Inconsistency and indeterminacy are different situations. For example, a proposition sometimes can be true or false, this is inconsistency. Sometimes we can not get accuracy results about a problem, si it is indeterminacy. So in an intuitionistic fuzzy set hesitancy specify uncetainity, and a neutrosophic set model inconsistency. Namely, in neutrosophic set results are accurate but it is inconsistent, in intuitionistic fuzz some results have incomplete information.
I hope that this explanation is useful
In neutrosophic sets all three measures (Truth, Falsehood, indeterminacy) are independent, how one effects another in decision making their sum < 1 we have incomplete information (we do not know all information), while if sum = 1 we have complete information as in intuitionistic fuzzy set.
Revolution of Neutrosophic Set from Classical Set as follows:
The classical set, I = ∅, inf T = sup T = 0 or 1, inf F = sup F = 0 or 1.
The fuzzy set, I = ∅, inf T = sup T ∈ [0, 1], inf F = sup F ∈ [0, 1] and sup T + sup F = 1.
The interval-valued fuzzy set, I = ∅, inf T, sup T, inf F, sup F ∈ [0, 1], sup T + inf F = 1 and inf T + sup F = 1.
The intuitionistic fuzzy set, I = ∅, inf T = sup T ∈ [0, 1], inf F = sup F ∈[0, 1] and sup T + sup F ≤ 1.
The interval-valued intuitionistic fuzzy set, I = ∅, inf T, sup T, inf F, sup F ∈ [0, 1] and sup T + sup F ≤ 1.
An interval neutrosophic set (INS), An INS 𝐴 in 𝑋 is characterized by a truth-membership function 𝑇𝐴(𝑥), an indeterminacy-membership function 𝐼𝐴(𝑥), and a falsity-membership function 𝐹𝐴(𝑥). For each point 𝑥 in 𝑋, we have that
𝑇𝐴(𝑥) = [inf 𝑇𝐴(𝑥), sup 𝑇𝐴(𝑥)]
𝐼𝐴(𝑥) = [inf 𝐼𝐴(𝑥), sup 𝐼𝐴(𝑥)]
𝐹𝐴(𝑥) = [inf 𝐹𝐴(𝑥), sup 𝐹𝐴(𝑥)]
𝑇𝐴(𝑥), 𝐼𝐴(𝑥), 𝐹𝐴(𝑥) ⊆ [0, 1] and
0 ≤ sup 𝑇𝐴(𝑥) + sup 𝐼𝐴(𝑥) + sup 𝐹𝐴(𝑥) ≤ 3, 𝑥 ∈ 𝑋.
We only consider the sub unitary interval of [0, 1]. It is the subclass of an NS.
Therefore, all INSs are clearly NSs.
A neutrosophic set, A NS 𝐴 in 𝑋 is characterized by a truth-membership function 𝑇𝐴(𝑥), an indeterminacy-membership function 𝐼𝐴(𝑥), and a falsity membership function 𝐹𝐴(𝑥).
𝑇𝐴(𝑥), 𝐼𝐴(𝑥), and 𝐹𝐴(𝑥) are real standard or nonstandard subsets of ]0−, 1+[
That is,
𝑇𝐴(𝑥):𝑋 →]0−, 1+[
𝐼𝐴(𝑥): 𝑋 →]0−, 1+[
𝐹𝐴(𝑥): 𝑋 →]0−, 1+[
There is no restriction on the sum of 𝑇𝐴(𝑥), 𝐼𝐴(𝑥), and 𝐹𝐴(𝑥)
So 0−≤ sup 𝑇𝐴(𝑥) + sup 𝐼𝐴(𝑥) + sup 𝐹𝐴(𝑥) ≤ 3+.
Reference paper for details:
https://arxiv.org/abs/math/0409113
http://dx.doi.org/10.1155/2014/645953
Logically, indeterminacy function (I) is a complement of the member and non-member functions (T, F). That is, indeterminacy function implicitly exists in intuitionistic fuzzy sets and explicitly exists in neutrosophic sets. It should noted that indeterminacy function takes a value (>0) when available information of phenomenon or problem are incomplete. With respect to the independent and dependent of each other, they must dependent of each other, how do they are independent?! For example, it is not logic that someone expects that A will win with a chance of 40% and will fail with a chance of 70%. But the chances of winning and loss can be respectively: 40% and 50% or 30% and 70% or...(
I Have a question. Is Neutrosophic Sets and Neutrosophic Fuzzy Sets are same?
Dear Dr. Pratihar,
Perhaps you meant "Evolution of Neutrosophic Set from Classical Set". Also, although you have not mentioned i think, the symbols used are I = Indeterminancy, T = Truth and F = Falsity.
Then onwards, you have mentioned that inf T = sup T and inf F = sup F. (Is it correct?). Perhaps it should be inf T, inf F, sup T, inf T in [0, 1]. And so on...
However, your effort if nice.
Thank you.
In Neutrosophic sets truthness and falsity are independent whereas in intuitionistic sets it is dependent.
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