the degree of membership of in is ;
the degree of indeterminacy of in is ; and
the degree of non-membership of in is .
is called neutrosophic set, whereas are called neutrosophic components of the element with respect to .
Now let’s explain the previous notations:
A number is said to be infinitesimal if and only if for all positive integers one has . Let be a such infinitesimal number. The hyper-real number set is an extension of the real number set, which includes classes of infinite numbers and classes of infinitesimal numbers.
Generally, for any real number one defines which signifies a monad, i.e. a set of hyper-real numbers in non-standard analysis, as follows:
is infinitesimal ,
and similarly one defines , which is also a monad, as:
is infinitesimal .
A binad is a union of the above two monads, i.e.
For example: The non-standard finite number , where is its standard part and its non-standard part, and similarly the non-standard finite number , where is its standard part and its non-standard part.
Similarly for , etc.
Note that is called the non-standard unit interval.
More information on hyperreal intervals http://www.gallup.unm.edu/ smarandache/Introduction.pdfis available.
The superior sum of the neutrosophic components is defined as
which may be as high as 3 or .
While the inferior sum of the neutrosophic components is defined as
which may be as low as 0 or .
The notion of neutrosophic set was introduced by Florentin Smarandache in 1995 as a generalization of fuzzy set (especially of intuitionistic fuzzy set) when , of intuitionistic set when , and of paraconsistent set when .
The main distinctions between the neutrosophic set (NS) and intuitionistic fuzzy set (IFS) are the facts that (a) the sum of the scalar neutrosophic components (or their superior sum, , if the neutrosophic components are subsets) in NS is not necessarily 1 as in IFS but any number from to in order to allow the characterization of incomplete or paraconsistent information, and (b) in NS one uses the non-standard interval in order to make a difference between absolute membership, denoted by , and relative membership, denoted by , while in IFS one only uses the standard interval .
Let be a neutrosophic set.
One can say, by abuse of language, that any element neutrosophically belongs to any set, due to the flexibility of degrees of truth/indeterminacy/falsity involved, which each varies between and .
Thus the element means, the degree of membership of in is 0.1, the degree on indeterminacy (undecidability) is 0.2, and the degree of non-membership is 0.3 (as one sees, the sum of components is ¡ 1).
Similarly the element , with the sum of components ¿ 1.
Or the element , with the sum of components = 1.
More general, the element , means:
- the degree of membership is between 0.20-0.30 (one cannot find an exact approximation because of various sources used);
- the degree of indeterminacy related to the appurtenance of to is between 0.40-0.45 or between 0.50-0.51 (limits included);
- the degree of non-membership is 0.20 or 0.24 or 0.28.
- In technical applications, where there is no need for distinctions between absolute membership and relative membership, we can use standard subsets instead of non-standard subsets and respectively the unit interval instead of the non-standard unit interval .
- 1 F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, third edition, Xiquan, Phoenix, 2003. http://www.gallup.unm.edu/ smarandache/eBook-Neutrosophics2.pdfThe whole book is also online and can be downloaded here. .
- 2 F. Smarandache, J. Dezert, A. Buller, M. Khoshnevisan, S. Bhattacharya, S. Singh, F. Liu, Gh. C. Dinulescu-Campina, C. Lucas, C. Gershenson, Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics, The University of New Mexico, Gallup Campus, 1-3 December 2001. \htmladdnormallinkThe Proceedings are also online and can be downloaded here.http://arxiv.org/pdf/math.GM/0306384
- 3 Haibin Wang, Praveen Madiraju, Yanqing Zhang, Rajshekhar Sunderraman, Interval Neutrosophic Sets, International Journal of Applied Mathematics and Statistics, Vol. 3, No. M05, 1-18, 2005.