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.
More information on hyperreal intervals is available.

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 $]^-0, 1^+[$ .

Bibliography

1

F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, third edition, Xiquan, Phoenix, 2003. The 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. The Proceedings are also online and can be downloaded here.

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.