NEUTROSOPHIC SET, A GENERALIZATION OF FUZZY SET
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                      by Florentin Smarandache


                          - Fourth Part -


     Let's second generalize, in the same way, the fuzzy set.

     A) Definition:
Neutrosophic Set is a set such that an element belongs to the set
with a neutrosophic probability, i.e. t% is true that the element
is in the set, f% false, and i% indeterminate.

     B) Neutrosophic Set Operations:
Let A and B be two neutrosophic sets.
One can say, by language abuse, that any element neutrosophically
belongs to any set, due to the percentage of
truth/indeterminacy/falsity involved, which varies between 0 and
100.
For example: x(50,20,30) belongs to A (which means, with a 
probability of 50% x is in A, with a probability of 30% x is not 
in A, and the rest is undecidable), or y(0,0,100) belongs to A 
(which normally means y is not for sure in A), or z(0,100,0) 
belongs to A (which means one doesn't know absolutely anything 
about z's affiliation with A).

     Let 0 <= t1, t2, t' <= 1 represent the truth-probabilities, 
0 <= i1, i2, i' <= 1 the indeterminacy-probabilities, and 
0 <= f1, f2, f' <= 1 the falsity-probabilities of an element x to
be in the set A and in the set B respectively, and of an element
y to be in the set B, where t1 + i1 + f1 = 1, t2 + i2 + f2 = 1,
and t' + i' + f' = 1.

One notes, with respect to the given sets,  
   x = x(t1, i1, f1) belongs to A and x = x(t2, i2, f2) belongs to B,
by mentioning x's neutrosophic probability appurtenance.
And, similarly,  y = y(t', i', f') belongs to B.
                                          _                  _
Also, for any 0 <= x <= 1 one notes 1-x = x.  Of course 0 <= x <= 1. 
Let's t = (t1, t2), i = (i1, i2), f = (f1, f2).
Let W(a,b,c) = (1-a)/(b+c) and W(R) = W( R(t),R(i),R(f) ) for any
tridimensional vector R = ( R(t),R(i),R(f) ).  

     Complement of A:
                 _
Let N(x) = 1-x = x.  Therefore:
if x( t1, i1, f1 ) belongs to A, 
then x( N(t1), N(i1)W(N), N(f1)W(N) ) belongs to C(A).

     Intersection:

Let C(x,y) = xy, and C(z1,z2) = C(z) for any bidimensional vector
z = (z1, z2).  Therefore:
if x( t1, i1, f1 ) belongs to A, x( t2, i2, f2 ) belongs to B, 
then x( C(t), C(i)W(C), C(f)W(C) ) belongs to A ï B.

     Union:
                         _       _                
Let D1(x,y) = x+y-xy = x+xy = y+xy, and D1(z1,z2) = D1(z) for any
bidimensional vector z = (z1, z2).  Therefore:
if x( t1, i1, f1 ) belongs to A, x( t2, i2, f2 ) belongs to B, 
then x( D1(t), D1(i)W(D1), D1(f)W(D1) ) belongs to A U B.

     Difference:
                     _                
Let D(x,y) = x-xy = xy, and D(z1,z2) = D(z) for any bidimensional
vector z = (z1, z2).  Therefore:
if x( t1, i1, f1 ) belongs to A,  x( t2, i2, f2 ) belongs to B, 
then x( D(t), D(i)W(D), D(f)W(D) ) belongs to A \ B,
because A \ B = A ï C(B).

     Cartesian Product:

if x( t1, i1, f1 ) belongs to A,  y( t', i', f' ) belongs to B, 
then ( x( t1, i1, f1 ), y( t', i', f' ) ) belongs to A x B.


     C) Applications:
þ From a pool of refugees, waiting in a political refugee camp to
get the America visa of emigration, a% are accepted, r% rejected,
and p% in pending (not yet decided), a+r+p=100.  The chance of
someone in the pool to emigrate to USA is not a% as in classical
probability, but a% true and p% pending (therefore normally
bigger than a%) - because later, the p% pending refugees will be
distributed into the first two categories, either accepted or
rejected.
þ Another example, a cloud is a neutrosophic set, because its
borders are ambiguous, and each element (water drop) belongs with
a neutrosophic probability to the set (e.g. there are separated
water drops, around a compact mass of water drops, that we don't
know how to consider them: in or out of the cloud). 
We are not sure where the cloud ends nor where it begins, neither
if some elements are or are not in the set.  That's why the
percent of indeterminacy is required: for a more organic, smooth,
and especially accurate estimation.