in
, written
, if
.
(ii) Nano Upper-equal in
,written
, if
.
(iii) Nano almost equal in
.written,if
, if
and
.
Example 4.2. Let
and
, where
,
,
,
,
and
,
,
,
,
. Then
and
. Let
and
. Then
i.e.,
and
i.e.,
. Thus
is equal to
and
.
Definition 4.3. Let
be nano topological space induced by relational topology and
. We say that:
(i) X is nano lower-included in Y, written
, if
.
(ii) X is nano upper-included in Y, written
, if
.
(iii) X is nano included in Y, written
, if
and
.
Example 4.4. Consider
and
,where
,
,
,
. Then
and
. Let
and
, clearly
and we have
,
,
and
. Then
and
which implies that
.
Definition 4.5. Let
be approximation space and X in
. Then the family of all
forms a nano topology on
, which can be defined by
(i)
.
(ii)
.
(iii)
.
Then
which is known as power set of lower approximation, power set of upper approximation and power set of boundary region of X.
Example 4.6. Consider the approximation space
in Example 4.4 and let
and
. Then
,
and hence
,
and
.
Clearly,
.
Proposition 4.7. If
be nano topological space induced by relational topology and
. Then the following conditions are hold:
(i) If
, then
.
(ii) If
, then
.
(iii) If
and
, then
.
(iv) If
and
, then
.
Proof:
(i) Let
, then
. But
, then
and
Math_193#. Hence
.
(ii) By the proof(ii) is similar way as in proof (i).
(iii) Let
and
, then
and
. Thus
, which implies that
Math_201#. Thus
.
iv) By the proof is similar way as in (iii).
Proposition 4.8. Let
be approximation space and
be nano topological space induced by relational topology and
. Then
(i)
if and only if
.
(ii) If
or
, then
.
(iii) If
or
, then
.
Proof:
(i) By the proof is directly from the definition 4.0.
(ii) Let
or
, then
or
. Then
, that is
.
(iii) By the similar way as in (ii).
Proposition 4.9. Let
be relational topology based on nano topological space and
. Then
(i) If
, then
and
.
(ii)
, and
if and only if
.
(iii)
and
if and only if
.
(iv)
and
if and only if
.
Proof:
(i) The proof is directly from the definition obvious.
(ii)
and
iff
and
iff
iff
.
(iii) and (iv) by similar way as in (ii).
Proposition 4.10. If
be approximation space and
be nano topological space induced by relational topology, and
.Then
(i)
if and only if
.
(ii)
if and only if
(iii)
.
(iv) If
and
, then
.
(v) If
and
and
, then
.
(vi) If
and
, then
.
(vii) If
and
, then
.
(viii)
and
, then
.
Proof:
(i) Let
iff
iff
iff
Math_269# iff
.
(ii) By the proof is similar way as in (i).
(iii) Since
and
. Then
and
and hence
(iv) Let
,
and
,then
,
Math_282# and
. Thus
and then
.
(v) and (vi) By the proof is similar way as in(iv).
(vi) Let
and
, then
and
Math_290# and hence
and
Math_293#. That is
.
(vii) By the proof is similar way as in (vii).
Proposition 4.11. Let
be nano topological space induced by relational topology on
and
be approximation space. Let
. Then
(i) If
and
, then
.
(ii) If
and
, then
.
(iii) If
and
, then
.
Proof: The proof is similar way as in proposition 4.10
Proposition 4.12. Let
be nano topological space induced by relational topology on
. Then
,
,
.
Proof: Let
, then
and hence
and
. Thus
, which implies that
Math_319# and
. Thus
,
.
Proposition 4.13. Let
be nano topological space induced by relational topology on
and
be approximation space. Let
. Then
(i) If
then
.
(ii) If
then
.
(iii) If
then
.
(iv) If
if and only if
.
(v) If
if and only if
.
(vi) If
if and only if
.
Proof:
(i) Let
, then
. Now let
, then
, that is
. Thus by
and then
. Hence
and then
.
(ii) and (iii) by the proof is same way as in (i).
(iii) If
iff
and
iff
and
iff
.
(iv) and (vi) by the proof is same way as in (iv).
Proposition 4.14. If
be relational topology based on nano topological space and
be approximation space. Let
. Then
(i)
,
and
.
(ii) If
, then
,
and
.
Proof:
(i) Since
are ordering relations. Then
and
and hence
,
and
.
(ii) Let
, then
and
. Hence
,
and
.
5. Application
In this section, we discuss the application of computer networking topology refers to the layout or design of the connected devices to applied in nano topological space. It is well known that the computation of lower and upper approximations and boundary region is an approximation space
. The network topologies [2] can be physical (or) logical. The physical topology of a network refers to the configuration (or) the layout of cables, computers and others.The physical topology means that the physical design of a network including the devices,location and cable installation. Logical topology means that pass the information between the computers. Logical Topology refers to the fact that how data actually transfers in a network as opposed to its design. The main types of physical topologies are bus topology, ring topology, star topology and mesh topology. An applied example of new operators in nano topology in the main types of physical topologies are presented. We will show the above discussion by the following example.
Example 5.1. Let
be a set of four different network topology, where
is a bus topology,
is a ring topology,
is a star topology,
is a mesh topology and
be the attributes of network topology, where
= The method of transfer data = {Broadcast, Multicast, Unicast} =
,
= Cable Type = {Twistedpaircable, Thin Coaxialcable, Thick Coaxialcable, Fiber opticcable} =
and
= Bandwidth Capacity, {10 Mbit/s, 10 - 100 Mbit/s, 10 Mbit/s - 40 Gbit/s} =
.
Data for Network Topology
If
. with
;
;
;
and
;
;
;
and we get
,
. Now we have to calculate the nano approximate equal, nano inclusion, nano power set as the following example: Let
and
. Then
Math_423# and
and
. Hence
that is
and
that is
. Thus
although
.
Next, we show that the nano inclusion. Let
and
Math_434#. Clearly,
we have
and
and
. Hence
that is
and
that is
.Thus
although
. Then we have to prove that nano power set.Let
Then
and
and
. We obtain that
and
and
.
Then the observations are follows:
(i) If two different set X and Y which are not equal that is
in ordinary set theory can be approximate equal in nano topological space that is
i.e.,
,
i.e.,
thus
.
(ii) If the nano inclusion of sets does not imply the inclusion of sets.
(iii) If the concept of power set P(X) in ordinary set theory differs from the concept of nano power set in nano topological space and it is clear that
,
,
.
6. Conclusion
This paper systematically studied that any kind of binary relation can be used to relational topology induced by nano topological space. The main aspect of this paper is to introduce nano topology by using the class of all after-composed set and the class of all fore-composed set. Now, we define the new operators in nano equality, nano inclusion and nano power set are now clear with respect to any relation in nano topology. We have proved that there exist similar properties. We will investigate the application of network topology.
Acknowledgements
The authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper and funding for this work was supported by University Grants Commission, New Delhi.