n-Valued Refined Neutrosophic Crisp Sets 

Ahmed B. AL-Nafee^1*, Said Broumi^2,3  and Luay A. Al-Swidi⁴

^1*Ministry of Education Open Educational College, Department of Mathematics, Babylon, Iraq.

"Ahm_math_88@yahoo.com"

²Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, B.P 7955, Morocco.

" broumisaid78@gmail.com

³Regional Center for the Professions of Education and Training,Casablanca-Settat, Morocco

⁴Department of Mathematics, University of Babylon, Iraq, pure.leal.abd@uobabylon.edu.iq.

"pure.leal.abd@uobabylon.edu.iq"

(* Correspondence: Ahm_math_88@yahoo.com )

 Abstract

The main purpose of this manuscript is to expand the notion of neutrosophic crisp set (NCS) by presenting the notion of n-valued refined neutrosophic crisp set with some illustration examples. We also establish some of its set-theoretical operations.

1. Introduction

  Smarandache presented in [1, 2] the concept of "neutrosophic set" which is a ^'generalization of the concept of "fuzzy set" and the concept"intuitionistic fuzzy" set"to handle with uncertainty and imprecision by incurporating degrees of non-membership and indeterminacy as independent components. Salama and _'Smarandache^' presented in [3, 4] the concept of"neutrosophic crisp set"(NCS) which is a ^'generalization of the concept of crisp set and the concept of "intuitionistic set" [5]. Thereafter, many researchers have applied the notion of NCSs in topology, image processing and decision-making problems (for example see [6, 7, 8, 9, 10], [11], [12-21]).

  Rajashi et al introduced in [22] the notion of (QSVNS) that involves "truth", "falsity", "unknown" and "contradiction" depending on four-valued logics [23,24]. Shawkat introduced in [25] the notion of "n-values refined neutrosophic soft set" which is a ^'generalization of the notion of "neutrosophic soft set". Thereafter, Shawkat and Ayman used this concept to solve decision-making problems [26]. Rama and Surapati introduced in [27] the notion of "pentapartitioned neutrosophic set" by splitting indeterminacy to independent components namely "contradiction", "ignorance, and "unknown"-membership based on [24]. 

  In [24], Smarandache split truth into many types of truths, indeterminacy into several types of indeterminacies_'and falsehood into several types of falsities_'and proposed "n-symbol-valued refined neutrosophic logic" (NSVRNL). In this manuscript, we use the NSVRNL and propose n-valued refined neutrosophic crisp set. We also establish some of its set-theoretical operations. The proposed set is generalization of existent set NCS.  

2. Preliminaries

  In this section, we recall some basic concepts relevant to the upcoming sections."For more details on the concepts presented in the section (return to the refs, [4], [24])^'.

I. Neutrosophic Crisp Set (NCS) [4].

  Let  be a non-empty fixed set. The form  is called (NCS). Hear A, B and C, represent the set of  "memberships","indeterminacies" and "non-memberships" respectively^'of elements^'of  to , where  are subsets of . 

II. n-Valued_'Refined^'Neutrosophic_'Logic [24].  

  The neutrosophic_'logic_''value of a given proposition has_'the values, A =(truth), B =(indeterminacy), and C = (falsehood). Smarandache_'have defined_'two types  n-valued_'logic, symbolic_'and numerical.

1 - The n-symbol-valued_'refined^'neutrosophic_'logic (NSVRNL).     

  In general: C can be split into many types of falsities C₁; C₂;… C_s, B into several types of indeterminacies_': B₁; B₂;… B_r, and A into several types of truths: A₁; A₂;… A_P, where_, all p,r,s _'are integers_', and n = p+r+s. All subcomponents A_i; B_j; C_k are symbols_'for all i {1,2,…p}, for all j {1,2,.., r}, and for all k {1,2,.., s}.

2 - The n- numerical-valued_'refined^'neutrosophic_'logic (NNVRNL).

  In same_'way, but all_'subcomponents_' A_i; B_j; C_k are not_'symbols, but subset of [0,1], for_'all i {1,2,..,p}, for all j {1,2,..,r} and for all k {1,2,.., s}. In this manuscript, we use the NSVRNL and propose n-valued refined neutrosophic crisp set as follows:

3. n-Valued Refined Neutrosophic Crisp Set ( -set).

  In 'this section, we define the notion of -set and give some illustration examples.

3.1. Definition 

  Let  be a nonempty fixed set. The form  = < A₁,..., A_p; B₁,..., B_r; C₁,..., C_s > is called an n-valued refined neutrosophic crisp set (for short, -set ),where A₁, ..., A_p; B₁, ..., B_r and C₁, ..., C_s are subsets of . Here, A₁,..., A_p; B₁,..., B_r and C₁,..., C_s, are called membership sets; indeterminacy sets and non-membership sets, respectively of -set , where all p, r, s are integers, and n= p+ r+s. The set of all -sets over  will be denoted by . The followingspecial cases can be obtained from the above definition.

Case (I): Quadripartitioned neutrosophic crisp set or ( -set).

3.2. Example 

  Let  = {h_1, h_1, h_3, h₄, h_5,} be a nonempty fixed set. Let the indeterminacy B be refined (split) as B₁ = unknown and B₂ = contradiction. A₁, B₁, B₂and C₁ are subsets of ,

n = 4 = 1+ 2+1. Then, we can write the -set  as follows: = < {h₂}; {h₅}, {h_3, h₄}; {h₁} >.

Case (II): Pentapartitioned neutrosophic crisp set or ( -set).

3.3. Example 

  Let  = {h_1, h_1, h_3, h₄, h_5,} be a nonempty fixed set. Let the indeterminacy B be refined (split) as B₁ = unknown, B₂ = ignorance and B₃ = contradiction. A₁, B₁, B₂, B₃ and C₁ are subsets of ,

n = 5 = 1 + 3 + 1. Then, we can write the -set  as follows:  = < {h₁}; {h₂}, {h₄}, {h₅}; {h₃, h₁} >.

Case (III): Hexapartitioned neutrosophic crisp set or ( -set).

3.4. Example 

  Let  = {h_1, h_1, h_3, h₄, h₅} be a nonempty fixed set. Let the truth A be refined (split) as A₁= absolute truth and A₂= relative truth, the indeterminacy B be refined (split) as B₁ = absolute indeterminacy and B₂ = relative indeterminacy and the falsity C be refined (split) as C₁ = absolute falsity and C₂ = relative falsity. A₁, A₂, B₁, B₂, C₁ and C₂ are subsets of , n = 6 = 2 + 2 + 2. Then, we can write the -set  as follows:  

                           = < {h₁}, {h₃}; {h₄}, {_h5}; {h₂}, {h_5, h₃} >.

3.5. Definition

  Let  = < A₁,..., A_p; B₁,..., B_r; C₁,..., C_s >   . If A_i B_j = empty, A_i C_k = empty and B_j C_k = empty, where i  {1,2,,,,p}, j  {1,2,,,,r} and k  {1,2,,,,s}, then the -set  is called star -set (for short, -set). 

3.6. Definition

  Let  = < A₁,..., A_p; B₁,..., B_r; C₁,..., C_s >   . If the union of all subcomponents, A₁,..., A_p; B₁,..., B_r; C₁,..., C_s equals  and A_i B_j = empty, A_i C_k = empty and B_j C_k = empty and, where i  {1,2,,,,p}, j  {1,2,,,,r} and k  {1,2,,,,s}, then the -set  is called an 2star- -set (for short, -set). 

3.7. Definition

  Let  = < A₁,..., A_p; B₁,..., B_r; C₁,..., C_s >   . If the union of all subcomponents, A₁,..., A_p; B₁,..., B_r; C₁,..., C_s equals  and A_i B_j C_k = empty, where i {1,2,,,,p}, j  {1,2,,,,r} and k  {1,2,,,,s}, then the -set  is called an 3star -set (for short, -set). 

The set of all -sets, -sets and -sets over  will be denoted by  ,  and , respectively.

4. Operations of n-Valued Refined Neutrosophic Crisp Set.

  Since our goal is to build the tools to develop n-valued refined neutrosophic crisp set."we will organize the existing definitions into three types in each type these operations will be consistent and functional". As follows: 

(( Operations of -set, Type I)) 

4.1. Definition

  Let  = < A₁,..., A_p; B₁,..., B_r; C₁,..., C_s > and W = < X₁,..., X_p; Y₁,..., Y_r; Z₁,..., Z_s >   . Then,

§       The -empty set, denoted by  is defined as follows: 

                                    = < ₁,..., _p; ₁,..., _r; ₁,..., _s >. 

§       The - absolute set, denoted by  is defined as follows: 

                                    = < ₁,..., _p; ₁,..., _r; ₁,..., _s >.

§       The inclusion between  and W, denoted by    W is defined as follows:

     W = < A₁  X₁,...,A_P  X_P; Y₁  B₁,...,Y_r  B_r; Z₁  C₁,...,Z_s  C_s >.

§       The union between  and W, denoted by    W is defined as follows:

     W = < A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; C₁  Z₁,...,C_s  Z_s >.

§       The intersection between  and W, denoted by Q  W is defined as follows:

     W = < A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; C₁  Z₁,...,C_s  Z_s >.

§       The complement of , denoted by ^C is defined as follows:

                                    ^C = < C₁,..., C_s; ₁,..., _r; A₁,..., A_p >.

§        and W are equal, denoted by    W if    W and W   . 

 (( Operations of -set, Type II)) 

4.2. Definition

  Let  = < A₁,..., A_p; B₁,..., B_r; C₁,..., C_s > and W = < X₁,..., X_p; Y₁,..., Y_r; Z₁,..., Z_s >   . Then,

§       The -empty set, denoted by  is defined as follows: 

                                    = < ₁,..., _p; ₁,..., _r; ₁,..., _s >. 

§       The - absolute set, denoted by  is defined as follows: 

                                    = < ₁,..., _p; ₁,..., _r; ₁,..., _s >. 

§       The inclusion between  and W, denoted by    W is defined as follows:

     W = < A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; Z₁  C₁,...,Z_s  C_s >.

§       The union between  and W, denoted by    W is defined as follows:

     W = < A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; C₁  Z₁,...,C_s  Z_s >.

§       The intersection between  and W, denoted by    W is defined as follows:

     W = < A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; C₁  Z₁,...,C_s  Z_s >.

§       The complement of , denoted by ^C is defined as follows:

                                    ^C = < ₁,..., _p; ₁,..., _r; ₁,..., _s >.

§        and W are equal, denoted by    W if    W and W   .  

(( Operations of -set, Type III)) 

4.3. Definition

  Let  = < A₁,..., A_p; B₁,..., B_r; C₁,..., C_s > and W = < X₁,..., X_p; Y₁,..., Y_r; Z₁,..., Z_s >   . Then,

§       The -empty set, denoted by  is defined as follows: 

                                    = < ₁,..., _p; ₁,..., _r; ₁,..., _s>. 

§       The - absolute set, denoted by  is defined as follows: 

                                    = < ₁,..., _p; ₁,..., _r; ₁,..., _s >. 

§       The inclusion between  and W, denoted by    W is defined as follows:

     W = < A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; C₁  Z₁,...,C_s  Z_s >.

§       The union between  and W, denoted by    W is defined as follows:

     W = < A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; C₁  Z₁,...,C_s  Z_s >.

§       The intersection between  and W, denoted by    W is defined as follows:

     W = < A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; C₁  Z₁,...,C_s  Z_s >.

§       The complement of , denoted by ^C is defined as follows: ^C = < ₁,..., _p; ₁,..., _r; ₁,..., _s >.

§        and W are equal, denoted by    W if    W and W   .  

4.4. Example 

  Consider example 1 in case 1. 

Let  = < {h₂}; {h₅}, {h_3, h₄}; {h_1, h₄} > and W = < {h_2, h₁}; {h₅}, {h₃}; {h₁} >   . Then,

v  Type I. 

§          W = < {h₂} {h_2, h₁}; {h₅} {h₅}, {h₃}  {h_3, h₄}; {h₁}  {h_1, h₄} >

                    = < {h_2, h₁}; {h₅}, {h₃}; {h₁} >.

§          W = < {h₂} {h_2, h₁}; {h₅} {h₅}, {h₃}  {h_3, h₄}; {h₁}  {h_1, h₄} >

                    = < {h₂}; {h₅}, {h_3, h₄}; {h_1, h₄} >.

§       ^C = < {h_1, h₄}; {h_1, h_2, h_3, h₄, h₆}, {h_1, h_2, h₅, h₆}; {h₂} >.

v  Type II. 

§          W = < {h₂} {h_2, h₁}; {h₅} {h₅}, {h₃}  {h_3, h₄}; {h₁}  {h_1, h₄} >

                    = < {h_2, h₁}; {h₅}, {h_3, h₄}; {h₁} >.

§          W = < {h₂} {h_2, h₁}; {h₅} {h₅}, {h₃}  {h_3, h₄}; {h₁}  {h_1, h₄} >

                    = < {h₂}; {h₅}, {h₃}; {h_1, h₄} >.

§       ^C = < {h_1, h_3, h_4, h₅, h₆}; {h_1, h_2, h_3, h₄, h₆}, {h_1, h_2, h₅, h₆}; {h_2, h_3, h₅, h₆} >.

v  Type III. 

§          W = < {h₂} {h_2, h₁}; {h₅} {h₅}, {h₃}  {h_3, h₄}; {h₁}  {h_1, h₄} >

                    = < {h_2, h₁}; {h₅}, {h_3, h₄}; {h_1, h₄} >.

§          W = < {h₂} {h_2, h₁}; {h₅} {h₅}, {h₃}  {h_3, h₄}; {h₁}  {h_1, h₄} >

                    = < {h₂}; {h₅}, {h₃}; {h₁} >.

§       ^C = < {h_1, h_3, h_4, h₅, h₆}; {h_1, h_2, h_3, h₄, h₆}, {h_1, h_2, h₅, h₆}; {h_2, h_3, h₅, h₆} >.

4.5. Corollary 

§       For, Type I and Type II. If  and  are two -sets, then    does not have to be an -set.

§       For, Type I and Type II. If  and  are two -sets, then    does not have to be an -set.

§       For, Type I and Type II. If  and  are two -sets, then    does not have to be an -set.

§       For, Type I and Type II. If  are two -sets, then    does not have to be an -set.

§       For, Type I. If  is an -set, then ^C does not have to be an -set.

Note that:

For, Type I and Type II.  = < {h₁, h₂}; {h_3, h₄},  ; {h₅}> and  = < {h₁}; {h_2, h₃},  ; {h₄, h₅} > are two -sets. But,    is not an -set.

For, Type I and Type II.    = < {h₁, h₂, h₆}; {h_3, h₄},  ; {h₄, h₅, h₆}> and  = < {h₁, h₆}; {h_2, h₆},  ; {h₃, h₄, h₅} > are two -sets. But,    is not an -set.

For, Type I and Type II.  = < {h₁, h₂}; {h_3, h₄},  ; {h₅}> and  = < {h₁}; {h_2, h₃},  ; {h₄, h₅} > are two -sets. But,    is not an -set.

For, Type I and Type II.    = < {h₁, h₂, h₆}; {h_3, h₄},  ; {h₂, h₅, h₆}> and  = < {h₁, h₆}; {h_2, h₆},  ; { h_2, h₃, h₄, h₅} > are two -sets. But,    is not an -set.

4.6. Proposition

  Let , £ and W . Then,

§       .

§       .

§         .

§       If  W and W  £, then  £.   

Proof. It is clear.

4.7. Proposition

  Let , £ and W   . Then,

§        W = W   . 

§        W = W   .

§          = ,    = .

§          = ,    = .

§          = ,    = .

§       ( W) £ =  (W £).

§       ( W) £ =  (W £).

§       ( ^C)^C = .

§       For, Type I and Type II, the equalities   ^C    and   ^C    are in general not true.

§       For, Type III, ^C   ,   ^C   , (in general).

Proof. It is clear.

4.8. Proposition

  Let  and W   . Then,

§       ( W)^C = ^C  W^C.

§       ( W)^C = ^C  W^C.

Proof. 

For, Type I.

( W)^C = (< A₁  X₁,..., A_P  X_P; B₁  Y₁,..., B_r  Y_r; C₁  Z₁,...,C_s  Z_s >)^C

  = < C₁  Z₁,...,C_s  Z_s; (B₁  Y₁)^C,...,(B_r  Y_r)^C; A₁  X₁,..., A_P  X_P >

  = < C₁  Z₁,...,C_s  Z_s; (B₁)^C  (Y₁)^C,..., (B_r)^C  (Y_r)^C ; A₁  X₁,..., A_P  X_P >

  = < C₁,..., C_s; (B₁)^C,...,(B_r)^C; A₁,..., A_P >  < Z₁,..., Z_s; (Y₁)^C,...,(Y_r)^C; X₁,..., X_P >

  = (< A₁,..., A_p; B₁,..., B_r; C₁,..., C_s >)^C  (< X₁,..., X_p; Y₁,..., Y_r; Z₁,..., Z_s >)^C = ^C  W^C.

Similarly, we can prove (2). 

For, Type II.

 ( W)^C = (< A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; C₁  Z₁,...,C_s  Z_s >)^C

  = < (A₁  X₁)^C,...,(A_P  X_P)^C; (B₁  Y₁)^C,...,(B_r  Y_r)^C; (C₁  Z₁)^C,...,(C_s  Z_s)^C >

  = < (A₁)^C  (X₁)^C,...,(A_P)^C  (X_P)^C; (B₁)^C  (Y₁)^C,...,(B_r)^C  (Y_r)^C; (C₁)^C  (Z₁)^C,..., (C_s)^C  (Z_s)^C >

  = < (A₁)^C,...,(A_P)^C; (B₁)^C,...,(B_r)^C; (C₁)^C,..., (C_s)^C >  < (X₁)^C,...,(X_P)^C; (Y₁)^C,...,(Y_r)^C; (Z₁)^C,..., (Z_s)^C >

  = (< A₁,..., A_p; B₁,..., B_r; C₁,..., C_s >)^C  (< X₁,..., X_p; Y₁,..., Y_r; Z₁,..., Z_s >)^C = ^C  W^C.

Similarly, we can prove (2). 

For, Type III.

 ( W)^C = (< A₁  X₁,...,A_P  X_P; B₁  Y₁,...,B_r  Y_r; C₁  Z₁,...,C_s  Z_s >)^C

  = < (A₁  X₁)^C,...,(A_P  X_P)^C; (B₁  Y₁)^C,...,(B_r  Y_r)^C; (C₁  Z₁)^C,...,(C_s  Z_s)^C >

  = < (A₁)^C  (X₁)^C,...,(A_P)^C  (X_P)^C; (B₁)^C  (Y₁)^C,...,(B_r)^C  (Y_r)^C; (C₁)^C  (Z₁)^C,..., (C_s)^C  (Z_s)^C >

  = < (A₁)^C,...,(A_P)^C; (B₁)^C,...,(B_r)^C; (C₁)^C,..., (C_s)^C >  < (X₁)^C,...,(X_P)^C; (Y₁)^C,...,(Y_r)^C; (Z₁)^C,..., (Z_s)^C >

  = (< A₁,..., A_p; B₁,..., B_r; C₁,..., C_s >)^C  (< X₁,..., X_p; Y₁,..., Y_r; Z₁,..., Z_s >)^C =  W^C.

Similarly, we can prove (2). 

4.9. Definition  

  Let :  K be a mapping,  = < A₁,..., A_p; B₁,..., B_r; C₁,..., C_s > ,

                             W = < X₁,..., X_p; Y₁,..., Y_r; Z₁,..., Z_s > . Then,

o      The image_'of  under_' , denoted_'by ( ) , is defined_'as: 

  ( ) = < f(A₁),..., f(A_p); f(B₁),..., f(B_r); f(C₁),..., f(C_s) >.

o      The preimage of W under , denoted by (W) , is defined as: 

  (W) = < (X₁),..., (X_p); (Y₁),..., (Y_r); (Z₁),..., (Z_s) >.

4.10. Corollary 

  Let :  K be a mapping,

                        = < A₁,..., A_p; B₁,..., B_r; C₁,..., C_s > ,

                       ₁ = < A₁₁,..., A_1p; B₁₁,..., B_1r; C₁₁,..., C_1s > ,

                       W = < X₁,..., X_p; Y₁,..., Y_r; Z₁,..., Z_s > ,

                       W₁ = < X₁₁,..., X_1p; Y₁₁,..., Y_1r; Z₁₁,..., Z_1s > . Then,

§       W₁  W   (W₁)   (W).

§       ₁      f( ₁)  f( ).

§       f( )), and if f is an injective, then  = f( )).

§       (W))  W, and if f is a surjective, then (W)) = W.

§       (W₁  W) = (W₁) (W).

§       (W₁  W) = (W₁) (W).

§       ( ₁   ) = ( ₁) ( ).

§       ( ₁   )   ( ₁) ( ) and, if f is an injective, then ( ₁   ) = ( ₁) ( ).

§       ( ) = , ( ) = . 

§       ( ) = , and ( ) = , if f is a surjective.

Proof. Straightforward_.

6. Conclusion

  We used the NSVRNL and proposed n-valued refined neutrosophic crisp set with some of its set-theoretical operations.  We can propose a group decision,making method based on -sets (In particular, n=4 or 5 or 6), and give algorithm of proposed,method. "We hope that the results of this study will be useful for resear-chers to present additional new studies on the n-valued refined neutrosophic crisp sets".

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