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NEUTROSOPHIC SET, A GENERALIZATION OF FUZZY SET ----------------------------------------------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 eleme

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NEUTROSOPHIC SET, A GENERALIZATION OF FUZZY SET-----------------------------------------------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 setwith a neutrosophic probability, i.e. t% is true that the elementis 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 neutrosophicallybelongs to any set, due to the percentage oftruth/indeterminacy/falsity involved, which varies between 0 and100.For example: x(50,20,30) belongs to A (which means, with aprobability of 50% x is in A, with a probability of 30% x is notin 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 anythingabout z's affiliation with A).Let 0 <= t1, t2, t' <= 1 represent the truth-probabilities,0 <= i1, i2, i' <= 1 the indeterminacy-probabilities, and0 <= f1, f2, f' <= 1 the falsity-probabilities of an element x tobe in the set A and in the set B respectively, and of an elementy 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 anytridimensional 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 vectorz = (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 B.
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Union:_ _Let D1(x,y) = x+y-xy = x+xy = y+xy, and D1(z1,z2) = D1(z) for anybidimensional 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 bidimensionalvector 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 C(B).
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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:From a pool of refugees, waiting in a political refugee camp to
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get the America visa of emigration, a% are accepted, r% rejected,and p% in pending (not yet decided), a+r+p=100. The chance ofsomeone in the pool to emigrate to USA is not a% as in classicalprobability, but a% true and p% pending (therefore normallybigger than a%) - because later, the p% pending refugees will bedistributed into the first two categories, either accepted orrejected.Another example, a cloud is a neutrosophic set, because its
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borders are ambiguous, and each element (water drop) belongs witha neutrosophic probability to the set (e.g. there are separatedwater drops, around a compact mass of water drops, that we don'tknow how to consider them: in or out of the cloud).We are not sure where the cloud ends nor where it begins, neitherif some elements are or are not in the set. That's why thepercent of indeterminacy is required: for a more organic, smooth,and especially accurate estimation.

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