Research Article A Cubic Set Theoretical Approach to Linear Space S. Vijayabalaji 1 and S. Sivaramakrishnan 2 1 Department of Mathematics, University College of Engineering, Panruti (A Constituent College of Anna University Chennai), Panruti, Tamilnadu 607 106, India 2 Department of Mathematics, Krishnasamy College of Engineering and Technology, Cuddalore, Tamilnadu 607 109, India Correspondence should be addressed to S. Vijayabalaji; balaji1977harshini@gmail.com Received 12 January 2015; Revised 28 March 2015; Accepted 7 April 2015 Academic Editor: Ademir F. Pazoto Copyright © 2015 S. Vijayabalaji and S. Sivaramakrishnan. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te main motivation of this paper is to introduce the notion of cubic linear space. Tis inspiration is received from the structure of cubic sets. Te notions of R-intersection, R-union, P-intersection, and P-union of cubic linear spaces are defned and we provide some results on these. We further introduce the notion of internal cubic linear space and external cubic linear space and establish some results on them. 1. Introduction Te notion of fuzzy sets introduced by Zadeh [1] in 1965 laid the foundation for the development of fuzzy Mathematics. Tis theory has a wide range of application in several branches of Mathematics such as logic, set theory, group theory, semigroup theory, real analysis, measure theory, and topology. Afer a decade, the notion of interval-valued fuzzy sets was introduced by Zadeh [2] in 1975, as an extension of fuzzy sets, that is, fuzzy sets with interval-valued membership functions. Lubczonok and Murali [3] introduced an inter- esting theory of fags and fuzzy subspaces of vector spaces. Katsaras and Liu [4] introduced the concepts of fuzzy vector and fuzzy topological vector spaces. Fuzzy bases of vector spaces and fuzzy vector spaces have been studied in [5, 6]. Nanda [7] introduced the notion of fuzzy feld and fuzzy linear space over a fuzzy feld. Wenxiang and Lu [8] redefned the concepts of fuzzy feld and fuzzy linear space. Vijayabalaji et al. [9] introduced the notion of interval-valued fuzzy linear subspace and interval-valued fuzzy -normed linear space. Tey have also proved that the intersection of two interval-valued fuzzy linear spaces is again an interval-valued fuzzy linear space. Atanassov [10] introduced the notion of intuitionistic fuzzy sets as a generalization of fuzzy sets. Jun et al. [11] have introduced a remarkable theory, name- ly, the theory of cubic sets. Tis structure is comprised of an interval-valued fuzzy set and a fuzzy set. In the same paper they introduced the notion of cubic subalgebras/ideals in BCK/BCI algebras and investigated some of their properties. Moreover, Jun et al. [12] introduced the notion of cubic sub- groups. Tey also studied images or inverse images of cubic subgroups. Furthermore, Jun et al. [13] introduced the concept of an internal cubit set and an external cubic set. Recently, Yaqoob et al. [14] introduced the notion of cubic  ideals of -algebras. Attracted by the theory of cubic sets we introduce the notion of cubic linear space. Te concept of -intersection, - union, -intersection, and -union of cubic linear space are introduced and some properties are studied. We prove that the -intersection of two cubic linear spaces is again a cubic linear space. It is shown by means of counter examples that the -union, -intersection, and -union of two cubic linear spaces need not be a cubic linear space. We also introduce the notions of internal cubic linear space and external cubic linear space. It is established that the -intersection of two internal (resp., external) cubic linear spaces is again an internal (resp., external) cubic linear space. We conclude the paper by providing examples to show that the -intersection, -union, and the -union of two internal (resp., external) cubic linear spaces are not internal (resp., external) cubic linear spaces. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2015, Article ID 523129, 8 pages http://dx.doi.org/10.1155/2015/523129