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Introduction

Problems with uncertainties are a major issue in many fields such as economics, engineering, environment and so on. One of the theories that deal with uncertainties is the soft set theory which was introduced by Molodtsov in 1999 (Molodstov, 1999) as a new mathematical tool. In the soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields including game theory, operations research, probability theory, and measurement theory.

In the literature, different kinds of operations of soft sets are defined and used in the works on the soft set theory and its applications. Well known operations of soft sets and their properties are given by Maji et al. (Maji et al, 2003). However, some of these definitions and their properties have a few gaps, aspointed out by Ali et al. (Ali et al, 2009) and Yang (Yang, 2008). To make some modifications to the operations of soft sets and fill in these gaps, Ali et al. (Ali et al, 2009). Cagman and Enginoglu (Cagman & Enginoglu, 2010), Pei and Miao (Pei & Miao, 2005), and Sezgin and Atagun (Sezgin & Atagun, 2011) have made contributions.

In 2011, Sezgin and Atagun (Sezgin & Atagun, 2011) discussed the fundamental theorems about operations of soft sets i.e; union and intersection of soft sets and other operations, see (Perović et al, 2008). In that paper, they defined union and intersection operations of soft sets both with restricted and extended conditions but defined the difference operation only with the restricted condition. In 2019, Sezgin, Ahmad and Mehmood (Sezgin et al, 2019) defined the extended difference operation.

Here in this paper, we have defined a new operation on soft sets called extended symmetric difference and also proved some of its properties. Moreover, we have also proved the interesting results which show the relationships between extended symmetric difference and other operations. The main objective of this paper is to make the soft set theory more effective and solid by enhancing the conceptual feature of operations on soft sets. This paper is one theoretical study of soft sets.

Preliminaries

In this section, we recall some basic notions in the soft set theory. Let U be an initial universe set and ${\text{E}}_{\text{U}}$ be the set of all possible parameters under consideration with respect to U. The power set of U is denoted by and A is a subset of E. Usually, parameters are attributes, characteristics, or properties of objects in U. In what follows, ${\text{E}}_{\text{U}}$ (simply denoted by E) always means the universe set of parameters with respect to U, unless otherwise specified.

Molodtsov (Molodtsov, 1999) defined the soft set in the following manner:

Definition 2.1. (Molodtsov, 1999) Let U be an initial universe set, E be a set of parameters, the power set of U. A pair (F,E) is called a soft set over U, where F is a mapping of E into the set of all subsets of the set U.

In other words, a soft set over U is a parameterized family of subsets of U. For , may be considered as the set of e-elements of the soft set (F,E) or as the set of e-approximate elements of the soft set.

Definition 2.2. (Maji et al, 2003) For two soft sets (F,A) and (G,B) over a common universe U, we say that (F,A) is a soft subset of (G,B), denoted by, if it satisfies:

• ,

• for all , and G(e) are identical approximations.

Similarly, (F,A) is called a soft superset of (G,B) if (G,B) is a soft subset of (F,A). This relation is denoted by.

Definition 2.3. (Maji et al, 2003) Two soft sets (F,A) and (G,B) over a common universe U are called soft equal if and .

Definition 2.4. (Ali et al, 2009) The relative complement of a soft set (F,A) is denoted by and is defined by , where is a mapping given by .

Clearly , and . It is worth noting that in the above definition of complement, the parameter set of the complement is still the original parameter set A (Ali et al, 2009).

Definition 2.5. (Maji et al, 2003) A soft (F,A) set over U is said to be a null soft set denoted by $\text{Ф}$, if for all, (null set).

Since some researchers are in some conflict about a null soft set due to its notation, like others (Sezgin & Atagun, 2011), so we prefer to use ${\text{Ф}}_{\text{А}}$ instead of $\text{Ф}$ for the null soft set of (F,A) as Ali et al. (Ali et al, 2009) used.

Definition 2.6. (Maji et al, 2003) A soft set (F,A) over U is said to be an absolute soft set denoted by.

Note that we use the notation instead of as in (Ali et al, 2009) throughout this paper.

Definition 2.7. (Ali et al, 2009) Let (F,A) and (G,B) be two soft sets over a common universe U such that. The restricted intersection of (F,A) and (G,B) s denoted by and is defined as, where and for all .

Definition 2.8. (Ali et al, 2009) Let (F,A) and (G,B) be two soft sets over a common universe U such that. The restricted difference of (F,A) and (G,B) is denoted by and is defined as , where and for all .

Definition 2.9. (Ali et al, 2009) Let (F,A) and (G,B) be two soft sets over a common universe U such that. The restricted union of (F,A) and (G,B) is denoted by and is defined as, where and for all .

Definition 2.10. (Sezgin & Atagun, 2011) Let (F,A) and (G,B) be two soft sets over a common universe U such that . The restricted symmetric difference of (F,A) and (G,B) is denoted by and is defined as , where .

Definition 2.11. (Ali et al, 2009) The extended union of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C), where , and for all ;

We write .

Definition 2.12. (Ali et al, 2009) The extended intersection of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C), where , and for all ;

We write .

Definition 2.13. (Sezginet al, 2019) The extended difference of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C), where , and for all ;

Thus, the relation is sown by .

Extended symmetric difference

For the fundamental properties and theorems related to operations of soft sets such as restricted union, extended union, restricted intersection, extended intersection, restricted difference, extended difference, we refer the readers to the papers Ali et al. (Ali et al, 2009), Cagman and Enginoglu (Cagman & Enginoglu, 2010), Pai and Miao (Pei & Miao, 2005), Sezgin and Atagun (Sezgin & Atagun, 2011) and Sezgin, Ahmad and Mehmood (Sezgin et al, 2019).

Now we are ready to give the definition of extended symmetric difference of soft sets and its basic properties.

Definition 3.1. The extended symmetric difference of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C), where , and for all ;

Thus, the relation is shown by

Example 1. Let E be the universe set of parameters, A and B be the subsets of E such that

Let (F,A) and (G,B) be two soft sets over the same universe U={h₁,h₂,h₃,h₄} such that

Now let us determine , where

for all . Since , then,

Theorem 3.2. The properties of the extended symmetric difference operation

Proof.

(a) Let , where

for all

Since , it follows that This means that F and H are the same mappings. This completes the proof.

(b) Let , where

for all

Hence, . This completes the proof.

(c) Let where

for all .

Since,

for all

(d) The proof can be illustrated by similar techniques used to prove (a), (b) and (c), and is therefore omitted.

(e) For the left-hand side of the property, let where

for all .

For the right-hand side of the property, let , where

for all . Suppose that , where

Assume that, , where

Now, we have

for all It shows that H and K are the identical mapping when we are assuming the attributes of operations about the set theory. Hence the proof is completed.

(f) For the left-hand side of the property, let , where

for all . For the right-hand side of the property, let , where

for all . And suppose that , where for all .

Assume that , where

i.e.

for all . This leads that H and K are the identical mapping. Hence this completes the proof.

(g) For the left-hand side of the property, let , where

for all .

For the right-hand side of the property, let

And suppose that, where

for all .

Assume that, , where

i.e.

for all .

This leads that H and K are the identical mapping.

Hence this completes the proof.

Now we will illustrate Theorem 3.2.(parts (e) and (f)) with a corresponding example.

Example 2. Let E be the universe set of parameters, A and B be the subsets of E such that

Let (F,A) and (G,B) be two soft sets over the same universe U={h₁,h₂,h₃,h₄} such that

Now let us determine , where

for all .

Since , then,

On the other side, we can easily determine that it is

Based on Definition 2.9. we have that

Therefore,

Similarly, we can easily determine that it is

Based on Definition 2.13. we have that

Therefore,

Conclusion

In this paper, we have illustrated a brief analytical review of operations of soft sets. We have defined the extended symmetric difference of soft sets and also proved some of its properties. Moreover, we have shown the relationship between extended symmetric difference and some other operations of soft sets.

References

Ali, M.I., Feng, F., Liu, X., Min, W.K. & Shabir, M. 2009. On some new operations in soft set theory. Computers & Mathematics with Applications, 57(9), pp.1547-1553. Available at: https://doi.org/10.1016/j.camwa.2008.11.009.

Cagman, N. & Enginoglu, S. 2010. Soft set theory and uni-int decision making. European Journal of Operational Research, 207(2), pp.848-855. Available at: https://doi.org/10.1016/j.ejor.2010.05.004.

Maji, P.K., Biswas, R. & Roy, A.R. 2003. Soft set theory. Computers & Mathematics with Applications, 45(4-5), pp.555-562. Available at: https://doi.org/10.1016/S0898-1221(03)00016-6.

Molodtsov, D. 1999. Soft set theory First results. Computers & Mathematics with Applications, 37(4-5), pp.19-31. Available at: https://doi.org/10.1016/S0898-1221(99)00056-5.

Pei, D. & Miao, D. 2005. From soft sets to information systems. In: 2005 IEEE International Conference on Granular Computing, Beijing, China, 2, pp.617-621, July 25-27. Available at: https://doi.org/10.1109/GRC.2005.1547365.

Perović, A., Jovanović, A. & Veličković, B. 2008. Teorija skupova - matematička teorija beskonačnosti. Belgrade: University of Belgrade, Faculty of Mathematics (in Serbian) [online]. Available at: http://elibrary.matf.bg.ac.rs/handle/123456789/473?locale-attribute=en [Accessed: 20 August 2021]. ISBN: 978-86-7589-058-4.

Sezgin, A., Ahmad, S. & Mehmood, A. 2019. A New Operation on Soft Sets: Extended Difference of Soft Sets. Journal of New Theory, 27, pp.33-42 [online]. Available at: https://dergipark.org.tr/en/pub/jnt/issue/43609/535589 [Accessed: 20August 2021].

Sezgin, A. & Atagun, A.O. 2011. On operations of soft sets. Computers & Mathematics with Applications, 61(5), pp.1457-1467. Available at: https://doi.org/10.1016/j.camwa.2011.01.018.

Yang, C-F. 2008. A note on “Soft Set Theory” [Comput. Math. Appl. 45 (4–5) (2003) 555–562]. Computers & Mathematics with Applications, 56(7), pp.1899-1900. Available at: https://doi.org/10.1016/j.camwa.2008.03.019.

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