# Neutrosophic crisp probability theory & decision making process.

1 Introduction

Neutrosophy has laid the foundation for a whole family of new mathematical theories generalizing both their classical and fuzzy counterparts [1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 42] such as the neutrosophic set theory. The fundamental concepts of neutrosophic set, introduced by Smarandache in [37, 38, 39, 40], and Salama et al. in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], provides a natural foundation for treating mathematically the neutrosophic phenomena which pervasively exist in our real world and for building new branches of neutrosophic mathematics. In this paper, we introduce and study the probability of neutrosophic crisp sets. After giving the fundamental definitions and operations, we obtain several properties, and discuss the relationship between neutrosophic crisp sets and others.

2 Terminology

We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in [37, 38, 39, 40], and Salama et al. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Smarandache introduced the neutrosophic components T, I, F--which represent the membership, indeterminacy and non-membership values respectively, which are included into the nonstandard unit interval.

2.1 Example 2.1 [37, 39]

Let us consider a neutrosophic set, a collection of possible locations (positions) of particle x and let A and B two neutrosophic sets.

One can say, by language abuse, that any particle x neutrosophically belongs to any set, due to the percentages of truth/indeterminacy/falsity involved, which varies between [sup.-]0 and 1[sup.+]

For example: x (0.5, 0.2, 0.3) belongs to A (which means a probability of 50% that the particle x is in A, a probability of 30% that x is not in A, and the rest is undecidable); or y (0, 0, 1) belongs to A (which normally means y is not for sure in A); or z (0,1, 0) belongs to A (which means one does know absolutely nothing about z affiliation with A).

More general, x((0.2-0.3), (0.4-0.45) [union] [0.50-0.51, {0.2,0.24, 0.28}) belongs to the set, which means: with a probability in between 20-30%, the particle x is in a position of A (one cannot find an exact approximation because of various sources used); with a probability of 20% or 24% or 28%, x is not in A; the indeterminacy related to the appurtenance of x to A is in between 40-45% or between 50-51% (limits included).

The subsets representing the appurtenance, indeterminacy, and falsity may overlap, and, in this case, n-sup = 30% + 51% + 28% > 100.

Definition 2.1 [14, 15, 21]

A neutrosophic crisp set (NCS for short) A = <[A.sub.1], [A.sub.2], [A.sub.3]> can be identified to an ordered triple <[A.sub.1], [A.sub.2], [A.sub.3]> which are subsets on x, and every crisp set in X is obviously a NCS having the form <[A.sub.1], [A.sub.2], [A.sub.3]>.

Definition 2.2 [21]

The object having the form A = <[A.sub.1], [A.sub.2], [A.sub.3]> is called

1) Neutrosophic Crisp Set with Type I if it satisfies [A.sub.1] [intersection] [A.sub.2] = [phi], [A.sub.1] [intersection] [A.sub.3] = [phi] and [A.sub.2] [intersection] [A.sub.3] = [phi] (NCS-Type I for short).

2) Neutrosophic Crisp Set with Type II if it satisfies [A.sub.1] [intersection] [A.sub.2] = [phi], [A.sub.1] [intersection] [A.sub.3] = [phi] and [A.sub.2] [intersection] [A.sub.3] = [phi] and [A.sub.1] [union] [A.sub.2] [union] [A.sub.3] = X (NCS-Type II for short).

3) Neutrosophic Crisp Set with Type III if it satisfies [A.sub.1] [intersection] [A.sub.2] [intersection] [A.sub.3] = [phi] and [A.sub.1] [union] [A.sub.2] [union] [A.sub.3] = X (NCS-Type III for short).

Definition 2.3

1. Neutrosophic Set [7]: Let X be a non-empty fixed set. A neutrosophic set (NS for short) A is an object having the form A = <[[mu].sub.A](x), [[sigma].sub.A](x), [[v.sub.A](x)]>, where [[mu].sub.A](x), [[sigma].sub.A](x) and [v.sub.A](x)] represent the degree of membership function (namely [[mu].sub.A](x)),the degree of indeterminacy (namely [[sigma].sub.A](x)), and the degree of non-membership (namely [v.sub.A](x)) respectively of each element x [member of] X to the set A where

[0.sup.-][less than or equal to][[mu].sub.A](x),[[sigma].sub.A](x), [v.sub.A](x) [less than or equal to][1.sup.+]

and

[0.sup.-][less than or equal to][[mu].sub.A](x),[[sigma].sub.A](x), [v.sub.A](x) [less than or equal to][3.sup.+]

2. Neutrosophic Intuitionistic Set of Type 1 [8]: Let X be a non-empty fixed set. A neutrosophic intuitionistic set of type 1 (NISI for short) set A is an object having the form A = <[[mu].sub.A](x), [[sigma].sub.A](x), [[v.sub.A](x)]>, where [[mu].sub.A](x), [[simgma].sub.A](x) and [[v.sub.A](x)] which represent the degree of membership function (namely [[mu].sub.A](x)), the degree of indeterminacy (namely [[sigma].sub.A](x)), and the degree of non-membership (namely[[v.sub.A](x)) respectively of each element x [member of] X to the set A where

[0.sup.-][less than or equal to][[mu].sub.A](x),[[sigma].sub.A](x), [v.sub.A](x) [less than or equal to] [1.sup.+]

and the functions satisfy the condition

[[mu].sub.A](x)[LAMBDA][[sigma].sub.A](x)[conjunction][v.sub.A](x) [less than or equal to] 0.5

and

[0.sup.-][less than or equal to][[mu].sub.A](x),[[sigma].sub.A](x), [v.sub.A](x) [less than or equal to] [3.sup.+]

3. Neutrosophic Intuitionistic Set of Type 1 [41]: Let X be a non-empty fixed set. A neutrosophic intuitionistic set of Type 1 A (NIS2 for short) is an object having the form A = <[[mu].sub.A](x), [[sigma].sub.A](x), [[v.sub.A](x)]> where [[mu].sub.A](x), [[sigma].sub.A](x) and [v.sub.A](x) [[v.sub.A](x)] which represent the degree of membership function (namely [[mu].sub.A](x)), the degree of indeterminacy (namely [[sigma].sub.A](x)), and the degree of non-membership (namely[[v.sub.A](x)) respectively of each element x [member of] X to the set A where

0.5 [less than or equal to] [[mu].sub.A](x), [[sigma].sub.A](x), [v.sub.A](x)

and the functions satisfy the condition

[[mu].sub.A](x) [and] [[sigma].sub.A](x) [less than or equal to]0.5, [[mu].sub.A](x) [and] [v.sub.A](x)] [less than or equal to]0.5, [[sigma].sub.A](x) [and] [[v.sub.A](x)] [less than or equal to]0.5,

and

[sub.-]0 [less than or equal to] [[mu].sub.A](x) + [[sigma].sub.A](x) + [[v.sub.A](x)] [less than or equal to] 2[sup.+].

A neutrosophic crisp with three types the object A = <[A.sub.1], [A.sub.2], [A.sub.3]> can be identified to an ordered triple <[A.sub.1], [A.sub.2], [A.sub.3]> which are subsets on X, and every crisp set in X is obviously a NCS having the form <[A.sub.1], [A.sub.2], [A.sub.3]>. Every neutrosophic set A = <[[mu].sub.A](x), [[sigma].sub.A](x), [[v.sub.A](x)]> on x is obviously a NS having the form <[[mu].sub.A](x), [[sigma].sub.A](x), [[v.sub.A](x)]>.

Salama et al in [14, 15, 21] constructed the tools for developed neutrosophic crisp set and introduced the NCS [[phi].sub.N], [X.sub.N] in X

Remark 2.1

The neutrosophic intuitionistic set is a neutrosophic set, but the neutrosophic set is not a neutrosophic intuitionistic set in general. Neutrosophic crisp sets with three types are neutrosophic crisp set.

3 The Probability of Neutrosophic Crisp Sets

If an experiment produces indeterminacy, that is called a neutrosophic experiment. Collecting all results, including the indeterminacy, we get the neutrosophic sample space (or the neutrosophic probability space) of the experiment. The neutrosophic power set of the neutrosophic sample space is formed by all different collections (that may or may not include the indeterminacy) of possible results. These collections are called neutrosophic events.

In classical experimental, the probability is

(number of times event A occurs/V total number of trials)

Similarly, Smarandache in [16, 17, 18] introduced the Neutrosophic Experimental Probability, which is:

(number of times event A occurs/total number of trials, number of times indeterminacy occurs/total number of trials number of times event A does not occur/total number of trials)

Probability of NCS is a generalization of the classical probability in which the chance that an event A = <[A.sub.1], [A.sub.2], [A.sub.3])> to occur is:

P([A.sub.1]) true, P([A.sub.2]) indeterminate, P([A.sub.3]) false,

on a sample space X, or NP(A) = <{P([A.sub.1]),P([A.sub.2]),P([A.sub.3])>.

A subspace of the universal set, endowed with a neutrosophic probability defined for each of its subsets, forms a probability neutrosophic crisp space.

Definition 3.1

Let X be a non- empty set and Abe any type of neutrosophic crisp set on a space X, then the neutrosophic probability is a mapping NP: X [right arrow] [[0,1].sup.3], NP(A) = <{P([A.sub.1]),P([A.sub.2]),P([A.sub.3])>, that is the probability of a neutrosophic crisp set that has the properly that--

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 3.1

1. In case if A = <[A.sub.1], [A.sub.2], [A.sub.3])> NCS, then [sub.-]0[less than or equal to] P([A.sub.1]) + P([A.sub.2]) + P([A.sub.3]) [less than or equal to][3.sup.+].

2. In case if A = <[A.sub.1], [A.sub.2], [A.sub.3])> is NCS-Type I, then 0[less than or equal to] P([A.sub.1]) + P([A.sub.2]) + P([A.sub.3]) [less than or equal to]2.

3. The Probability of NCS-Type II is a neutrosophic crisp set where [sub.-]0[less than or equal to] P([A.sub.1]) + P([A.sub.2]) + P([A.sub.3]) [less than or equal to][2.sup.+].

4. The Probability of NCS-Type III is a neutrosophic crisp set where [sub.-]0[less than or equal to] P([A.sub.1]) + P([A.sub.2]) + P([A.sub.3]) [less than or equal to][3.sup.+].

Probability Axioms of NCS Axioms

1. The Probability of neutrosophic crisp set and NCS-Type III A on X NP(A) = <(P([A.sub.1]),P([A.sub.2]),P([A.sub.3])> where P([A.sub.1]) [less than or equal to]0, P([A.sub.2]) [less than or equal to]0, P([A.sub.3]) [less than or equal to]0, or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. The probability of neutrosophic crisp set and NCS-Type IIIs A on X NP(A) = <{P([A.sub.1]),P([A.sub.2]),P([A.sub.3])> where [sub.-]0[less than or equal to] P([A.sub.1]) + P([A.sub.2]) + P([A.sub.3]) [less than or equal to][3.sup.+].

3. Bounding the probability of neutrosophic crisp set and NCS-Type III NP(A) = <{P([A.sub.1]),P([A.sub.2]),P([A.sub.3])> where 1[less than or equal to] P([A.sub.1]) + P([A.sub.2]) + P([A.sub.3]) [less than or equal to]0.

4. Addition law for any two neutrosophic crisp sets or NCS-Type III

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]])

if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]])

Since our main purpose is to construct the tools for developing probability of neutrosophic crisp sets, we must introduce the following--

1. Probability of neutrosophic crisp empty set with three types [NP{<pN) for short) may be defined as four types:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. Probability of neutrosophic crisp universal and NCS-Type III universal sets (NP([X.sub.n]) for short) may be defined as four types--

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 3.2

NP([X.sub.N]) = [1.sub.N], NP([[phi].sub.N]) = [0.sub.N], where [1.sub.N],[O.sub.N] are in Definition 2.1 [6], or equals any type for [1.sub.N].

Definition 3.2 (Monotonicity)

Let x be a non-empty set, and NCSS a and b in the form A = <[A.sub.1], [A.sub.2], [A.sub.3])>, B = <[B.sub.1], [B.sub.2], [B.sub.3])> with

NP(A) = <{P([A.sub.1]),P([A.sub.2]),P([A.sub.3])>, NP(B) = <P([B.sub.1]),P([B.sub.2]),P([B.sub.3])>

then we may consider two possible definitions for subsets (A [subset or equal to] B)--

Type 1:

NP(A) [less than or equal to] NP(A) [??] P([A.sub.1]) [less than or equal to] P([B.sub.1]), P([A.sub.2]) [less than or equal to] P([B.sub.2) and P([A.sub.3]) [greater than or equal to] P([B.sub.3])

or Type 2:

NP(A) [less than or equal to] NP(A) [??] P([A.sub.1]) [less than or equal to] P([B.sub.1]), P([A.sub.2]) [greater than or equal to] P([B.sub.2) and P([A.sub.3]) [greater than or equal to] P([B.sub.3])

Definition 3.3

Let X be a non-empty set, and NCSs a and b in the form A = <[A.sub.1], [A.sub.2], [A.sub.3])>, B = <[B.sub.1], [B.sub.2], [B.sub.3])> be NCSs.

Then--

1. NP(A[intersection]B) may be defined two types as--

Type 1:

NP(A [intersection] B) = {P([A.sub.1] [intersection] [B.sub.1]), P([A.sub.2] [intersection] [B.sub.2]), P([A.sub.3] [union] [B.sub.3])} or

Type 2:

NP(A [intersection] B) = <P([A.sub.1] [intersection] [B.sub.1]), P([A.sub.2] [union] [B.sub.2]), P([A.sub.3] [union] [B.sub.3])>

2. NP(A [union] B) may be defined two types as:

Type 1:

NP(A [union] B) = <P([A.sub.1] [union [B.sub.1]), P([A.sub.2] [intersection] [B.sub.2]), P([A.sub.3] [intersection] [B.sub.3])>

or Type 1:

NP(A [union] B) = {P([A.sub.1] [union] [B.sub.1]), P([A.sub.2] [union] [B.sub.2]), P([A.sub.3] [intersection] [B.sub.3])}

3. NP([A.sup.c]) may be defined by three types:

Type 1:

NP([A.sup.c]) = <(P([A.sup.c.sub.1], (P([A.sup.c.sub.2], (P([A.sup.c.sub.3]> = <(1-[A.sub.1]), (1-[A.sub.2]), (1-[A.sub.3])>

or Type 2:

NP([A.sup.c]) = <(P([A.sub.3], (P([A.sup.C.sub.2], (P([A.sub.1])>

or Type 3:

NPI[A.sup.c]) = <(P([A.sub.3], (P([A.sub.2], (P([A.sub.1])>

Proposition 3.1

Let A and B in the A = <[A.sub.1], [A.sub.2], [A.sub.3])>, B = <[B.sub.1], [B.sub.2], [B.sub.3])>) be NCSs on a nonempty set X.

Then--

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 3.1

Let A and B in the form A = <[A.sub.1], [A.sub.2], [A.sub.3])>, B = <[B.sub.1], [B.sub.2], [B.sub.3])> are NCSs on a nonempty set x and p,[p.sub.N] are NCSs.

Then

NP(p) = <1/n(X), 1/n(X), 1/n(X)>; NP([p.sub.N]) = <0, 1/n(X), 1, 1/n(X)>.

Example 3.1

1. Let X = {a, b, c, d} and A, B are two neutrosophic crisp events on X defined by A = <{a}, {b,c}, {c,d}>, B = <{a,b}, {a,c}, {c}>, p = ({a},{c},{d}) then see that NP(A) = (0.25, 0.5, 0.5), NP(B) = (0.5, 0.5, 0.25), NP(p) = (0.25, 0.25, 0.25), one can compute all probabilities from definitions.

2. If A = <{[phi]}, {b,c}, {[phi]}> and B = <{[phi]}, {d}, {[phi]}>are neutrosophic crisp sets on X.

Then--

A [intersection] B = <{[phi]}, {[phi]}, {[phi]}> and NP(A [intersection] B) = (0, 0, 0) = [0.sub.N], A [intersection] B = <{[phi]}, {b, c, d,}, {[phi]}> and NP(A [intersection] B) = (0, 0.75, 0) [not equal to] [0.sub.N ].

Example 3.2

Let X = {a, b, c, d, e, f},

A = ({a, b, c, d], {e}, {f}), D = <{a, b} {e, c} {f, d}> be a NCS-Type l,

B = ({a, b, c}, {d}, {e}) be a NCT-Type I but not NCS-Type II, III,

C = ({a, b}, {c, d}, {e, f, a}) be a NCS-Type III, but not NCS-Type I, II,

E = ({a, b, c, d, e}, {c, d}, {e, a}),

F = ({a, b, c, d, e}, [PHI], {e, f, a, d, c, b}).

We can compute the probabilities for NCSs by the following:

NP(A) = <4/6, 1/6, 1/6>,

NP(B) = <2/6, 2/6, 2/6>,

NP(C) = <3/6, 1/6, 1/6>,

NP(D) = <2/6, 2/6, 3/6>,

NP(E) = <4/6, 2/6, 3/6>,

NP(F) = <5/6, 0, 6/6>,

Remark 3.3

The probabilities of a neutrosophic crisp set are neutrosophic sets.

Example 3.3

Let X = {a, b, c, d}, A = <{a, b}, {c}, {d}>, B = <{a},{c},{d, b}> are NCS-Type I on X and [U.sub.1] = <{a,b}, {c,d}, {a,d}>, [U.sub.2] = <{a, b, c},{c},{d})>are NCS-Type III on X; then we can find the following operations--

1. Union, intersection, complement, difference and its probabilities.

a) Type 1: A [intersection] B = <{a}, {c}, {d,b}>,NP (A [intersection] B) = <0.25, 0.25, 0.5}> and Type 1,3: A [intersection] B = <{a}, {c}, {d,b}>,NP (A [intersection] B) = <0.25, 0.25, 0.5}>.

2. NP(A-B) may be equals.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. Probabilities for events.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4. Probabilities for Products. The product of two events given by--

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 3.4

The following diagram represents the relation between neutrosophic crisp concepts and neutrosphic sets:

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A. A. Salama (1), Florentin Smarandache (2)

(1) Department of Math. and Computer Science, Faculty of Sciences, Port Said University, Egypt drsalama44@gmail.com

(2) Math & Science Department, University of New Mexico, Gallup, New Mexico, USA smarand@unm.edu
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