# neutrosophic set

Let $M$ be a subset of a universe of discourse $U$. Each element $x\in U$ has degrees of membership, indeterminacy, and non-membership in $M$, which are subsets of the hyperreal interval $]^{-}0,1^{+}[$. The notation $x(T,I,F)\in M$ means that

• the degree of membership of $x$ in $M$ is $T$;

• the degree of indeterminacy of $x$ in $M$ is $I$; and

• the degree of non-membership of $x$ in $M$ is $F$.

$M$ is called neutrosophic set, whereas $T,I,F$ are called neutrosophic components of the element $x$ with respect to $M$.

Now let’s explain the previous notations:
A number $\varepsilon$ is said to be infinitesimal if and only if for all positive integers $n$ one has $|\varepsilon|<\frac{1}{n}$. Let $\varepsilon>0$ 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 $a$ one defines ${}^{-}a$ which signifies a monad, i.e. a set of hyper-real numbers in non-standard analysis, as follows:
${}^{-}a=\{a-\varepsilon:\varepsilon\in R^{*},\varepsilon$ is infinitesimal $\}$,
and similarly one defines $a^{+}$, which is also a monad, as:
$a^{+}=\{a+\varepsilon:\varepsilon\in R^{*},\varepsilon$ is infinitesimal $\}$.
A binad ${}^{-}a^{+}$ is a union of the above two monads, i.e.
${}^{-}a^{+}=^{-}a\cup a^{+}$.
For example: The non-standard finite number $1^{+}=1+\varepsilon$, where $1$ is its standard part and $\varepsilon$ its non-standard part, and similarly the non-standard finite number ${}^{-}0=0-\varepsilon$, where $0$ is its standard part and $\varepsilon$ its non-standard part.
Similarly for $3^{+}=3+\varepsilon$, etc.
Note that $]^{-}0,1^{+}[$ is called the non-standard unit interval.

The superior sum of the neutrosophic components is defined as

 $n_{sup}=sup(T)+sup(I)+sup(F)\in]^{-}0,3^{+}[$

which may be as high as 3 or $3^{+}$.
While the inferior sum of the neutrosophic components is defined as

 $n_{inf}=inf(T)+inf(I)+inf(F)\in]^{-}0,3^{+}[$

which may be as low as 0 or ${}^{-}0$.

The notion of neutrosophic set was introduced by Florentin Smarandache in 1995 as a generalization of fuzzy set (especially of intuitionistic fuzzy set) when $n_{sup}=1$, of intuitionistic set when $n_{sup}<1$, and of paraconsistent set when $n_{sup}>1$.

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, $n_{sup}$, if the neutrosophic components are subsets) in NS is not necessarily 1 as in IFS but any number from ${}^{-}0$ to $3^{+}$ in order to allow the characterization of incomplete or paraconsistent information, and (b) in NS one uses the non-standard interval $]^{-}0,1^{+}[$ in order to make a difference between absolute membership, denoted by $1^{+}$, and relative membership, denoted by $1$, while in IFS one only uses the standard interval $[0,1]$.

An example:
Let $A$ 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 ${}^{-}0$ and $1^{+}$.
Thus the element $x(0.1,0.2,0.3)\in A$ means, the degree of membership of $x$ in $A$ 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 $y(0.6,0.2,0.5)\in A$, with the sum of components ¿ 1.
Or the element $z(0.7,0.1,0.2)\in A$, with the sum of components = 1.
More general, the element $w((0.20-0.30),(0.40-0.45)\cup[0.50-0.51],\{0.20,0.24,0.28\})\in A$, 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 $w$ to $A$ 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.

A remark:
- 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 $[0,1]$ instead of the non-standard unit interval $]^{-}0,1^{+}[$.

## References

• 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.
Title neutrosophic set NeutrosophicSet 2013-03-22 15:21:49 2013-03-22 15:21:49 para0doxa (5174) para0doxa (5174) 9 para0doxa (5174) Definition msc 03E70