Skip to main content
SpringerLink
Log in

Arithmetic Operations of Neutrosophic Sets, Interval Neutrosophic Sets and Rough Neutrosophic Sets

  • Chapter
  • First Online:
Fuzzy Multi-criteria Decision-Making Using Neutrosophic Sets

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 369))

Abstract

In approximation theory, neutrosophic set and logic show an important role. They are generalizations of intuitionistic fuzzy set and logic respectively. Based on neutrosophy, which is a new branch of philosophy, every idea X, has an opposite denoted as anti (X) and their neutral which is denoted as neut (X). These are the main features of neutrosophic set and logic. This chapter is based on the basic concepts of neutrosophic set as well as some of their hybrid structures. In this chapter, we define and study the notion of neutrosophic set and their basic properties. Moreover, interval-valued neutrosophic set are studied with some of their properties. Finally, we define rough neutrosophic sets .

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Zadeh, L.A.: Fuzzy sets. Inform. Control 8, 338–353 (1965)

    Article  Google Scholar 

  2. Turksen, I.B.: Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 20, 191–210 (1968)

    Article  MathSciNet  Google Scholar 

  3. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1986), 87–96 (1986)

    Article  Google Scholar 

  4. Smarandache, F.: Neutrosophy: Neutrosophic Set, Logic, Probability and Statistics. ProQuest Learning, Ann Arbor, Mchigan (1998)

    MATH  Google Scholar 

  5. Pawlak, Z.: Rough sets. Int. J. Comput. Inform. Sci. 11, 341–356 (1982)

    Article  Google Scholar 

  6. Broumi, S., Smarandache, F.: Rough neutrosophic sets. Ital. J. Pure Appl. Math. N. 32, 493–502 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Dubios, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen. Syst. 17, 191–208 (1990)

    Article  Google Scholar 

  8. Thomas, K.V., Nair, L.S.: Rough intuitionistic fuzzy sets in a lattice. Int. Math. Forum 6(27), 1327–1335 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Atanassov, K.T., Gargov, G.: Interval valued intuitionistic fuzzy sets, fuzzy sets and systems. Int. J. Gen. Syst. 393(31), 343–349 (1998)

    MATH  Google Scholar 

  10. Broumi, S., Smarandache, F.: New Operations on Interval Neutrosophic Set (2013), accepted

    Google Scholar 

  11. Wang, H., Smarandache, F., Zhang, Y.Q., Sunderraman, R.: Interval Neutrosophic Sets and Logic: Theory and Applications in Computing. Hexis, Phoenix, AZ (2005)

    MATH  Google Scholar 

  12. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning- Part I. Inf. Sci. 7, 199–249 (1975)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of New Mexico, 705 Gurley Ave., Gallup, NM, 87301, USA

    Florentin Smarandache

  2. University of Southern Queensland, 4300, Toowoomba, QLD, Australia

    Mumtaz Ali

  3. Abdul Wali Khan University, Mardan, 23200, Pakistan

    Mohsin Khan

Authors
  1. Florentin Smarandache
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mumtaz Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohsin Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Florentin Smarandache .

Editor information

Editors and Affiliations

  1. Department of Industrial Engineering, Istanbul Technical University, Macka, Istanbul, Turkey

    Cengiz Kahraman

  2. Department of Industrial Engineering, Istanbul Okan University, Akfirat-Tuzla, Istanbul, Turkey

    İrem Otay

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Smarandache, F., Ali, M., Khan, M. (2019). Arithmetic Operations of Neutrosophic Sets, Interval Neutrosophic Sets and Rough Neutrosophic Sets. In: Kahraman, C., Otay, İ. (eds) Fuzzy Multi-criteria Decision-Making Using Neutrosophic Sets. Studies in Fuzziness and Soft Computing, vol 369. Springer, Cham. https://doi.org/10.1007/978-3-030-00045-5_2

Download citation

Publish with us

Policies and ethics

Search

Navigation