# Neutrosophic vague set theory.

Acknowledgement

We would like to acknowledge the financial support received from Shaqra University. With our sincere thanks and appreciation to Professor Smarandache for his support and his comments.

1 Introduction

Many scientists wish to find appropriate solutions to some mathematical problems that cannot be solved by traditional methods. These problems lie in the fact that traditional methods cannot solve the problems of uncertainly in economy, engineering, medicine, problems of decision-making and others. There have been a great amount of research and applications in the literature concerning some special tools like probability theory, fuzzy set theory [13], rough set theory [19], vague set theory [18], intuitionistic fuzzy set theory [10, 12] and interval mathematics [11, 14].

Since Zadeh published his classical paper almost fifty years ago, fuzzy set theory has received more and more attention from researchers in a wide range of scientific areas, especially in the past few years.

The difference between a binary set and a fuzzy set is that in a "normal" set every element is either a member or a non-member of the set; it either has to be A or not A.

In a fuzzy set, an element can be a member of a set to some degree and at the same time a non-member of the same set to some degree. In classical set theory, the membership of elements in a set is assessed in binary terms: according to a bivalent condition, an element either belongs or does not belong to the set.

By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the closed unit interval [0, 1].

Fuzzy sets generalise classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the later only take values 0 or 1. Therefore, a fuzzy set A in an universe of discourse X is a function A:X [right arrow] [0,1], and usually this function is referred to as the membership function and denoted by [[mu].sub.A(x)].

The theory of vague sets was first proposed by Gau and Buehrer [18] as an extension of fuzzy set theory and vague sets are regarded as a special case of context-dependent fuzzy sets.

A vague set is defined by a truth-membership function tv and a false-membership function [f.sub.v], where [t.sub.v](x) is a lower bound on the grade of membership of x derived from the evidence for x, and [f.sub.v](x) is a lower bound on the negation of x derived from the evidence against x. The values of [t.sub.v](x) and [f.sub.v](x) are both defined on the closed interval [0, l] with each point in a basic set X, where [t.sub.v](x) + [f.sub.v](x) [less than or equal to] 1.

In 1995, Smarandache talked for the first time about neutrosophy, and in 1999 and 2005 [4, 6] defined the neutrosophic set theory, one of the most important new mathematical tools for handling problems involving imprecise, indeterminacy, and inconsistent data.

In this paper, we define the concept of a neutrosophic vague set as a combination of neutrosophic set and vague set. We also define and study the operations and properties of neutrosophic vague set and give examples.

2 Preliminaries

In this section, we recall some basic notions in vague set theory and neutrosophic set theory. Gau and Buehrer have introduced the following definitions concerning its operations, which will be useful to understand the subsequent discussion.

Definition 2.1 ([18]). Let x be a vague value, x = [[t.sub.x], 1 - [f.sub.x]], where [t.sub.x] [member of] [0,1], [f.sub.x] [member of] [0,1], and 0 [less than or equal to] [t.sub.x] [less than or equal to] l - [f.sub.x] [less than or equal to] l. If [t.sub.x] = l and [f.sub.x] = 0 (i.e., x = [1,1]). then x is called a unit vague value. If [t.sub.x] = 0 and [f.sub.x] = i(i.e., x = [0,0]), then x is called a zero vague value.

Definition 2.2 ([18]). Let x and y be two vague values, where x = [[t.sub.x], 1 - [f.sub.x]] and y = [[t.sub.y], 1 - [f.sub.y]]. If [t.sub.x] = [t.sub.y] and [f.sub.x] = [f.sub.y], then vague values x and y are called equal (i.e. [[t.sub.x], l - [f.sub.x]] = [[t.sub.y], l - [f.sub.y]]).

Definition 2.3 ([18]). Let A be a vague set of the universe U. If [for all][u.sub.i] [member of] U, [t.sub.A]([u.sub.t]) = 1 and [f.sub.A]([u.sub.t]) = 0, then A is called a unit vague set, where 1 [less than or equal to] i [less than or equal to] n. If [for all][u.sub.i] [member of] U, [t.sub.A] ([u.sub.i]) = 0 and [f.sub.A]([u.sub.i]) = 1, then A is called a zero vague set, where 1 [less than or equal to] i [less than or equal to] n.

Definition 2.4 ([18]). The complement of a vague set A is denoted by [A.sup.c] and is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 2.5 ([18]). Let A and B be two vague sets of the universe U. If [for all][u.sub.i] [member of] U, [[t.sub.A] ([u.sub.i]), l - [f.sub.A]([u.sub.i])] = [[t.sub.B] ([u.sub.i]), l - [f.sub.B]([u.sub.i])]. then the vague set A and B are called equal, where 1 [less than or equal to] i [less than or equal to] n.

Definition 2.6 ([18]). Let A and B be two vague sets of the universe U. If [for all][u.sub.i] [member of] U, [[t.sub.A] ([u.sub.i]) [less than or equal to] [[t.sub.B] ([u.sub.i]) and 1 - [[f.sub.A] ([u.sub.i]) [less than or equal to] [[f.sub.B] ([u.sub.i]), then the vague set A are included by B, denoted by A [subset or equal to] B, where 1 [less than or equal to] i [less than or equal to] n.

Definition 2.7 ([18]). The union of two vague sets A and B is a vague set C, written as C = A [union] B, whose truth-membership and false-membership functions are related to those of A and B by

[t.sub.c] = max{[t.sub.A],[t.sub.B]), 1 - [f.sub.C] = max(l - [f.sub.A], l - [f.sub.B]) = l - min([f.sub.A], [f.sub.B]).

Definition 2.8 ([18]). The intersection of two vague sets A and B is a vague set C, written as C = A[intersection]B, whose truth-membership and false-membership functions are related to those of A and B by

[t.sub.C] = min([t.sub.A],[t.sub.B]), [f.sub.c] = min{1 - [f.sub.A], 1 - [f.sub.B]) = 1 - max([f.sub.A], [f.sub.B]).

In the following, we recall some definitions related to neutrosophic set given by Smarandache. Smarandache defined neutrosophic set in the following way:

Definition 2.9 [6] A neutrosophic set A on the universe of discourse X is defined as

A = {< x, [T.sub.A] (x), [I.sub.A] (x), [F.sub.A] (x) >, x [member of] X]

where T,I,F: [[X.sub.[right arrow]].sup.-] [0.1.sup.+][[and.sup.-] 0 [less than or equal to] [T.sub.A] (x) + [I.sub.A] (x) + [F.sub.A] (x) [less than or equal to] [3.sup.+].

Smarandache explained his concept as it follows: "For example, neutrosophic logic is a generalization of the fuzzy logic. In neutrosophic logic a proposition is T [equivalent to] true, I [equivalent to] indeterminate, and F [equivalent to] false. For example, let's analyze the following proposition: Pakistan will win against India in the next soccer game. This proposition can be (0.6,0.3,0.1), which means that there is a possibility of 60% [equivalent to] that Pakistan wins, 30% [equivalent to] that Pakistan has a tie game, and 10% [equivalent to] that Pakistan looses in the next game vs. India."

Now we give a brief overview of concepts of neutrosophic set defined in [8, 5, 17]. Let [S.sub.1] and [S.sub.2] be two real standard or non-standard subsets, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 2.10 (Containment) A neutrosophic set A is contained in the other neutrosophic set B, A [subset or equal to] B, if and only if

inf [T.sub.A] (x) [less than or equal to] inf [T.sub.B] (x), sup [T.sub.A] (x) [less than or equal to] sup [T.sub.B] (x),

inf [I.sub.A] (x) [greater than or equal to] inf [I.sub.B] (x), sup [I.sub.A] (x) [greater than or equal to] sup [I.sub.B] (x),

inf [F.sub.A] (x) [greater than or equal to] inf [F.sub.B] (x), sup [F.sub.A] (x) [greater than or equal to] sup [F.sub.B] (x), for all x [member of] X.

Definition 2.11 The complement of a neutrosophic set A is denoted by [bar.A] and is defined by

[T.sub.[bar.A]] (x) = {[1.sup.+]} [bar.[direct sum]] [T.sub.[??]] (x),

[I.sub.[bar.A]] (x) = {[1.sup.+]} [bar.[direct sum]] [I.sub.[??]] (x),

[F.sub.[bar.A]] (x) = {[1.sup.+]} [bar.[direct sum]] [F.sub.[??]] (x), for all x [member of] X.

Definition 2.12 (Intersection) The intersection of two neutrosophic sets A and B is a neutrosophic set C, written as C = A [intersection] B, whose truth-membership, indeterminacy-membership and falsity-membership functions are related to those of A and B by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 2.11 (Union) The union of two neutrosophic sets A and B is a neutrosophic set C written as C = A [union] B, whose truth-membership, indeterminacy-membership and falsity-membership functions are related to those of A and B by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3 Neutrosophic Vague Set

A vague set over U is characterized by a truth-membership function [t.sub.v] and a false-membership function [f.sub.v], [t.sub.v] : U [right arrow] [0,1] and [f.sub.v] : U [right arrow] [0,1] respectively where [t.sub.v] ([u.sub.i]) is a lower bound on the grade of membership of [u.sub.i] which is derived from the evidence for [u.sub.i], [f.sub.v] ([u.sub.i]) is a lower bound on the negation of [u.sub.i] derived from the evidence against [u.sub.i] and [t.sub.v] ([u.sub.i]) + [f.sub.v] ([u.sub.i]) [less than or equal to] 1. The grade of membership of [u.sub.i] in the vague set is bounded to a subinterval [[t.sub.v] ([u.sub.i]), 1 - [f.sub.v] ([u.sub.i])] of [0,1]. The vague value [[t.sub.v] ([u.sub.i]), 1 - [f.sub.v] ([u.sub.i])] indicates that the exact grade of membership [[mu].sub.v] ([u.sub.i]) of [u.sub.i] maybe unknown, but it is bounded by [t.sub.v] ([u.sub.i]) [less than or equal to] [f.sub.v] ([u.sub.i]) where [t.sub.v] ([u.sub.i]) [less than or equal to]1 . Let U be a space of points (objects), with a generic element in U denoted by u. A neutrosophic sets (N-sets) A in U is characterized by a truth-membership function [T.sub.A], an indeterminacy-membership function [I.sub.A] and a falsity-membership function [F.sub.A]. [T.sub.A] (u); [I.sub.A] (u) and [F.sub.A] (u) are real standard or nonstandard subsets of [0, 1]. It can be written as:

A = {< u, ([T.sub.A] (u), [I.sub.A] (u), [F.sub.A] (u)) >:u [member of] U, [T.sub.A] (u), [I.sub.A] (u), [F.sub.A] (u)[member of][0,1]}.

There is no restriction on the sum of [T.sub.A] (u); [I.sub.A] (u) and [F.sub.A] (u), so:

0 [less than or equal to] sup [T.sub.A] (u) + sup [I.sub.A] (u) + sup [F.sub.A] (u) [less than or equal to] 3.

By using the above information and by adding the restriction of vague set to neutrosophic set, we define the concept of neutrosophic vague set as it follows.

Definition 3.1 A neutrosophic vague set [A.sub.NV] (NVS in short) on the universe of discourse X written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whose truth-membership, indeterminacy-membership and falsity-membership functions is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[T.sup.+] = 1 - [F.sup.-], [F.sup.+] = 1 - [T.sup.-], and

[sup.-]0 [less than or equal to] [T.sup.-] + [I.sup.-] + [F.sup.-] [less than or equal to] [2.sup.+],

when X is continuous, a NVS [A.sub.NV] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When X is discrete, a NVS [A.sub.NV] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In neutrosophic logic, a proposition is T [equivalent to] true, I [equivalent to] indeterminate, and F [equivalent to] false such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Also, vague logic is a generalization of the fuzzy logic where a proposition is T [equivalent to] true and F [equivalent to] false, such that: [t.sub.v]([u.sub.i]) + [f.sub.v] ([u.sub.i]) [less than or equal to] 1, he exact grade of membership [[mu].sub.v] ([u.sub.i]) of [u.sub.i] maybe unknown, but it is bounded by

[t.sub.v] ([u.sub.i]) [less than or equal to] [[mu].sub.v]([u.sub.i]) [less than or equal to] [f.sub.v]([u.sub.i]).

For example, let's analyze the Smarandache's proposition using our new concept: Pakistan will win against India in the next soccer game. This proposition can be as it follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which means that there is possibility of 60% to 90% [equivalent to] that Pakistan wins, 30% to 40% [equivalent to] that Pakistan has a tie game, and 40% to 60% [equivalent to] that Pakistan looses in the next game vs. India.

Example 3.1 Let u = {[u.sub.1], [u.sub.2], [u.sub.3]} be a set of universe we define the NVS [A.sub.NV] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Definition 3.2 Let [[PSI].sub.NV] be a NVS of the universe U where [for all][u.sub.i] [member of] U,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then [[PSI].sub.NV] is called a unit NVS, where 1 [less than or equal to] i [less than or equal to] n.

Let [[PHI].sub.NV] be a NVS of the universe U where [for all][u.sub.i] [member of] U,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then [[PHI].sub.NV] is called a zero NVS, where 1 [less than or equal to] i [less than or equal to] n.

Definition 3.3 The complement of a NVS [A.sub.NV] is denoted by [A.sup.c] and is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Example 3.2 Considering Example 3.1, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Definition 3.5 Let [A.sub.NV] and [B.sub.NV] be two NVSs of the universe U. If [for all][u.sub.i] [member of] U,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then the NVS [A.sub.NV] and [B.sub.NV] are called equal, where 1 [less than or equal to] i [less than or equal to] n.

Definition 3.6 Let [A.sub.NV] and [B.sub.NV] be two NVSs of the universe U. If [for all][u.sub.i] [member of] U,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then the NVS [A.sub.NV] are included by [B.sub.NV], denoted by [A.sub.NV] [subset or equal to][B.sub.NV], where 1 [less than or equal to] i [less than or equal to] n.

Definition 3.7 The union of two NVSs [A.sub.NV] and [B.sub.NV] is a NVS [C.sub.NV], written as [C.sub.NV] = [A.sub.NV] [union] [B.sub.NV], whose truth-membership, indeterminacy-membership and false-membership functions are related to those of [A.sub.NV] and [B.sub.NV] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 3.8 The intersection of two NVSs [A.sub.NV] and [B.sub.NV] is a NVS [H.sub.NV], written as [H.sub.NV] = [A.sub.NV] [intersection] [B.sub.NV], whose truth-membership, indeterminacy-membership and false-membership functions are related to those of [A.sub.NV] and [B.sub.NV] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 3.3 Let u = {[u.sub.1], [u.sub.2], [u.sub.3]} be a set of universe and let [A.sub.NV] and [B.sub.NV] define as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then we have [c.sub.NV] = [A.sub.NV] [union] [B.sub.NV] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover, we have [H.sub.NV] = [A.sub.NV] [intersection] [B.sub.NV] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.1 Let P be the power set of all NVS defined in the universe X. Then <P; [[union].sub.NV], [[intersection].sub.NV]> is a distributive lattice.

Proof Let A, B, C be the arbitrary NVSs defined on X. It is easy to verify that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4 Conclusion

In this paper, we have defined and studied the concept of a neutrosophic vague set, as well as its properties, and its operations, giving some examples.

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Shawkat Alkhazaleh (1)

(1) Department of Mathematics, Faculty of Science and Art Shaqra University, Saudi Arabia shmk79@gmail.com
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