Degree of Dependence and Independence of the (Sub)Components of Fuzzy Set and Neutrosophic Set

2006·0.39 MB·

by  Florentin Smarandache

Neutrosophic Sets and Systems, Vol. 11, 2016 95
University of New Mexico
Degree of Dependence and Independence
of the (Sub)Components of Fuzzy Set and Neutrosophic Set
Florentin Smarandache1
1 University of New Mexico, Mathematics & Science Department, 705 Gurley Ave., Gallup, NM 87301, USA, E-mail: [email protected]
Abstract. We have introduced for the first time the in our 2006 book’s fifth edition [1]. Now we extend it for
degree of dependence (and consequently the degree of the first time to the refined neutrosophic set considering
independence) between the components of the fuzzy set, the degree of dependence or independence of
and also between the components of the neutrosophic set subcomponets.
Keywords: neutrosophy, neutrosophic set, fuzzy set, degree of dependence of (sub)components, degree of independence of
(sub)components.
1 Refined Neutrosophic Set. For example, if 𝑦 and 𝑦 are 100% dependent, then
1 2
We start with the most general definition, that of a 0≤𝑦 +𝑦 ≤1, (7)
1 2
n-valued refined neutrosophic set 𝐴. An element 𝑥 from
while other variables 𝑦 ,…,𝑦 are 100% independent of
𝐴 belongs to the set in the following way: 3 𝑛
each other and also with respect to 𝑦 and 𝑦 , then
1 2
𝑥(𝑇 ,𝑇 ,…,𝑇 ; 𝐼 ,𝐼 ,…,𝐼 ; 𝐹 ,𝐹 ,…,𝐹)∈𝐴, (1)
1 2 𝑝 1 2 𝑟 1 2 𝑠 0≤𝑦_3+⋯+𝑦_𝑛≤𝑛−2, (8)
where 𝑝,𝑟,𝑠≥1 are integers, and 𝑝+𝑟+𝑠=𝑛≥3,
thus
where
0≤𝑦 +𝑦 +𝑦 +⋯+𝑦 ≤𝑛−1. (9)
𝑇 ,𝑇 ,…,𝑇 ; 𝐼 ,𝐼 ,…,𝐼 ; 𝐹 ,𝐹 ,…,𝐹 (2) 1 2 3 𝑛
1 2 𝑝 1 2 𝑟 1 2 𝑠
are respectively sub-membership degrees, sub-indeter- 3 Fuzzy Set.
minacy degrees, and sub-nonmembership degrees of
Let 𝑇 and 𝐹 be the membership and respectively the
element x with respect to the n-valued refined
nonmembership of an element 𝑥(𝑇,𝐹) with respect to a
neutrosophic set A.
fuzzy set 𝐴, where 𝑇,𝐹 are crisp numbers in [0,1].
Therefore, one has n (sub)components.
If 𝑇 and 𝐹 are 100% dependent of each other, then
Let’s consider all of them being crisp numbers in
one has as in classical fuzzy set theory
the interval [0,1].
0≤𝑇+𝐹 ≤1. (10)
2 General case.
But if 𝑇 and 𝐹 are 100% independent of each other
Now, in general, let’s consider n crisp-components (that we define now for the first time in the domain of
(variables): fuzzy setand logic), then
𝑦 ,𝑦 ,…,𝑦 ∈[0,1]. (3) 0≤𝑇+𝐹 ≤2. (11)
1 2 𝑛
If all of them are 100% independent two by two, We consider that the sum 𝑇+𝐹 =1 if the
then their sum: information about the components is complete, and
𝑇+𝐹 <1 if the information about the components is
0≤𝑦 +𝑦 + …+𝑦 ≤𝑛. (4)
1 2 𝑛 incomplete.
But if all of them are 100% dependent (totally Similarly, 𝑇+𝐹 =2 for complete information, and
interconnected), then 𝑇+𝐹 <2 for incomplete information.
For complete information on T and F, one has
0≤𝑦 +𝑦 + …+𝑦 ≤1. (5)
1 2 𝑛 𝑇+𝐹 ∈[1,2].
When some of them are partially dependent and
partially independent, then 4 Degree of Dependence and Degree
of Independence for two Components.
𝑦 +𝑦 + …+𝑦 ∈(1,𝑛). (6)
1 2 𝑛
Florentin Smarandache, Degree of Dependence and Independence of the (Sub)Components of Fuzzy Set and
Neutrosophic Set
96 Neutrosophic Sets and Systems, Vol. 11, 2016
In general (see [1], 2006, pp. 91-92), the sum of two information (sum > 1), or complete information (sum =
components x and y that vary in the unitary interval [0, 1).
1]is: If all three components T, I, F are dependent, then
similarly one leaves room for incomplete information
0 ≤ 𝑥+𝑦 ≤ 2 – 𝑑°(𝑥,𝑦), (12)
(sum < 1), or complete information (sum = 1).
where 𝑑°(𝑥,𝑦) is the degree of dependence between x The dependent components are tied together.
and y. Three sources that provide information on T, I, and
F respectively are independent if they do not
Therefore 2 – 𝑑°(𝑥,𝑦) is the degree of
communicate with each other and do not influence each
independence between x and y.
other.
Of course, 𝑑°(𝑥,𝑦) ∈ [0,1], and it is zero when x
Therefore, max{T+I+F} is in between 1 (when the
and y are 100% independent, and 1 when x and y are
degree of independence is zero) and 3 (when the degree
100% dependent.
of independence is 1).
In general, if T and F are 𝑑% dependent [and
consequently (100−𝑑)% independent], then
7 Examples of Neutrosophic Set with
0≤𝑇+𝐹 ≤2−𝑑/100. (13) Partially Dependent and Partially
Independent Components.
5 Example of Fuzzy Set with Partially
Dependent and Partially Independent The max{T+I+F} may also get any value in (1, 3).
a) For example, suppose that T and F are 30%
Components.
dependent and 70% independent (hence T + F ≤ 2-0.3 =
As an example, if 𝑇 and 𝐹 are 75% (= 0.75) 1.7), while I and F are 60% dependent and 40%
dependent, then independent (hence I + F ≤ 2-0.6 = 1.4). Then max{T +
I + F} = 2.4 and occurs for T = 1, I = 0.7, F = 0.7.
0≤𝑇+𝐹 ≤2−0.75=1.25. (14)
b) Second example: suppose T and I are 100%
dependent, but I and F are 100% independent. Therefore
6 Neutrosophic Set
T + I ≤ 1 and I + F ≤ 2, then T + I + F ≤ 2.
Neutrosophic set is a general framework for
unification of many existing sets, such as fuzzy set 8 More on Refined Neutrosophic Set
(especially intuitionistic fuzzy set), paraconsistent set,
intuitionistic set, etc. The main idea of NS is to The Refined Neutrosophic Set [4], introduced for
characterize each value statement in a 3D-Neutrosophic
the first time in 2013. In this set the neutrosophic
Space, where each dimension of the space represents
component (T) is split into the subcomponents (T , T ,
respectively the membership/truth (T), the 1 2
…, T ) which represent types of truths (or sub-truths),
nonmembership/falsehood (F), and the indeterminacy p
with respect to membership/nonmembership (I) of the the neutrosophic component (I) is split into the
statement under consideration, where T, I, F are subcomponents (I , I , …, I) which represents types of
1 2 r
standard or non-standard real subsets of ]-0, 1+[ with not indeterminacies (or sub-indeterminacies), and the
necessarily any connection between them.
neutrosophic components (F) is split into the
For software engineering proposals the classical
subcomponents (F , F , …, F) which represent types of
unit interval [0, 1] is used. 1 2 s
falsehoods (or sub-falsehoods), such that p, r, s are
For single valued neutrosophic set, the sum of the
integers ≥ 1 and p + r + s = n ≥ 4. (18)
components (T+I+F) is (see [1], p. 91):
When n = 3, one gets the non-refined neutrosophic
0 ≤ T+I+F ≤ 3, (15)
set. All T, I , and F subcomponents are subsets of [0,
j k l
when all three components are independent; 1].
Let’s consider the case of refined single-valued
0 ≤ T+I+F ≤ 2, (16)
neutrosophic set, i.e. when all n subcomponents are
when two components are dependent, while the third crisp numbers in [0, 1].
one is independent from them; Let the sum of all subcomponents be:
p r s
0 ≤ T+I+F ≤ 1, (17) S T I F (19)
j k l
when all three components are dependent. 1 1 1
When all subcomponents are independent two by
When three or two of the components T, I, F are
two, then
independent, one leaves room for incomplete
0 ≤ S ≤ n. (20)
information (sum < 1), paraconsistent and contradictory
Florentin Smarandache, Degree of Dependence and Independence of the (Sub)Components of Fuzzy Set and
Neutrosophic Set
Neutrosophic Sets and Systems, Vol. 11, 2016 97
If m subcomponents are 100% dependent, 2 ≤ m ≤ Conclusion.
n, no matter if they are among T, I , F or mixed, then
j k l We have introduced for the first time the degree of
0 ≤ S ≤ n – m +1 (21)
dependence/independence between the components of
and one has S = n – m + 1 when the information is
fuzzy set and neutrosophic set. We have given easy
complete, while S < n – m + 1 when the information is
examples about the range of the sum of components,
incomplete.
and how to represent the degrees of dependence and
independence of the components. Then we extended it
9 Examples of Refined Neutrosophic Set
to the refined neutrosophic set considering the degree of
with Partially Dependent and Partially
dependence or independence of subcomponets.
Independent Components.
References.
Suppose T is split into T , T , T , and I is not split,
1 2 3
while F is split into F1, F2. Hence one has: [1] Florentin Smarandache, Neutrosophy. Neutrosophic
{T , T , T ; I; F , F }. (22) Probability, Set, and Logic, ProQuest Information &
1 2 3 1 2
Therefore a total of 6 (sub)components. Learning, Ann Arbor, Michigan, USA, 105 p.,
a) If all 6 components are 100% independent two [2] 1998, 2000, 2003, 2005, 2006;
by two, then: http://fs.gallup.unm.edu/eBook-neutrosophics5.pdf
0 ≤ T + T + T + I + F +F ≤ 6 (23) (fifth edition).
1 2 3 1 2
b) Suppose the subcomponets T1, T2, and F1 are [3] Florentin Smarandache, Degree of Dependence and
100% dependent all together, while the others Independence of Neutrosophic Logic Components
are totally independent two by two and Applied in Physics, 2016 Annual Spring Meeting of
independent from T , T , F , therefore: the American Physical Society (APS) Ohio-Region
1 2 1
0 ≤ T + T + F ≤ 1 (24) Section, Dayton, Ohio, USA, 08-09 April 08 2016.
1 2 1
whence [4] Vasile Pătraşcu, Penta and Hexa Valued
0 ≤ T + T + T + I + F +F ≤ 6 – 3 + 1 = 4. (25) Representation of Neutrosophic Information,
1 2 3 1 2
One gets equality to 4 when the information is Technical Report TI.1.3.2016 10 March 2016, DOI:
complete, or strictly less than 4 when the information is 10.13140/RG.2.1.2667.1762.
incomplete. [5] Florentin Smarandache, Refined Literal Indeter-
c) Suppose in another case that T and I are 20% minacy and the Multiplication Law of Sub-Indeter-
1
dependent, or d°(T , I) = 20%, while the others minacies, Neutrosophic Sets and Systems, 58-63,
1
similarly totally independent two by two and Vol. 9, 2015.
independent from T1 and I, hence [6] Florentin Smarandache, n-Valued Refined
0 ≤ T + I ≤ 2 – 0.2 = 1.8 (26) Neutrosophic Logic and Its Applications in Physics,
1
whence Progress in Physics, 143-146, Vol. 4, 2013.
0 ≤ T + T + T + I + F +F ≤ 1.8 + 4 = 5.8, (27) http://fs.gallup.unm.edu/n-
1 2 3 1 2
since 0 ≤ T + T + F +F ≤ 4. (28) valuedNeutrosophicLogic-PiP.pdf
2 3 1 2
Similarly, to the right one has equality for complete
information, and strict inequality for incomplete Received: January 15, 2016. Accepted: March 25, 2016
information.
Florentin Smarandache, Degree of Dependence and Independence of the (Sub)Components of Fuzzy Set and
Neutrosophic Set