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

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

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

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the degree of nonmembership 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 $\epsilon $ is said to be infinitesimal if and only if for all positive integers $n$ one has $$. Let $\epsilon >0$ be a such infinitesimal number. The hyperreal number set is an extension^{} of the real number set, which includes classes of infinite^{} numbers and classes of infinitesimal numbers.
Generally, for any real number $a$ one defines ${}^{}a$ which signifies a monad, i.e. a set of hyperreal numbers in nonstandard analysis, as follows:
${}^{}a=\{a\epsilon :\epsilon \in {R}^{*},\epsilon $ is infinitesimal $\}$,
and similarly one defines ${a}^{+}$, which is also a monad, as:
${a}^{+}=\{a+\epsilon :\epsilon \in {R}^{*},\epsilon $ is infinitesimal $\}$.
A binad ${}^{}a^{+}$ is a union of the above two monads, i.e.
${}^{}a^{+}{=}^{}a\cup {a}^{+}$.
For example: The nonstandard finite number ${1}^{+}=1+\epsilon $, where $1$ is its standard part and $\epsilon $ its nonstandard part, and similarly the nonstandard finite number ${}^{}0=0\epsilon $, where $0$ is its standard part and $\epsilon $ its nonstandard part.
Similarly for ${3}^{+}=3+\epsilon $, etc.
Note that $]{}^{}0,1{}^{+}[$ is called the nonstandard unit interval.
More information on hyperreal intervals http://www.gallup.unm.edu/ smarandache/Introduction.pdfis available.
The superior sum of the neutrosophic components is defined as
$${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{}^{+}[$$ 
which may be as low as 0 or ${}^{}0$.
The notion of neutrosophic set was introduced by Florentin Smarandache in 1995 as a generalization^{} of fuzzy set (especially of intuitionistic fuzzy set) when ${n}_{sup}=1$, of intuitionistic set when $$, 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 nonstandard 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 nonmembership 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 $w((0.200.30),(0.400.45)\cup [0.500.51],\{0.20,0.24,0.28\})\in A$, means:
 the degree of membership is between 0.200.30 (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 0.400.45 or between 0.500.51 (limits included);
 the degree of nonmembership 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 nonstandard subsets and respectively the unit interval $[0,1]$ instead of the nonstandard 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/eBookNeutrosophics2.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. DinulescuCampina, 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, 13 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, 118, 2005.
Title  neutrosophic set 

Canonical name  NeutrosophicSet 
Date of creation  20130322 15:21:49 
Last modified on  20130322 15:21:49 
Owner  para0doxa (5174) 
Last modified by  para0doxa (5174) 
Numerical id  9 
Author  para0doxa (5174) 
Entry type  Definition 
Classification  msc 03E70 