Multi-Granulation Picture Hesitant Fuzzy Rough Sets
Abstract
:1. Introduction
2. Preliminaries
3. Picture Hesitant Fuzzy Rough Sets
- 1.
- Inclusion: if, and only if, , for all ;
- 2.
- Union of and : ;
- 3.
- Intersection of and : ;
- 4.
- Complement of : .
- 1.
- The inverse relation is involutive, i.e., ;
- 2.
- The inverse relation is monotone with respect to inclusion, i.e., implies ;
- 3.
- The inverse relation preserves unions, i.e., ;
- 4.
- The inverse relation preserves intersections, i.e., ;
- 5.
- There is distributivity of unions with respect to intersections, i.e., ;
- 6.
- There is distributivity of intersections with respect to unions, i.e., .
{0.1,0.3,0.4},{0.2,0.3},{0.1,0.25} | {0.25,0.35},{0.1,0.15,0.2},{0.05,0.35} | {0.2,0.22},{0.11,0.29},{0.2,0.4} | |
{0.1,0.2},{0.13,2},{0.3,0.35} | {0.15,0.2,0.35},{0.2,0.25}, {0.15,0.3} | {0.1,0.4},{0.1,0.3}, {0.2,0.3} |
- 1.
- impliesimplies
- 2.
- 3.
- We can prove that with a similar reasoning.
- Similarly, and .Hence, .In a similar way, one can prove .
- Similarly, and .Hence, .In a similar way, one can prove . □
- 1.
- 2.
- .
4. Multi-Granulation Picture Hesitant Fuzzy Rough Sets
4.1. Optimistic Multi-Granulation Picture Hesitant Fuzzy Rough Sets
- 1.
- 2.
- .
- Additionally,.
- Additionally,□
- 1.
- ;
- 2.
- ;
- 3.
- ;
- 4.
- .
4.2. Pessimistic Multi-Granulation Picture Hesitant Fuzzy Rough Sets
- 1.
- 2.
- .
- Additionally,.
- Additionally,. □
- 1.
- ;
- 2.
- ;
- 3.
- ;
- 4.
- .
- 3.
- 4.
- The argument for claim 3 can be adapted to prove claim 4. □
5. The Relationship among PHFRS, the Optimistic MGPHFRS, and Pessimistic MGPHFRS
- 1.
- ;
- 2.
- ;
- 3.
- ;
- 4.
- .
- The proof is similar to the argument for claim 1.
- The argument for claim 3 can be adapted to prove claim 4. □
- 1.
- ;
- 2.
- .
6. Comparison with Existing Structures
7. Conclusions and Future Work
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
RS | Rough Set |
FS | Fuzzy Set |
PHFRS | Picture Hesitant Fuzzy Rough Set |
MGPHFRS | Multi-Granulation Picture Hesitant Fuzzy Rough Set |
OMGPHFRS | Optimistic Multi-Granulation Picture Hesitant Fuzzy Rough Set |
PMGPHFRS | Pessimistic Multi-Granulation Picture Hesitant Fuzzy Rough Set |
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Mathew, B.; John, S.J.; Alcantud, J.C.R. Multi-Granulation Picture Hesitant Fuzzy Rough Sets. Symmetry 2020, 12, 362. https://doi.org/10.3390/sym12030362
Mathew B, John SJ, Alcantud JCR. Multi-Granulation Picture Hesitant Fuzzy Rough Sets. Symmetry. 2020; 12(3):362. https://doi.org/10.3390/sym12030362
Chicago/Turabian StyleMathew, Bibin, Sunil Jacob John, and José Carlos R. Alcantud. 2020. "Multi-Granulation Picture Hesitant Fuzzy Rough Sets" Symmetry 12, no. 3: 362. https://doi.org/10.3390/sym12030362
APA StyleMathew, B., John, S. J., & Alcantud, J. C. R. (2020). Multi-Granulation Picture Hesitant Fuzzy Rough Sets. Symmetry, 12(3), 362. https://doi.org/10.3390/sym12030362