T -soft equality relation

The desire of generalizing some set-theoretic properties to the soft set theory motivated many researchers to define various types of soft operators. For example, they redefined the complement of a soft set, and soft union and intersection between two soft sets in a way that satisfies De Morgan's laws. In this paper, we introduce and study the concepts of $T$-soft subset and $T$-soft equality relations. Then, we utilize them to define the concepts of $T$-soft union and $T$-soft intersection for arbitrary family of soft sets. By $T$-soft union, we successfully keep some classical properties via soft set theory. We conclude this work by giving and investigating new types of soft linear equations with respect to some soft equality relations. Illustrative examples are provided to elucidate main obtained results.

Keywords:

T -soft subset, T -soft equality, T -soft union T -soft intersection, soft linear equations,

___

[1] Abbas M, Ali B, Romaguera S. On generalized soft equality and soft lattice structure. Filomat 2014; 28 (6): 1191- 1203.
[2] Abbas M, Ali MI, Romaguera S. Generalized operations in soft set theory via relaxed conditions on parameters. Filomat 2017; 31(19): 5955-5964.
[3] Aktas H, Cağman N. Soft sets and soft groups. Information Sciences 2007; 77: 2726-2735.
[4] Ali MI, Feng F, Liu X, Min WK, Shabir M. On some new operations in soft set theory. Computers and Mathematics with Applications 2009; 57: 1547-1553.
[5] Al-shami TM, El-Shafei ME. Partial belong relation on soft separation axioms and decision making problem: two birds with one stone. Soft Computing 2020; 24: 5377-5387.
[6] Al-shami TM, El-Shafei ME, Abo-Elhamayel M. On soft topological ordered spaces. Journal of King Saud UniversityScience 2019; 31 (4): 556-566.
[7] Cağman N. Contributions to the theory of soft sets. Journal of New Results in Science, 4 (2014) 33-41.
[8] Cağman N, Enginoğ S. Soft set theory and uni − int decision making. European Journal of Operational Research 2010; 207 (2): 848-855.
[9] Das S, Samanta SK. Soft real sets, soft real numbers and their properties, The Journal of Fuzzy Mathematics, 20 (3) (2012) 551-576.
[10] Das S, Samanta SK. On soft complex sets and soft complex numbers. The Journal of Fuzzy Mathematics 2013; 21 (1): 195-216.
[11] El-Shafei ME, Abo-Elhamayel M, Al-shami TM. Partial soft separation axioms and soft compac spaces. Filomat 2018; 32 (13): 4755-4771.
[12] El-Shafei ME, Al-shami TM. Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem. Computational and Applied Mathematics 2020; 39 (3) doi: 10.1007/s40314-020- 01161-3
[13] Feng F, Jun YB, Zhao XZ. Soft semirings. Computers and Mathematics with Applications 2008; 56: 2621-2628.
[14] Feng F, Li YM. Soft subsets and soft product operations. Information Science 2013; 232: 1468-1470.
[15] Feng F, Li YM, Davvaz B, Ali MI. Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Computing 2010; 14: 899-911.
[16] Gong K, Xia Z, Zhang X. The bijective soft set with its operations. Computers and Mathematics with Applications 2010; 60: 2270-2278.
[17] Jiang Y, Tang Y, Chen Q, Wang J, Tang S. Extending soft sets with description logics. Computers and Mathematics with Applications 2009; 59: 2087-2096.
[18] Jun YB, Yang X. A note on the paper ”Combination of interval-valued fuzzy set and soft set” [Computers and Mathematics with Applications 58 (2009) 521-527]. Computers and Mathematics with Applications 2011; 61: 1468- 1470.
[19] Li F. Notes on the soft operations. ARPN Journal of Systems and Software 2011; 1(6): 205-208.
[20] Liu X, Feng F, Jun YB. A note on generalized soft equal relations. Computers and Mathematics with Applications 2012; 64: 572-578.
[21] Maji PK, Biswas R, Roy R. An application of soft sets in a decision making problem. Computers and Mathematics with Applications 2002; 44: 1077-1083.
[22] Maji PK, Biswas R, Roy R. Soft set theory. Computers and Mathematics with Applications 45 (2003) 555-562.
[23] Min WK. Similarity in soft set theory. Applied Mathematics Letters 2012; 25: 310-314.
[24] Molodtsov D. Soft set theory-first results. Computers and Mathematics with Applications 1999; 37: 19-31.
[25] Pawlak Z. Rough sets. International Journal of Computer & Information Sciences 1982; 11: 341-356.
[26] Qin K, Hong Z. On soft equality. Journal of Computational and Applied Mathematics 2010; 234: 1347-1355.
[27] Sezgin A, Atagün AO. On operations of soft sets. Computers and Mathematics with Applications 2011; 61: 1457- 1467.
[28] Sezgin A, Atagün AO. Soft groups and normalistic soft groups. Computers and Mathematics with Applications 2011; 62: 685-698.
[29] Sezgin A, Atagün AO, Aygün E. A note on soft near-rings and idealistic soft near-rings. Filomat 2011; 25 (1): 53-68.
[30] Shabir M, Naz M. On soft topological spaces. Computers and Mathematics with Applications 2011; 61: 1786-1799.
[31] Yang CF. A note on soft set theory [Computers and Mathematics with Applications. 45 (4-5) (2003) 555-562]. Computers and Mathematics with Applications 2008; 56: 1899-1900.
[32] Yang HL, Guo ZL. Kernels and closures of soft set relations, and soft set relation mappings. Computers and Mathematics with Applications 2011; 61: 651-662.
[33] Zadeh LA. Fuzzy Sets. Information and Control 1965; 8: 338-353.
[34] Zhu P, Wen Q. Operations on soft sets revisited. Journal of Applied Mathematics 2013; Article ID 105752: 7 pages