Neutrosophic Sets and Systems. Book series, vol. 1

2013·2.9 MB·English

by  Florentin Smarandache

Neutrosophic Sets and Systems
An International Book Series in Information Science and Engineering
Editor-in-Chief: Associate Editor:
Mumtaz Ali, Quaid-e-azam University Islamabad, Pakistan.
Prof. Florentin Smarandache Editorial Board:
Department of Mathematics and Science Dmitri Rabounski and Larissa Borissova, independent researchers.
University of New Mexico W. B. Vasantha Kandasamy, IIT, Chennai, Tamil Nadu, India.
705 Gurley Avenue Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. A.
Gallup, NM 87301, USA A. Salama, Faculty of Science, Port Said University, Egypt.
E-mail: [email protected] Yanhui Guo, School of Science, St. Thomas University, Miami, USA.
Home page: http://fs.gallup.unm.edu/NSS Francisco Gallego Lupiaňez, Universidad Complutense, Madrid, Spain.
Peide Liu, Shandong Universituy of Finance and Economics, China.
Pabitra Kumar Maji, Math Department, K. N. University, WB, India.
S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia.
Jun Ye, Shaoxing University, China.
Ştefan Vlăduţescu, University of Craiova, Romania.
Volume 1 2013
Contents
Florentin Smarandache. Neutrosophic Measure and Yanhui Guo, and Abdulkadir Şengür. A Novel Image
Neutrosophic Integral …………………………………. 3 Segmentation Algorithm Based on Neutrosophic Fil-
tering and Level Set …………………………………… 46
Jun Ye. Another Form of Correlation Coefficient
between Single Valued Neutrosophic Sets and Its Mul-
A. A. Salama. Neutrosophic Crisp Points & Neutro-
tiple Attribute Decision-Making Method …..…………. 8
sophic Crisp Ideals…………………………………... 50
Muhammad Shabir, Mumtaz Ali, Munazza Naz, and
Said Broumi, and Florentin Smarandache. Several
Florentin Smarandache. Soft Neutrosophic Group…….. 13
Similarity Measures of Neutrosophic Sets ……………. 54
Fu Yuhua. Neutrosophic Examples in Physics ………... 26
Pingping Chi, and Peide Liu. An Extended TOPSIS
A. A. Salama, and Florentin Smarandache. Filters via Method for the Multiple Attribute Decision Making
Neutrosophic Crisp Sets ………………………………. 34 Problems Based on Interval Neutrosophic Set ………... 63
Florentin Smarandache, and Ştefan Vlăduțescu.
Communication vs. Information, an Axiomatic Neutro-
sophic Solution ………………………………………... 38
EuropaNova ASBL
3E clos du Parnasse Brussels, 1000
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Copyright © Neutrosophic Sets and Systems
Neutrosophic Sets and Systems, Vol. 1, 2013
Neutrosophic Sets and Systems
An International Book Series in Information Science and
Engineering Copyright Notice
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Information for Authors and Subscribers
Neutrosophic Sets and Systems has been created for (T), a degree of indeterminacy (I), and a degree of falsity (F),
publications on advanced studies in neutrosophy, neutrosophic where T, I, F are standard or non-standard subsets of ]-0, 1+[.
set, neutrosophic logic, neutrosophic probability, neutrosophic Neutrosophic Probability is a generalization of the classical
statistics that started in 1995 and their applications in any field, probability and imprecise probability.
such as the neutrosophic structures developed in algebra, Neutrosophic Statistics is a generalization of the classical
geometry, topology, etc. statistics.
The submitted papers should be professional, in good What distinguishes the neutrosophics from other fields is the
English, containing a brief review of a problem and obtained <neutA>, which means neither <A> nor <antiA>.
results. <neutA>, which of course depends on <A>, can be
Neutrosophy is a new branch of philosophy that studies the indeterminacy, neutrality, tie game, unknown, contradiction,
origin, nature, and scope of neutralities, as well as their ignorance, imprecision, etc.
interactions with different ideational spectra.
This theory considers every notion or idea <A> together with All submissions should be designed in MS Word format using
its opposite or negation <antiA> and with their spectrum of our template file:
neutralities <neutA> in between them (i.e. notions or ideas http://fs.gallup.unm.edu/NSS/NSS-paper-template.doc.
supporting neither <A> nor <antiA>). The <neutA> and <antiA>
ideas together are referred to as <nonA>. A variety of scientific books in many languages can be
Neutrosophy is a generalization of Hegel's dialectics (the last one downloaded freely from the Digital Library of Science:
is based on <A> and <antiA> only). http://fs.gallup.unm.edu/eBooks-otherformats.htm.
According to this theory every idea <A> tends to be neutralized
and balanced by <antiA> and <nonA> ideas - as a state of To submit a paper, mail the file to the Editor-in-Chief. To order
equilibrium. printed issues, contact the Editor-in-Chief. This book is non-
In a classical way <A>, <neutA>, <antiA> are disjoint two commercial, academic edition. It is printed from private
by two. But, since in many cases the borders between notions are donations.
vague, imprecise, Sorites, it is possible that <A>, <neutA>,
<antiA> (and <nonA> of course) have common parts two by Information about the neutrosophics you get from the UNM
two, or even all three of them as well. website:
Neutrosophic Set and Neutrosophic Logic are generalizations http://fs.gallup.unm.edu/neutrosophy.htm.
of the fuzzy set and respectively fuzzy logic (especially of
intuitionistic fuzzy set and respectively intuitionistic fuzzy
logic). In neutrosophic logic a proposition has a degree of truth
Copyright © Neutrosophic Sets and Systems
Neutrosophic Sets and Systems, Vol. 1, 2013
Neutrosophic Measure and Neutrosophic Integral
Florentin Smarandache
University of New Mexico, Math & Science Division, 705 Gurley Ave., Gallup, NM 87301, USA, E-mail: [email protected]
Abstract. Since the world is full of indeterminacy, the of indeterminacies, depending on the problem we need to
neutrosophics found their place into contemporary solve. Indeterminacy is different from randomness.
research. We now introduce for the first time the notions Indeterminacy can be caused by physical space materials
of neutrosophic measure and neutrosophic integral. and type of construction, by items involved in the space,
Neutrosophic Science means development and or by other factors. Neutrosophic measure is a
applications of neutrosophic logic/set/measure/integral/ generalization of the classical measure for the case when
probability etc. and their applications in any field. It is the space contains some indeterminacy. Neutrosophic
possible to define the neutrosophic measure and Integral is defined on neutrosophic measure. Simple
consequently the neutrosophic integral and neutrosophic examples of neutrosophic integrals are given.
probability in many ways, because there are various types
Keywords: neutrosophy, neutrosophic measure, neutrosophic integral, indeterminacy, randomness, probability.
1 Introduction to Neutrosophic Measure for any A⊆ X and A∈Σ,
1.1 Introduction m(A) means measure of the determinate part of A;
Let <A> be an item. <A> can be a notion, an attribute, m(neutA) means measure of indeterminate part of A;
an idea, a proposition, a theorem, a theory, etc. and m(antiA) means measure of the determinate part of
And let <antiA> be the opposite of <A>; while antiA;
ν
<neutA> be neither <A> nor <antiA> but the neutral (or where is a function that satisfies the following two
indeterminacy, unknown) related to <A>. properties:
For example, if <A> = victory, then <antiA> = defeat, a) Null empty set:ν(Φ)=(0,0,0).
while <neutA> = tie game. σ
b) Countable additivity (or -additivity): For all
If <A> is the degree of truth value of a proposition, { }
countable collections A of disjoint neutrosophic
then <antiA> is the degree of falsehood of the proposition, n n∈L
Σ
while <neutA> is the degree of indeterminacy (i.e. neither sets in , one has:
trueA nlosro ,f aifl s<eA) o>f =th veo ptirnogp ofosirt iao nca. ndidate, <antiA> = voting νn∈LAn=n∈Lm(An), n∈Lm(neutAn),n∈Lm(antiAn)−(n−1)m(X)
against that candidate, while <neutA> = not voting at all, where X is the whole neutrosophic space,
or casting a blank vote, or casting a black vote. In the case and (2)
when <antiA> does not exist, we consider its measure be m(antiA )−(n−1)m(X)=m(X)−m(A )=m(∩antiA ).
null {m(antiA)=0}. And similarly when <neutA> does not n∈L n n∈L n n∈L n
exist, its measure is null { m(neutA) = 0}.
1.3 Neutrosophic Measure Space
1.2 Definition of Neutrosophic Measure
A neutrosophic measure space is a triplet (X,Σ,ν).
We introduce for the first time the scientific notion of
neutrosophic measure.
1.4 Normalized Neutrosophic Measure
Let X be a neutrosophic space, and Σ a
σ-neutrosophic algebra over X . A neutrosophic A neutrosophic measure is called normalized if
measure ν is defined by for neutrosophic set A∈Σ by 4ν(X)=(m(X),m(neutX),m(antiX))=(x1,x2,x3),
ν:Xν→(AR)3=, (m(A), m(neutA),m(antiA)), (1) 4 with x1+x2+axn3d= x1, ≥ 0,x ≥0,x ≥0. (3)
1 2 3
with antiA = the opposite of A, and neutA = the neutral Where, of course, X is the whole neutrosophic measure
(indeterminacy) neither A nor anti A (as defined above); space.
1.5 Finite Neutrosophic Measure Space
Florentin Smarandache, Neutrosophic Measure and Neutrosophic Integral
4 Neutrosophic Sets and Systems, Vol. 1, 2013
Let A⊂X. We say that ν(A)=(a ,a ,a ) is finite if all The neutrosophic measurable functions and their
1 2 3 neutrosophic measurable spaces form a neutrosophic
a , a , and a are finite real numbers.
1 A2 neutro3sophic measure space (X,Σ,ν) is called finite category, where the functions are arrows and the spaces
objects.
if ν(X)=(a,b,c) such that all a, b, and c are finite (rather We introduce the neutrosophic category theory, which
than infinite). means the study of the neutrosophic structures and of the
neutrosophic mappings that preserve these structures.
1.6 σ-Finite Neutrosophic Measure The classical category theory was introduced about
A neutrosophic measure is called σ-finite if X can be 1940 by Eilenberg and Mac Lane.
decomposed into a countable union of neutrosophically A neutrosophic category is formed by a class of
measurable sets of fine neutrosophic measure. neutrosophic objects X,Y,Z,... and a class of
Analogously, a set A in X is said to have a σ-finite neutrosophic morphisms (arrows) ν,ξ,ω,... such that:
neutrosophic measure if it is a countable union of sets with
( )
finite neutrosophic measure. a) If Hom X,Y represent the neutrosophic
1.7 Neutrosophic Axiom of Non-Negativity morphisms from X to Y , then Hom(X,Y)and
ν
We say that the neutrosophic measure satisfies the Hom(X',Y') are disjoint, except when X =X' and
axiom of non-negativity, if:
∀A∈Σ , Y =Y';
ν(A)=(a ,a ,a )≥0 if a ≥0,a ≥0, and a ≥0. (4) b) The composition of the neutrosophic morphisms
1 2 3 1 2 3 verify the axioms of
While a neutrosophic measure ν, that satisfies only i) Associativity: (ν ξ) ω=ν (ξ ω)
   
the null empty set and countable additivity axioms (hence
ii) Identity unit: for each neutrosophic object X
not the non-negativity axiom), takes on at most one of the
±∞ values. there exists a neutrosophic morphism denoted id , called
X
1.8 Measurable Neutrosophic Set and Measurable neutrosophic identity of X such that id ν=ν and
X
NeuTthreo smoepmhbiecr sS opfa Σce a re called measurable neutrosophic ξidX =ξ
sets, while (X,Σ) is called a measurable neutrosophic
space.
1.9 Neutrosophic Measurable Function
A function f :(X,Σ )→(Y,Σ ), mapping two
X Y
measurable neutrosophic spaces, is called neutrosophic
measurable function if ∀B∈Σ , f−1(B)∈Σ (the
Y X
inverse image of a neutrosophic Y -measurable set is a
neutrosophic X -measurable set). Fig. 2
1.10 Neutrosophic Probability Measure
ν 1.12 Properties of Neutrosophic Measure
As a particular case of neutrosophic measure is th
a) Monotonicity.
neutrosophic probability measure, i.e. a neutrosophic
measure that measures probable/possible propositions If A and A are neutrosophically measurable, with
1 2
−0≤ν(X)≤3+, (5) A ⊆ A , where
1 2
where X is the whole neutrosophic probability sample ν(A)=(m(A),m(neutA),m(antiA )),
space. 1 1 1 1
We use nonstandard numbers, such 1+ for example, to and ν(A )=(m(A ),m(neutA ),m(antiA )),
2 2 2 2
denominate the absolute measure (measure in all possible
then
worlds), and standard numbers such as 1 to denominate the m(A)≤m(A),m(neutA)≤m(neutA),m(antiA)≥m(antiA)
relative measure (measure in at least one world). Etc. 1 2 1 2 1 2
We denote the neutrosophic probability measure by (6)
NP for a closer connection with the classical probability Let ν(X)=(x1,x2,x3) and ν(Y)=(y1,y2,y3). We
P . say that ν(X)≤ν(Y), if x ≤ y ,x ≤ y , and x ≥ y .
1 1 2 2 3 3
1.11 Neutrosophic Category Theory
b) Additivity.
Florentin Smarandache, Neutrosophic Measure and Neutrosophic Integral
Neutrosophic Sets and Systems, Vol. 1, 2013 5
IfA1A2 =Φ, then ν(A1A2)=ν(A1)+ν(A2), ν(book)=(97,3,0) (13)
(7) where ν is the neutrosophic measure of the book
where we define number of pages.
(a ,b,c )+(a ,b ,c )=(a +a ,b +b ,a +b −m(X )) b) If a surface of 5 × 5 square meters has cracks of
1 1 1 2 2 2 1 2 1 2 3 3 0.1 × 0.2 square meters, then ν(surface)=(24.98,0.02,0),
(8)
where X is the whole neutrosophic space, and (14), where ν is the neutrosophic measure of the surface.
a +b −m(X)=m(X)−m(A)−m(B)=m(X)−a −a c) If a die has two erased faces then
=3m(a3ntiA∩antiB). 1 2 ν(die)=(4,2,0), (14)
(9) where ν is the neutrosophic measure of the die’s
number of correct faces.
1.13 Neutrosophic Measure Continuous from d) An approximate number N can be interpreted as
Below or Above
ν a neutrosophic measure N=d+i, where d is its
A neutrosophic measure is continuous from below
if, for A,A ,... neutrosophically measurable sets with determinate part, and i its indeterminate part. Its anti part
1 2
is considered 0.
An⊆An+1 for all n, the union of the sets An is For example if we don’t know exactly a quantity q,
neutrosophically measurable, and but only that it is between let’s say q∈[0.8,0.9], then
νn∞=1An=nlνi→m∞ν(An) (10) iqts= in0d.8e+teir,m winhaetree p 0a.r8t iis∈ t[h0e,0 d.1e]t.erminate part of q, and
And a neutrosophic measure is continuous from
We get a negative neutrosophic measure if we
above if for A,A ,... neutrosophically measurable sets,
1 2 approximate a quantity measured in an inverse direction on
with A ⊇ A for all n, and at least one A has finite the x-axis to an equivalent positive quantity.
n n+1 n For example, if r∈[−6,−4], then r=−6+i, where -6
neutrosophic measure, the intersection of the sets An and is the determinate part of r, and i∈[0,2] is its
neutrosophically measurable, and
indeterminate part. Its anti part is also 0.
ν∞ A =limν(A ). (11) e ) Let’s measure the truth-value of the proposition
n=1 n n→∞ n G = “through a point exterior to a line one can draw
only one parallel to the given line”.
1.14 Generalizations The proposition is incomplete, since it does not specify
Neutrosophic measure is a generalization of the fuzzy the type of geometrical space it belongs to. In an Euclidean
measure, because when m(neutA)=0 and m(antiA) is geometric space the proposition G is true; in a Riemannian
geometric space the proposition G is false (since there is
ignored, we get
no parallel passing through an exterior point to a given
ν(A)=(m(A),0,0)≡m(A) (12) line); in a Smarandache geometric space (constructed from
and the two fuzzy measure axioms are verified: mixed spaces, for example from a part of Euclidean
a) If A=Φ, then ν(A)=(0,0,0)≡0 subspace together with another part of Riemannian space)
the proposition G is indeterminate (true and false in the
b) If A⊆B, then ν(A)≤ν(B). same time).
The neutrosophic measure is practically a triple ν(G)=(1,1,1). (15)
classical measure: a classical measure of the determinate f) In general, not well determined objects, notions,
part of a neutrosophic object, a classical part of the ideas, etc. can become subject to the neutrosophic theory.
indeterminate part of the neutrosophic object, and another
classical measure of the determinate part of the opposite 2 Introduction to Neutrosophic Integral
neutrosophic object. Of course, if the indeterminate part
does not exist (its measure is zero) and the measure of the 2.1 Definition of Neutrosophic Integral
opposite object is ignored, the neutrosophic measure is Using the neutrosophic measure, we can define a
reduced to the classical measure. neutrosophic integral.
1.15 Examples The neutrosophic integral of a function f is written as:
Let’s see some examples of neutrosophic objects and  fdν (16)
neutrosophic measures. X
a) If a book of 100 sheets (covers included) has 3 where X is the a neutrosophic measure space,
missing sheets, then
Florentin Smarandache, Neutrosophic Measure and Neutrosophic Integral
6 Neutrosophic Sets and Systems, Vol. 1, 2013

and the integral is taken with respect to the determinant part “a ” and an indeterminate part ε, i.e.
ν 1
neutrosophic measure .
a = a +ε (23)
1
Indeterminacy related to integration can occur in
multiple ways: with respect to value of the function to be
where
integrated, or with respect to the lower or upper limit of
integration, or with respect to the space and its measure. ε∈[0,0.1]. (24)
2.2 First Example of Neutrosophic Integral:
Therefore
Indeterminacy Related to Function’s Values
Let fN: [a, b]  R (17) b b
where the neutrosophic function is defined as:  fdν=  f(x)dx−i (25)
X 1
fN (x) = g(x)+i(x) (18) a a
1
with g(x) the determinate part of fN(x), and i(x) the
indeterminate part of fN(x),where for all x in [a, b] one where the indeterminacy i1 belongs to the interval:
has: i(x)∈[0,h(x)],h(x)≥0. (19)
a1+0.1
i ∈[0,  f(x)dx]. (26)
1
a1
Or, in a different way:
b b
 fdν=  f(x)dx+i (27)
X 2
a a1+0.1
where similarly the indeterminacy i belongs to the
2
interval:
Therefore the values of the function fN(x) are i ∈[0,a1+0.1f(x)dx]. (28)
2
approximate, i.e. f (x)∈[g(x),g(x)+h(x)]. (20) a1
N
References
Similarly, the neutrosophic integral is an approxi-
[1] A. B. Author, C. D. Author, and E. F. Author. Journal,
mation:
volume (year), page.
b b b
 f (x)dν=g(x)dx+i(x)dx (21) [1] Florentin Smarandache, An Introduction to Neutrosophic
N Probability Applied in Quantum Physics, SAO/NASA ADS
a a a
Physics Abstract Service,
1.10 Second Example of Neutrosophic Integral: http://adsabs.harvard.edu/abs/2009APS..APR.E1078S
Indeterminacy Related to the Lower Limit [2] Florentin Smarandache, An Introduction to Neutrosophic
Suppose we need to integrate the function Probability Applied in Quantum Physics, Bulletin of the
American Physical Society, 2009 APS April Meeting,
f: X R 22)
Volume 54, Number 4, Saturday–Tuesday, May 2–5, 2009;
Denver, Colorado, USA,
on the interval [a, b] from X, but we are unsure about the http://meetings.aps.org/Meeting/APR09/Event/102001
lower limit a. Let’s suppose that the lower limit “a” has a [3] Florentin Smarandache, An Introduction to Neutrosophic
Probability Applied in Quantum Physics, Bulletin of Pure
and Applied Sciences, Physics, 13-25, Vol. 22D, No. 1,
2003.
[4] Florentin Smarandache, "Neutrosophy. / Neutrosophic
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Rehoboth, USA, 105 p., 1998.
[5] Florentin Smarandache, A Unifying Field in Logics:
Neutrosophic Logic, Neutrosophic Set, Neutrosophic
Probability and Statistics, 1999, 2000, 2003, 2005.
[6] Florentin Smarandache, Neutrosophic Physics as a new field
of research, Bulletin of the American Physical Society, APS
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March Meeting 2012, Volume 57, Number 1, Monday– [9] Florentin Smarandache, Neutrosophic Degree of
Friday, February 27–March 2 2012; Boston, Massachusetts, Paradoxicity of a Scientific Statement, Bulletin of the
http://meetings.aps.org/Meeting/MAR12/Event/160317 American Physical Society 2011 Annual Meeting of the
[7] Florentin Smarandache, V. Christianto, A Neutrosophic Four Corners Section of the APS Volume 56, Number 11.
Logic View to Schrodinger's Cat Paradox, Bulletin of the Friday–Saturday, October 21–22, 2011; Tuscon, Arizona,
American Physical Society 2008 Joint Fall Meeting of the http://meetings.aps.org/link/BAPS.2011.4CF.F1.37
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http://meetings.aps.org/link/BAPS.2011.SHOCK.F1.167
Received: November 12, 2013. Accepted: December 5, 2013
Florentin Smarandache, Neutrosophic Measure and Neutrosophic Integral
Neutrosophic Sets and Systems, Vol. 1, 2013 8
Another Form of Correlation Coefficient between Single
Valued Neutrosophic Sets and Its Multiple Attribute Decision-
Making Method
Jun Ye
Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing, Zhejiang 312000, P.R. China
E-mail: [email protected]
Abstract. A single valued neutrosophic set (SVNS), attribute decision making method using the correlation
which is the subclass of a neutrosophic set, can be coefficient of SVNSs under single valued neutrosophic
considered as a powerful tool to express the environment. Through the weighted correlation
indeterminate and inconsistent information in the process coefficient between each alternative and the ideal
of decision making. Then, correlation is one of the most alternative, the ranking order of all alternatives can be
broadly applied indices in many fields and also an determined and the best alternative can be easily
important measure in data analysis and classification, identified as well. Finally, two illustrative examples are
pattern recognition, decision making and so on. employed to illustrate the actual applications of the
Therefore, we propose another form of correlation proposed decision-making approach.
coefficient between SVNSs and establish a multiple
Keywords: Correlation coefficient; Single valued neutrosophic set; Decision making.
1 Introduction the correlation coefficient of SVNSs based on the
extension of the correlation coefficient of intuitionistic
To handle the indeterminate information and
fuzzy sets and proved that the cosine similarity measure of
inconsistent information which exist commonly in real
SVNSs is a special case of the correlation coefficient of
situations, Smarandache [1] firstly presented a
SVNSs, and then applied it to single valued neutrosophic
neutrosophic set from philosophical point of view, which is
multicriteria decision-making problems. Hanafy et al. [6]
a powerful general formal framework and generalized the
presented the centroid-based correlation coefficient of
concept of the classic set, fuzzy set, interval-valued fuzzy
neutrosophic sets and investigated its properties.
set, intuitionistic fuzzy set, interval-valued intuitionistic
Recently , S. Broumi and F. Smarandache [8] Correlation
fuzzy set, paraconsistent set, dialetheist set, paradoxist set,
coefficient of interval neutrosophic set and investigated its
and tautological set [1, 2]. In the neutrosophic set, a truth-
properties.
membership, an indeterminacy-membership, and a falsity-
In this paper, we propose another form of correlation
membership are represented independently. Its functions
coefficient between SVNSs and investigate its properties.
T (x), I (x) and F (x) are real standard or nonstandard
A A A Then, a multiple attribute decision-making method using
subsets of ]−0, 1+[, i.e., T (x): X → ]−0, 1+[, I (x): X → ]−0,
A A the correlation coefficient of SVNSs is established under
1+[, and F (x): X → ]−0, 1+[. Obviously, it will be difficult
A single valued neutrosophic environment. To do so, the rest
to apply in real scientific and engineering areas. Therefore,
of the paper is organized as follows. Section 2 briefly
Wang et al. [3] proposed the concept of a single valued
describes some concepts of SVNSs. In Section 3, we
neutrosophic set (SVNS), which is the subclass of a
develop another form of correlation coefficient between
neutrosophic set, and provided the set-theoretic operators
SVNSs and investigate its properties. Section 4 establishes
and various properties of SVNSs. Thus, SVNSs can be
a multiple attribute decision-making method using the
applied in real scientific and engineering fields and give us
correlation coefficient of SVNSs under single valued
an additional possibility to represent uncertainty,
neutrosophic environment. In Section 5, two illustrative
imprecise, incomplete, and inconsistent information which
examples are presented to demonstrate the applications of
exist in real world. However, the correlation coefficient is
the developed approach. Section 6 contains a conclusion
one of the most frequently used tools in engineering
and future research.
applications. Therefore, Hanafy et al. [4] introduced the
correlation of neutrosophic data. Then, Ye [5] presented
Jun Ye, Another Form of Correlation Coefficient between Single Valued Neutrosophic Sets
and Its Multiple Attribute Decision-Making Method
Neutrosophic Sets and Systems, Vol. 1, 2013 9
2 Some concepts of SVNSs Theorem 1. The correlation coefficient N(A, B) satisfies
the following properties:
Smarandache [1] firstly presented the concept of a
(1) N(A, B) = N(B, A);
neutrosophic set from philosophical point of view and gave
(2) 0 ≤ N(A, B) ≤ 1;
the following definition of a neutrosophic set.
(3) N(A, B) = 1, if A = B.
Definition 1 [1]. Let X be a space of points (objects), with
Proof. (1) It is straightforward.
a generic element in X denoted by x. A neutrosophic set A
(2) The inequality N(A, B) ≥ 0 is obvious. Thus, we
in X is characterized by a truth-membership function TA(x), only prove the inequality N(A, B) ≤ 1.
an indeterminacy-membership function I (x), and a falsity-
A
membership function F (x). The functions T (x), I (x) and N(A,B)=
A A A
FA(x) are real standard or nonstandard subsets of ]−0, 1+[, n [ ]
i.e., T (x): X → ]−0, 1+[, I (x): X → ]−0, 1+[, and F (x): X  T (x)⋅T (x)+I (x)⋅I (x)+F (x)⋅F (x)
A A A A i B i A i B i A i B i
→ ]−0, 1+[. There is no restriction on the sum of T (x), i=1
IA(x) and FA(x), so −0 ≤ sup TA(x) + sup IA(x) + sup FA(Ax) ≤ =TA(x1)⋅TB(x1)+TA(x2)⋅TB(x2)+...+TA(xn)⋅TB(xn)
3+. +I (x)⋅I (x)+I (x )⋅I (x )+...+I (x )⋅I (x )
A 1 B 1 A 2 B 2 A n B n
Obviously, it is difficult to apply in practical problems. +F (x)⋅F (x)+F (x )⋅F (x )+...+F (x )⋅F (x )
Therefore, Wang et al. [3] introduced the concept of a A 1 B 1 A 2 B 2 A n B n
SVNS, which is an instance of a neutrosophic set, to apply According to the Cauchy–Schwarz inequality:
in real scientific and engineering applications. In the
following, we introduce the definition of a SVNS [3]. (x ⋅y + x ⋅y +...+ x ⋅y )2
1 1 2 2 n n ,
Definition 2 [3]. Let X be a space of points (objects) with ≤(x2 + x2 +...+ x2)⋅(y2 + y2 +...+ y2)
generic elements in X denoted by x. A SVNS A in X is 1 2 n 1 2 n
characterized by a truth-membership function TA(x), an where (x , x , …, x ) ∈ Rn and (y , y , …, y ) ∈ Rn, we can
1 2 n 1 2 n
indeterminacy-membership function I (x), and a falsity-
A obtain

membership function F (x) for each point x in X, T (x),
A A
I (x), F (x) ∈ [0, 1]. Thus, A SVNS A can be expressed as [ ] n [ ]
A A (N(A,B)2 ≤T2(x)+I2(x )+F2(x )
{ } A i A n A n
A= x,T (x),I (x),F (x) |x∈X . i=1
A A A n [ ]
Then, the sum of TA(x), IA(x) and FA(x) satisfies the ⋅TB2(xi)++IB2(xi)+FB2(xi)
condition 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3. i=1
Definition 3 [3]. The complement of a SVNS A is denoted =N(A,A)⋅N(B,B).
by Ac and is defined as
[ ] [ ]
{ } Thus, N(A,B)≤ N(A,A)1/2⋅ N(B,B)1/2.
Ac = x,F (x),1−I (x),T (x) |x∈X . { }
A A A Then, N(A,B)≤max N(A,A),N(B,B) .
Definition 4 [3]. A SVNS A is contained in the other Therefore, N(A, B) ≤ 1.
SVNS B, A ⊆ B if and only if T (x) ≤ T (x), I (x) ≥ I (x),
A B A B (3) If A = B, there are T (x) = T (x), I (x) = I (x), and
A i B i A i B i
and FA(x) ≥ FB(x) for every x in X. F (x) = F (x) for any x ∈ X and i = 1, 2, …, n. Thus, there
A i B i i
Definition 5 [3]. Two SVNSs A and B are equal, written as
are N(A, B) = 1.
A = B, if and only if A ⊆ B and B ⊆ A.
In practical applications, the differences of importance
are considered in the elements in the universe. Therefore,
3 Correlation coefficient of SVNSs
we need to take the weights of the elements x (i = 1, 2,…,
i
Motivated by another correlation coefficient between n)into account. Let wi be the weight for each element xi (i
intuitionistic fuzzy sets [7], this section proposes another = 1, 2,…, n), wi ∈ [0, 1], and n w =1, then we have the
form of correlation coefficient between SVNSs as a i=1 i
following weighted correlation coefficient between the
generalization of the correlation coefficient of intuitionistic
SVNSs A and B:
fuzzy sets [7].
Definition 6. For any two SVNSs A and B in the universe W(A,B)=
of discourse X = {x1, x2,…, xn}, another form of n w[T (x)⋅T (x)+I (x)⋅I (x)+F (x)⋅F (x)]
correlation coefficient between two SVNSs A and B is i=1 i A i B i A i B i A i B i
defined by maxi=n1wi[TA2(xi)+IA2(xi)+FA2(xi)],i=n1wi[TB2(xi)+IB2(xi)+FB2(xi)]
N(A,B)= { C(A,B) } (1) (2)
maxC(A,A),C(B,B)
n [T (x)⋅T (x)+I (x)⋅I (x)+F (x)⋅F (x)] If w = (1/n, 1/n,…, 1/n)T, then Eq. (2) reduce to Eq.
= i=1 A i B i A i B i A i B i (1). Note that W(A, B) also satisfy the three properties of
maxi=n1[TA2(xi)+IA2(xi)+FA2(xi)],i=n1[TB2(xi)+IB2(xi)+FB2(xi)] Theorem 1.
Jun Ye, Another Form of Correlation Coefficient between Single Valued Neutrosophic Sets
and Its Multiple Attribute Decision-Making Method