# A Comparative Study of Some Soft Rough Sets

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 3. Relationships among Several Soft Rough Sets

#### 3.1. Relationships between F-Soft Rough Approximations and MSR Approximations

**Example**

**1.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 3.2. The Relationships between MSR Approximations and Pawlak’s Rough Approximations

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

- (1)
- For any $x\in U$, ${[x]}_{{R}_{S}}=\{y\in U;\phi (x)=\phi (y)\}$.
- (2)
- For any $X\subseteq U$, ${\underline{X}}_{\phi}=\underline{{R}_{S}}(X)$.
- (3)
- For any $X\subseteq U$, ${\overline{X}}_{\phi}=\overline{{R}_{S}}(X)$.

**Proof.**

**Definition**

**6.**

**Corollary**

**3.**

- (1)
- ${\underline{\mu}}_{\phi}(x)=\wedge \{\mu (y);y\in {[x]}_{{R}_{S}}\},and$
- (2)
- ${\overline{\mu}}_{\phi}(x)=\vee \{\mu (y);y\in {[x]}_{{R}_{S}}\}$

#### 3.3. The Relationships among Several Soft Rough Fuzzy Sets

**Definition**

**7.**

**Definition**

**8.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

- (1)
- ${\underline{sap}}_{P}^{\prime}(\mu )\subseteq {\underline{\mu}}_{\phi}$,
- (2)
- ${\overline{\mu}}_{\phi}\subseteq {\overline{sap}}_{P}^{\prime}(\mu )$.

**Proof.**

**Corollary**

**4.**

**Definition**

**9.**

**Theorem**

**9.**

- (1)
- ${\underline{Apr}}_{SF}(\mu )=\underline{{R}_{\mathcal{F}}}(\mu )$,
- (2)
- ${\overline{Apr}}_{SF}(\mu )=\overline{{R}_{\mathcal{F}}}(\mu )$.

## 4. $\mathit{F}$-Soft Rough Sets and Modal-Style Operators in FCA

**Definition**

**10.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

## 5. A New Generalization of F-Soft Rough Set: Soft Rough Soft Sets

**Definition**

**11.**

**Example**

**2.**

**Proposition**

**1.**

- (1)
- ${\underline{sapr}}_{{S}_{1}}(\mathcal{S})\subseteq \mathcal{S}\subseteq {\overline{sapr}}_{{S}_{1}}(\mathcal{S})$,
- (2)
- ${\underline{sapr}}_{{S}_{1}}({\tilde{N}}_{(U,A)})={\tilde{N}}_{(U,A)}={\overline{sapr}}_{{S}_{1}}({\tilde{N}}_{(U,A)})$,
- (3)
- ${\overline{sapr}}_{{S}_{1}}({\tilde{W}}_{(U,A)})={\tilde{W}}_{(U,A)}={\underline{sapr}}_{{S}_{1}}({\tilde{W}}_{(U,A)})$.

**Proof.**

**Proposition**

**2.**

- (1)
- $\mathcal{S}\subseteq \mathcal{T}\Rightarrow {\underline{sapr}}_{{S}_{1}}(\mathcal{S})\subseteq {\underline{sapr}}_{{S}_{1}}(\mathcal{T})$,
- (2)
- $\mathcal{S}\subseteq \mathcal{T}\Rightarrow {\overline{sapr}}_{{S}_{1}}(\mathcal{S})\subseteq {\overline{sapr}}_{{S}_{1}}(\mathcal{T})$,
- (3)
- ${\underline{sapr}}_{{S}_{1}}(\mathcal{S}\cap \mathcal{T})\subseteq {\underline{sapr}}_{{S}_{1}}(\mathcal{S})\cap {\underline{sapr}}_{{S}_{1}}(\mathcal{T})$,
- (4)
- ${\underline{sapr}}_{{S}_{1}}(\mathcal{S}\cup \mathcal{T})\supseteq {\underline{sapr}}_{{S}_{1}}(\mathcal{S})\cup {\underline{sapr}}_{{S}_{1}}(\mathcal{T})$,
- (5)
- ${\overline{sapr}}_{{S}_{1}}(\mathcal{S}\cup \mathcal{T})\supseteq {\overline{sapr}}_{{S}_{1}}(\mathcal{S})\cup {\overline{sapr}}_{{S}_{1}}(\mathcal{T})$,
- (6)
- ${\overline{sapr}}_{{S}_{1}}(\mathcal{S}\cap \mathcal{T})\subseteq {\overline{sapr}}_{{S}_{1}}(\mathcal{S})\cap {\overline{sapr}}_{{S}_{1}}(\mathcal{T})$.

**Proof.**

**Proposition**

**3.**

- (1)
- ${\underline{sapr}}_{{S}_{1}}(\mathcal{S})\subseteq {\underline{sapr}}_{{S}_{1}}({\overline{sapr}}_{{S}_{1}}(\mathcal{S}))$,
- (2)
- ${\overline{sapr}}_{{S}_{1}}(\mathcal{S})\supseteq {\overline{sapr}}_{{S}_{1}}({\underline{sapr}}_{{S}_{1}}(\mathcal{S}))$.

**Proof.**

**Example**

**3.**

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The process of computing ${f}_{{S}_{1}}({e}_{4})$ and ${f}^{{S}_{1}}({e}_{4})$ from $f({e}_{4})$ in Example 2.

U | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |
---|---|---|---|---|---|---|---|

A | |||||||

e_{3} | 1 | 0 | 0 | 0 | 0 | 1 | |

e_{4} | 0 | 1 | 1 | 0 | 0 | 0 | |

e_{5} | 0 | 0 | 0 | 0 | 0 | 0 | |

e_{6} | 0 | 0 | 0 | 0 | 1 | 0 | |

e_{7} | 0 | 0 | 0 | 1 | 1 | 1 |

U | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |
---|---|---|---|---|---|---|---|

A | |||||||

e_{1} | 1 | 1 | 0 | 1 | 0 | 1 | |

e_{2} | 0 | 1 | 1 | 0 | 0 | 0 | |

e_{3} | 0 | 0 | 0 | 1 | 1 | 1 | |

e_{4} | 1 | 1 | 1 | 1 | 0 | 1 |

U | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |
---|---|---|---|---|---|---|---|

A | |||||||

e_{1} | 1 | 0 | 0 | 0 | 0 | 1 | |

e_{2} | 0 | 1 | 1 | 0 | 0 | 0 | |

e_{3} | 0 | 0 | 0 | 1 | 1 | 1 | |

e_{4} | 1 | 1 | 1 | 0 | 0 | 1 |

U | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |
---|---|---|---|---|---|---|---|

A | |||||||

e_{1} | 1 | 1 | 1 | 1 | 1 | 1 | |

e_{2} | 0 | 1 | 1 | 0 | 0 | 1 | |

e_{3} | 1 | 0 | 0 | 1 | 1 | 1 | |

e_{4} | 1 | 1 | 1 | 1 | 1 | 1 |

U | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |
---|---|---|---|---|---|---|---|

A | |||||||

e_{1} | 1 | 1 | 1 | 1 | 0 | 1 | |

e_{2} | 0 | 1 | 1 | 0 | 0 | 0 | |

e_{3} | 0 | 0 | 0 | 1 | 1 | 1 | |

e_{4} | 1 | 1 | 1 | 1 | 0 | 1 |

U | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |
---|---|---|---|---|---|---|---|

A | |||||||

e_{1} | 1 | 1 | 0 | 1 | 0 | 1 | |

e_{2} | 1 | 1 | 1 | 0 | 0 | 0 | |

e_{3} | 0 | 0 | 0 | 1 | 1 | 1 | |

e_{4} | 1 | 1 | 1 | 1 | 0 | 1 |

U | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |
---|---|---|---|---|---|---|---|

A | |||||||

e_{1} | 1 | 1 | 0 | 1 | 0 | 1 | |

e_{2} | 0 | 1 | 1 | 0 | 0 | 0 | |

e_{3} | 1 | 0 | 0 | 1 | 1 | 1 | |

e_{4} | 1 | 1 | 1 | 1 | 0 | 1 |

U | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | |
---|---|---|---|---|---|---|---|

A | |||||||

e_{1} | 1 | 1 | 0 | 1 | 0 | 1 | |

e_{2} | 0 | 1 | 1 | 0 | 0 | 0 | |

e_{3} | 0 | 0 | 1 | 1 | 1 | 1 | |

e_{4} | 1 | 1 | 1 | 1 | 0 | 1 |

Various Hybrid Models | Relationships |
---|---|

F-soft rough approximations and modified soft rough approximations (MSR approximations) | ${\overline{X}}_{\phi}\subseteq {\overline{apr}}_{P}(X)$, ${\overline{apr}}_{P}(X)\subseteq {\overline{X}}_{\phi}$, ${\underline{X}}_{\phi}\subseteq {\underline{apr}}_{P}(X)$, if some specific conditions hold, respectively (see Theorems 1–3) |

F-soft rough sets in $(U,S)$ and Pawlak’s rough sets in $(U,{R}_{S})$ | F-soft rough sets in $(U,S)$ could be identified with Pawlak’s rough sets in $(U,{R}_{S})$, when the underlying soft set is a partition soft set (see Theorems 4 and 5) |

MSR approximations and Pawlak’s rough approximations | MSR approximation operator is a kind of Pawlak rough approximation operator (see Theorem 6) |

Z-lower, Z-upper soft rough approximation operators and Dubois and Prade’s lower and upper rough fuzzy approximation operators in [6] | Z-lower and Z-upper soft rough approximation operators are equivalent to Dubois and Prade’s lower and upper rough fuzzy approximation operators in [6] (see Corollary 3) |

The (classical) rough fuzzy sets and M-soft rough fuzzy sets | The (classical) rough fuzzy sets in Pawlak approximation space $(U,R)$ and M-soft rough fuzzy sets in soft approximation space $(U,S)$ are equivalent when the underlying soft set S is a partition soft set (see Theorem 7) |

Z-soft rough approximation operators and M-soft Rough approximation operators and F-soft rough approximation operators | ${\underline{sap}}_{P}(\mu )\subseteq {\underline{sap}}_{P}^{\prime}(\mu )\subseteq {\underline{\mu}}_{\phi}\subseteq \mu \subseteq {\overline{\mu}}_{\phi}\subseteq {\overline{sap}}_{P}^{\prime}(\mu )$ $\subseteq {\overline{sap}}_{P}(\mu )$ (see Theorem 8 and Corollary 4) |

The soft fuzzy rough approximation in Definition 9 and Dubois and Prade’s fuzzy rough approximation in [6] | The soft fuzzy rough approximation is a kind of Dubois and Prade’s fuzzy rough approximation in [6] (see Theorem 9) |

F-soft rough set and soft rough soft set | Soft rough soft set is an extension of F-soft rough set |

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**MDPI and ACS Style**

Liu, Y.; Martínez, L.; Qin, K.
A Comparative Study of Some Soft Rough Sets. *Symmetry* **2017**, *9*, 252.
https://doi.org/10.3390/sym9110252

**AMA Style**

Liu Y, Martínez L, Qin K.
A Comparative Study of Some Soft Rough Sets. *Symmetry*. 2017; 9(11):252.
https://doi.org/10.3390/sym9110252

**Chicago/Turabian Style**

Liu, Yaya, Luis Martínez, and Keyun Qin.
2017. "A Comparative Study of Some Soft Rough Sets" *Symmetry* 9, no. 11: 252.
https://doi.org/10.3390/sym9110252