Applied Mathematics
Vol.08 No.08(2017), Article ID:78799,9 pages
10.4236/am.2017.88089
Heuristic Approach to Establish New Operators via Nano Topology
Lellis Thivagar Maria Antony, Priyalatha Sundharambal Perumal Rajendran
School of Mathematics, Madurai Kamaraj University, Madurai (Dt), India
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: June 9, 2017; Accepted: August 27, 2017; Published: August 30, 2017
ABSTRACT
This article presents a new kind of class of all after composed set and fore composed set using the arbitrary binary relation based on nano topological space. We express the notion of nano equality, and nano inclusion and nano power set regarding binary relation based on nano topology. Also, we discuss their properties. Finally, the real life application of network topology is studied.
Keywords:
Nano Topology, After Composed Set, Nano Equality, Nano Inclusion, Nano Power Set
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.
Cite this paper
Antony, L.T.M. and Rajendran, P.S.P. (2017) Heuristic Approach to Establish New Operators via Nano Topology. Applied Mathematics, 8, 1186-1194. https://doi.org/10.4236/am.2017.88089
References
- 1. Thivagar, M.L. and Richard, C. (2013) On Nano Forms of Weakly Open Sets. International journal of Mathematics and statistics Invention, 1, 31-37.
- 2. Martin, M.J. (2000) Understanding the Network a Practical Guide to Internet Working. New Riders Publishing, 11, 341-356.
- 3. Abo-Tabl, E.A. (2011) A Comparison of Two Kinds Definitions of Rough Approximation Based on Similarity Relation. Information Sciences, 181, 2587-2596. https://doi.org/10.1016/j.ins.2011.01.007
- 4. Abo-Tabl, E.A. (2013) Rough Sets and Topological Spaces Based on Similarity. International Journal of Machine Learning and Cybernetics, 4, 451-458. https://doi.org/10.1007/s13042-012-0107-7
- 5. Dubois, D. and Prade, H. (1990) Rough Fuzzy Sets and Fuzzy Rough Sets. International Journal of General Systems, 17, 191-208. https://doi.org/10.1080/03081079008935107
- 6. Zhu, W. (2009) Relationship between Generalized Rough Sets Based on Binary Relation and Covering. Journal of Information Science, 2, 210-225. https://doi.org/10.1016/j.ins.2008.09.015
- 7. Lin, T.Y. (1998) Granular Computing on Binary Relation, I: Data Mining and Neighborhood Systems. Rough Sets in Knowledge Discovery Physica-Verlag, 2, 107-121.
- 8. Pawlak, Z. (1982) Rough Sets. International journal of computer and Information Sciences, 2, 341-356.