New Distance and Similarity Measures of Interval Neutrosophic Sets

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1 New Distace ad Similarity Measures of Iterval Neutrosophic Sets Said Broumi Abstract: I this paper we proposed a ew distace ad several similarity measures betwee iterval eutrosophic sets. Keywords: Neutrosophic set, Iterval eutrosohic set, Similarity measure. I. INTRODCTION The eutrsophic set, fouded by F.Smaradache [], has capability to deal with ucertaity, imprecise, icomplete ad icosistet iformatio which exist i the real world. Neutrosophic set theory is a powerful tool i the formal framework, which geeralizes the cocepts of the classic set, fuzzy set [2], iterval-valued fuzzy set [3], ituitioistic fuzzy set [4], iterval-valued ituitioistic fuzzy set [5], ad so o. After the pioeerig work of Smaradache, i 2005 Wag [6] itroduced the otio of iterval eutrosophic set (INS for short) which is a particular case of the eutrosophic set. INS ca be described by a membership iterval, a o-membership iterval, ad the idetermiate iterval. Thus the iterval value eutrosophic set has the virtue of beig more flexible ad practical tha sigle value eutrosophic set. Ad the Iterval Neutrosophic Set provides a more reasoable mathematical framework to deal with idetermiate ad icosistet iformatio. May papers about eutrosophic set theory have bee doe by various researchers [7, 8, 9, 0,, 2, 3, 4, 5, 6, 7, 8, 9, 20]. A similarity measure for eutrosophic set (NS) is used for estimatig the degree of similarity betwee two eutrosophic sets. Several researchers proposed some similarity measures betwee NSs, such as S. Broumi ad F. Smaradache [26], Ju Ye [, 2], P. Majumdar ad S.K.Smata [23]. I the literature, there are few researchers who studied the distace ad similarity measure of IVNS. I 203, Ju Ye [2] proposed similarity measures betwee iterval eutrosophic set based o the Hammig ad Euclidea distace, ad developed a multicriteria decisio makig method based o the similarity degree. S. Broumi ad F. Smaradache [0] proposed a ew similarity measure, called cosie similarity measure of iterval valued eutrosophic sets. O the basis of umerical computatios, S. Broumi ad F. Smaradache foud out that their similarity measures are stroger ad more robust tha Ye s measures. We all kow that there are various distace measures i mathematics. So, i this paper, we will exted the geeralized distace of sigle valued eutrosophic set proposed by Ye [2] to the case of iterval eutrosophic set ad we ll study some ew similarity measures. This paper is orgaized as follows. I sectio 2, we review some otios of eutrosophic set aditerval valued eutrosophic set. I sectio 3, some ew similarity measures of iterval valued eutrosophic sets ad their proofs are itroduced. Fially, the coclusios are stated i sectio 4. II. PREIMIAIRIES This sectio gives a brief overview of the cocepts of eutrosophic set, ad iterval valued eutrosophic set. A. Neutrosophic Sets ) Defiitio [] et X be a uiverse of discourse, with a geeric elemet i X deoted by x, the a eutrosophic set A is a object havig the form: A = {< x: T A (x), I A (x), F A (x)>, x X}, where the fuctios T, I, F : X ] 0, + [ defie respectively the degree of membership (or Truth), the degree of idetermiacy, ad the degree of o-membership (or Falsehood) of the elemet x X to the set A with the coditio: 0 T A (x)+i A (x)+f A (x) 3 +. () From philosophical poit of view, the eutrosophic set takes the value from real stadard or o-stadard subsets of ] 0, + [. Therefore, istead of ] 0, + [ we eed to take the iterval [0, ] for techical applicatios, because ] 0, + [ will 249

2 be difficult to apply i the real applicatios such as i scietific ad egieerig problems. For two NSs, A NS = {<x, T A (x), I A (x), F A (x)> x X} (2) ad B NS ={ <x, T B (x), I B (x), F B (x)> x X } the two relatios are defied as follows: () A NS B NS if ad oly if T A (x) T B (x), I A (x) I B (x), F A (x) F B (x). (2) A NS = B NS if ad oly if, T A (x)=t B (x), I A (x) =I B (x), F A (x) =F B (x). B. Iterval Valued Neutrosophic Sets I actual applicatios, sometimes, it is ot easy to express the truth-membership, idetermiacy-membership ad falsitymembership by crisp value, ad they may be easier to be expressed by iterval umbers. Wag et al. [6] further defied iterval eutrosophic sets (INS) shows as follows: ) Defiitio [6] et X be a uiverse of discourse, with geeric elemet i X deoted by x. A iterval valued eutrosophic set (for short IVNS) A i X is characterized by truth-membership fuctiot A (x), idetemiacy-membership fuctio I A (x), ad falsity-membership fuctio F A (x). For each poit x i X, we have that T A (x), I A (x), F A (x) [ 0,]. For two IVNS, A IVNS ={<x,[t A (x),t A (x)], [I A (x), I A (x)], [F A (x), F A (x)]> x X } (3) ad B IVNS = {<x, [T B (x),t B (x)], [I B (x), I B (x)], [F B (x), F B (x)] > x X } the two relatios are defied as follows: () A IVNS B IVNS if ad oly if T A (x) T B (x),t A (x) T B (x), I A (x) I B (x), I A (x) I B (x), F A (x) F B (x), F A (x) F B (x). (2) A IVNS = B IVNS if ad oly if T A (x i ) = T B (x i ), T A (x i ) = T B (x i ), I A (x i ) = I B (x i ),I A (x i ) = I B (x i ), F A (x i ) = F B (x i ) ad F A (x i ) = F B (x i ) for ay x X. C. Defitio i. 0 S(A, B). ii. S(A, B) = S(B, A). iii. S(A, B) = if A= B, i.e T A (x i ) = T B (x i ), T A (x i ) = T B (x i ), I A (x i ) = I B (x i ), I A (x i ) = I B (x i ) ad F A (x i ) = F B (x i ), F A (x i ) = F B (x i ), for i =, 2,.,. iv. A B C S(A,B) mi (S(A,B), S(B,C). III. NEW DISTANCE MEASRE OF INTERVA VAED NETROSOPHIC SETS et A ad B be two sigle eutrosophic sets, the J. Ye [] proposed a geeralized sigle valued eutrosophic weighted distace measure betwee A ad B as follows: d (A, B) = { 3 w i[ T A (x i ) T B (x i ) + I A (x i ) I B (x i ) + F A (x i ) F B (x i ) ]} (4) where > 0 ad T A (x i ), I A (x i ), F A (x i ), T B (x i ), I B (x i ), F B (x i ) [ 0, ]. Based o the geometrical distace model ad usig the iterval eutrosophic sets, we exteded the distace (4) as follows: d (A, B) = { 6 w i[ T A (x i ) T B (x i ) + T A (x i ) T B (x i ) + I A (x i ) I B (x i ) + I A (x i ) I B (x i ) + F A (x i ) F B (x i ) + F A (x i ) F B (x i ) ]}. (5) The ormalized geeralized iterval eutrosophic distace is d (A, B) = { 6 F B (x i ) + F A (x i ) F B (x i ) ]} w i[ T A (x i ) T B (x i ) + T A (x i ) T B (x i ) + I A (x i ) I B (x i ) + I A (x i ) I B (x i ) + F A (x i ). If w={,,, }, the distace (6) is reduced to the followig distaces: d (A, B) = { 6 F B (x i ) + F A (x i ) F B (x i ) ]} d (A, B) = { 6 F B (x i ) + F A (x i ) F B (x i ) ]} [ T A (x i ) T B (x i ) + T A (x i ) T B (x i ) + I A (x i ) I B (x i ) + I A (x i ) I B (x i ) + F A (x i ). [ T A (x i ) T B (x i ) + T A (x i ) T B (x i ) + I A (x i ) I B (x i ) + I A (x i ) I B (x i ) + F A (x i ). (6) (7) (8) Particular case. 250

3 (i) If = the the distaces (7) ad (8) are reduced to the followig Hammig distace ad respectively ormalized Hammig distace defied by Ye Ju []: d H (A, B) = { [ T 6 A (x i ) T B (x i ) + T A (x i ) T B (x i ) + I A (x i ) I B (x i ) + I A (x i ) I B (x i ) + F A (x i ) F B (x i ) + F A (x i ) F B (x i ) ]}, (9) d NH (A, B) = { [ T 6 A (x i ) T B (x i ) + T A (x i ) T B (x i ) + I A (x i ) I B (x i ) + I A (x i ) I B (x i ) + F A (x i ) F B (x i ) + F A (x i ) F B (x i ) ]}. (0) (ii) If = 2 the the distaces (7) ad (8) are reduced to the followig Euclidea distace ad respectively ormalized Euclidea distace defied by Ye Ju [2]: d E (A, B) = { [ T 6 A (x i ) T B (x i ) 2 + T A (x i ) T B (x i ) 2 + I A (x i ) I B (x i ) 2 + I A (x i ) I B (x i ) 2 + F A (x i ) F B (x i ) 2 + F A (x i ) F B (x i ) 2 ]} d NE (A, B) = { 6 F B (x i ) 2 + F A (x i ) F B (x i ) 2 ]} IV. 2, [ T A (x i ) T B (x i ) 2 + T A (x i ) T B (x i ) 2 + I A (x i ) I B (x i ) 2 + I A (x i ) I B (x i ) 2 + F A (x i ) 2. NEW SIMIARITY MEASRES OF INTERVA VAED NETROSOPHIC SET A. Similarity measure based o the geometric distace model Based o distace (4), we defie the similarity measure betwee the iterval valued eutrosophic sets A ad B as follows: S DM (A, B) = - { 6, F B (x i ) + F A (x i ) F B (x i ) ]} [ T A (x i ) T B (x i ) + T A (x i ) T B (x i ) + I A (x i ) I B (x i ) + I A (x i ) I B (x i ) + F A (x i ) where > 0 ad S DM (A, B) is the degree of similarity of A ad B. If we take the weight of each elemet x i X ito accout, the w (A, B)=- { 6 w i [ T A (x i ) T B (x i ) + T A (x i ) T B (x i ) + I A (x i ) I B (x i ) + I A (x i ) I B (x i ) + F A (x i ) S DM F B (x i ) + F A (x i ) F B (x i ) ]}. If each elemets has the same importace, i.e. w = {,,, }, the similarity (4) reduces to (3). By (defiitio C) it ca easily be kow that S DM (A, B) satisfies all the properties of the defiitio.. Similarly, we defie aother similarity measure of A ad B, as: S( A, B) = [ ( T A (x i ) T B(xi ) + TA (xi ) T B (xi ) + I A (xi ) I B (xi ) + I A (xi ) I B (xi ) + F A (xi ) F B (xi ) + F A (xi ) F B (xi ) ) ( T A (x i )+T B (x i ) + T A (x i )+T B (x i ) + I A (x i )+I B (x i ) + I A (x i )+I B (x i ) + F A (x i )+F B (x i ) + F A (x i )+F B (x i ) ) If we take the weight of each elemet x i X ito accout, the S( A, B) = [ w i( T A(xi ) T B(xi ) + TA (xi ) T B (xi ) + I A (xi ) I B (xi ) + I A (xi ) I B (xi ) + F A (xi ) F B (xi ) + F A (xi ) F B (xi ) ) w i ( T A (x i )+T B (x i ) + T A (x i )+T B (x i ) + I A (x i )+I B (x i ) + I A (x i )+I B (x i ) + F A (x i )+F B (x i ) + F A (x i )+F B (x i ) ) () (2) (3) (4) ]. (5) ]. (6) It also has bee proved that all coditios of the defiitio are satisfied. If each elemets has the same importace, ad the the similarity (6) reduces to (5). B. Similarity measure based o the iterval valued eutrosophic theoretic approach: I this sectio, followig the similarity measure betwee two eutrosophic sets defied by P. Majumdar i [24], we exted Majumdar s defiitio to iterval valued eutrosophic sets. 25

4 we defie a similarity measure betwee A ad B as follows: S TA (A, B)= {mi{t A (x i ),T B(xi )}+mi{t A(xi ),T B(xi )} +mi{i A(xi ),I B(xi )}+mi{i A(xi ),I B(xi )}+ mi{f A(xi ),F B(xi )}+mi{f A(xi ),F B(xi )} {max{t A (x i ),T B (x i )}+max{t A (x i ),T B (x i )} +max{i A (x i ),I B (x i )}+max{i A (x i ),I B (x i )}+ max{f A (x i ),F B (x i )}+max{f A (x i ),F B (x i )} ) Propositio iv. 0 S TA (A, B). v. S TA (A, B) = S TA (A, B). vi. S(A, B) = if A = B i.e. T A (x i ) = T B (x i ), T A (x i ) = T B (x i ), I A (x i ) = I B (x i ), I A (x i ) = I B (x i ) ad F A (x i ) = F B (x i ), F A (x i ) = F B (x i ) for i =, 2,.,. iv. A B C S TA (A, B) mi (S TA (A, B), S TA (B, C)). Proof. Properties (i) ad (ii) follow from the defiitio. (iii) It is clearly that if A = B S TA (A, B) = {mi{t A (x i ), T B (x i )} + mi{t A (x i ), T B (x i )} +mi{i A (x i ), I B (x i )} + mi{i A (x i ), I B (x i )} + mi{f A (x i ), F B (x i )} + mi{f A (x i ), F B (x i )} = {max{t A (x i ), T B (x i )} + max{t A (x i ), T B (x i )} +max{i A (x i ), I B (x i )} + max{i A (x i ), I B (x i )} + max{f A (x i ), F B (x i )} + max{f A (x i ), F B (x i )} {[mi{t A (x i ), T B (x i )} max{t A (x i ), T B (x i )}] + [mi{t A (x i ), T B (x i )} max{t A (x i ), T B (x i )}] + [mi{i A (x i ), I B (x i )} max{i A (x i ), I B (x i )}] + [mi{i A (x i ), I B (x i )} max{i A (x i ), I B (x i )}] + [mi{f A (x i ), F B (x i )} max{f A (x i ), F B (x i )}] + [mi{f A (x i ), F B (x i )} max{f A (x i ), F B (x i )]} = 0. Thus for each x, oe has that [mi{t A (x i ), T B (x i )} max{t A (x i ), T B (x i )}] = 0 [mi{t A (x i ), T B (x i )} max{t A (x i ), T B (x i )}] = 0 [mi{i A (x i ), I B (x i )} max{i A (x i ), I B (x i )}] = 0 [mi{i A (x i ), I B (x i )} max{i A (x i ), I B (x i )}] = 0 [mi{f A (x i ), F B (x i )} max{f A (x i ), F B (x i )}] = 0 [mi{f A (x i ), F B (x i )} max{f A (x i ), F B (x i )]}= 0 hold. Thus T A (x i ) = T B (x i ), T A (x i ) = T B (x i ), I A (x i ) = I B (x i ), I A (x i ) = I B (x i ), F A (x i ) = F B (x i ) ad F A (x i ) = F B (x i ) A=B (iv) Now we prove the last result. et A B C, the we have T A (x) T B (x) T C (x), T A (x) T B (x) T C (x), I A (x) I B (x) I C (x), I A (x) I B (x) I C (x), F A (x) F B (x) F C (x), F A (x) F B (x) F C (x) for all x X. Now T A (x) +T A (x) +I A (x) +I A (x) +F B (x)+f B (x) T A (x) +T A (x) +I A (x) +I A (x)+f C (x)+f C (x) ad T B (x) +T B (x) +I B (x) +I B (x) +F A (x)+f A (x) T C (x) +T C (x) +I C (x) +I C (x)+f A (x)+f A (x). S(A,B) = T A (x) +T A(x) +IA(x) +IA(x) +FB(x)+FB (x) T B (x) +T B (x) +I B (x) +I B (x) +F A (x)+f A (x) Agai, similarly we have T A (x) +TA(x) +IA(x) +IA(x)+FC (x)+fc (x) = S(A,C). T C (x) +T C (x) +I C (x) +I C (x)+f A (x)+f A (x) T B (x) +T B (x) +I B (x) +I B (x)+f C (x)+f C (x) T A (x) +T A (x) +I A (x) +I A (x)+f C (x)+f C (x) T C (x) +T C (x) +I C (x) +I C (x)+f A (x)+f A (x) T C (x) +T C (x) +I C (x) +I C (x)+f B (x)+f B (x) (x) +TA (x) +IA (x) +IA (x)+fc (x)+fc (x) S(B,C) = T B (x) +T B(x) +IB(x) +IB(x)+FC (x)+fc (x) T A = S(A,C) T C (x) +T C (x) +I C (x) +I C (x)+f B (x)+f B (x) T C (x) +T C (x) +I C (x) +I C (x)+f A (x)+f A (x) S TA (A, B) mi (S TA (A, B), S TA (B, C)). Hece the proof of this propositio. If we take the weight of each elemet x i X ito accout, the (7) S(A, B)= w i{mi{t A(xi ),T B(xi )}+mi{t A(xi ),T B(xi )} +mi{i A(xi ),I B(xi )}+mi{i A(xi ),I B(xi )}+ mi{f A(xi ),F B(xi )}+mi{f A(xi ),F B(xi )} w i {max{t A (x i ),T B (x i )}+max{t A (x i ),T B (x i )} +max{i A (x i ),I B (x i )}+max{i A (x i ),I B (x i )}+ max{f A (x i ),F B (x i )}+max{f A (x i ),F B (x i )} 252

5 Particularly, if each elemet has the same importace, the (8) is reduced to (7), clearly this also satisfies all the properties of the defiitio. C. Similarity measure based o matchig fuctio by usig iterval eutrosophic sets: Che [24] ad Che et al. [25] itroduced a matchig fuctio to calculate the degree of similarity betwee fuzzy S MF (A,B) = (8) sets. I the followig, we exted the matchig fuctio to deal with the similarity measure of iterval valued eutrosophic sets. we defie a similarity measure betwee A ad B as follows: ((T A (x i ) T B (x i )) + (T A (x i ) T B (x i )) + (I A (x i ) I B (x i )) + (I A (x i ) I B (x i )) + (F A (x i ) F B (x i )) + (F A (x i ) F B (x i ))) max( i= (T A (x i ) 2 + T A (x i ) 2 + I A (x i ) 2 + I A (x i ) 2 + F A (x i ) 2 + F A (x i ) 2 ), i= (T B (x i ) 2 + T B (x i ) 2 + I B (x i ) 2 + I B (x i ) 2 + F B (x i ) 2 + F B (x i ) 2 )) (9) Proof. i. 0 S MF (A,B). The iequality S MF (A,B) 0 is obvious. Thus, we oly prove the iequality S(A, B). S MF (A,B)= ((T A (x i ) T B (x i )) + (T A (x i ) T B (x i )) + (I A (x i ) I B (x i )) + (I A (x i ) I B (x i )) + (F A (x i ) F B (x i )) + (F A (x i ) F B (x i ))) = T A (x ) T B (x )+T A (x 2 ) T B (x 2 )+ +T A (x ) T B (x )+T A (x ) T B (x )+T A (x 2 ) T B (x 2 )+ +T A (x ) T B (x )+ I A (x ) I B (x )+I A (x 2 ) I B (x 2 )+ +I A (x ) I B (x )+I A (x ) I B (x )+I A (x 2 ) I B (x 2 )+ +I A (x ) I B (x )+ F A (x ) F B (x )+F A (x 2 ) F B (x 2 )+ +F A (x ) F B (x )+F A (x ) T B (x )+F A (x 2 ) F B (x 2 )+ +F A (x ) F B (x ). Accordig to the Cauchy Schwarz iequality: (x y + x 2 y x y ) 2 (x 2 + x x 2 ) (y 2 + y y 2 ) where (x, x 2,, x ) R ad (y, y 2,, y ) R we ca obtai [S MF (A, B)] 2 (T A (x i ) 2 + T A (x i ) 2 + I A (x i ) 2 + I A (x i ) 2 + F A (x i ) 2 + F A (x i ) 2 ) (T B (x i ) 2 + T B (x i ) 2 + I B (x i ) 2 + I B (x i ) 2 + F B (x i ) 2 + F B (x i ) 2 ) = S(A, A) S(B, B) Thus S MF (A,B) [S(A, A)] 2 [S(B, B)] 2. The S MF (A,B) max{s(a,a), S(B,B)]. Therefore S MF (A, B). If we take the weight of each elemet x i X ito accout, the S w MF (A,B)= w i ((T A(xi ) T B(xi ))+(T A (xi ) T B (xi ))+(I A(xi ) I B(xi ))+(I A (xi ) I B (xi ))+(F A(xi ) F B(xi ))+(F A (xi ) F B (xi ))) max( i= w i (T A (x i ) 2 +T A (x i ) 2 + I A (x i ) 2 + I A (x i ) 2 + F A (x i ) 2 +F A (x i ) 2 ), i= w i (T B (x i ) 2 +T B (x i ) 2 + I B (x i ) 2 + I B (x i ) 2 + F B (x i ) 2 +F B (x i ) 2 )) (20) Particularly, if each elemet has the same importace, the the similarity (20) is reduced to (9). Clearly this also satisfies all the properties of defiitio. The larger the value of S(A,B), the more the similarity betwee A ad B. V. COMPARISON OF NEW SIMIARITY MEASRE OF IVNS WITH THE EXISTING MEASRES. et A ad B be two iterval valued eutrosophic sets i the uiverse of discourse X = {x, x 2,.., x }.The ew similarity S TA (A, B) of IVNS ad the existig similarity measures of iterval valued eutrosophic sets (examples ad 2) itroduced i [0, 2, 23] are listed as follows: Piaki similarity I: this similarity measure was proposed as cocept of associatio coefficiet of the eutrosophic sets as follows S PI = {mi{t A (x i ),T B (x i )}+mi{i A (x i ),I B (x i )}+ mi{f A (x i ),F B (x i )}} {max{t A (x i ),T B (x i )}+max{i A (x i ),I B (x i )}+ max{f A (x i ),F B (x i )}} (2) Broumi ad Smaradache cosie similarity: 253

6 C N ( A, B)= (T A(xi ) + T A (xi )) (T B(xi )+T B (xi ))+(I A(xi ) + I A (xi )) (I B(xi ) + I B (xi )) +(F A(xi ) + F A (xi )) (F B(xi ) + F B (xi )) (T A(xi ) + T A (xi )) 2 +(I A (x i )+I A (x i )) 2 +(F A(xi ) + F A (xi )) 2 (T B(xi ) + T B (xi )) 2 +(I B (x i )+I B (x i )) 2 +(F B(xi ) + F B (xi )) 2 (22) Ye similarity S ye (A, B) = - 6 [ ift A (x i ) ift B (x i ) + supt A (x i ) supt B (x i ) + ifi A (x i ) ifi B (x i ) + supi A (x i ) supi B (x i ) + iff A (x i ) iff B (x i ) + supf A (x i ) supf B (x i ) ]. (23) Example et A = {<x, (a, 0.2, 0.6, 0.6), (b, 0.5, 0.3, 0.3), (c, 0.6, 0.9, 0.5)>} ad B = {<x, (a, 0.5, 0.3, 0.8), (b, 0.6, 0.2, 0.5), (c, 0.6, 0.4, 0.4)>}. Piaki similarity I = 0.6. S ye (A, B) = 0.38 (with w i =). Cosie similarity C N (A, B) = S TA (A, B) = 0.8. Example 2: et A= {<x, (a, [ 0.2, 0.3 ],[0.2, 0.6], [0.6, 0.8]), (b, [ 0.5, 0.7 ], [0.3, 0.5], [0.3, 0.6]), (c, [0.6, 0.9],[0.3, 0.9], [0.3, 0.5])>} ad B={<x, (a, [ 0.5, 0.3 ],[0.3, 0.6], [0.6, 0.8]), (b, [ 0.6, 0.8 ],[0.2, 0.4], [0.5, 0.6]), (c, [ 0.6, 0.9],[0.3, 0.4], [0.4, 0.6])>}. Piaki similarity I = NA. S ye (A, B) = 0.7 (with w i =). Cosie similarity C N ( A, B) = S TA (A, B) = 0.9. O the basis of computatioal study Ju Ye [2] has show that their measure is more effective ad reasoable. A similar kid of study with the help of the proposed ew measure based o theoretic approach, it has bee doe ad it is foud that the obtaied results are more refied ad accurate. It may be observed from the above examples that the values of similarity measures are closer to with S TA (A, B) which is this proposed similarity measure. VI. CONCSIONS Few distace ad similarity measures have bee proposed i literature for measurig the distace ad the degree of similarity betwee iterval eutrosophic sets. I this paper, we proposed a ew method for distace ad similarity measure for measurig the degree of similarity betwee two weighted iterval valued eutrosophic sets, ad we have exteded the work of Piaki, Majumdar ad S. K. Samat ad Che. The results of the proposed similarity measure ad existig similarity measure are compared. I the future, we will use the similarity measures which are proposed i this paper i group decisio makig VII. ACKNOWEDGMENT The authors are very grateful to the aoymous referees for their isightful ad costructive commets ad suggestios, which have bee very helpful i improvig the paper. VIII. REFERENCES [] F. Smaradache, A ifyig Field i ogics. Neutrosophy: Neutrosophic Probability, Set ad ogic. Rehoboth: America Research Press, (998). [2]. A. Zadeh, "Fuzzy Sets". Iformatio ad Cotrol, 8, (965), pp [3] Turkse, Iterval valued fuzzy sets based o ormal forms.fuzzy Sets ad Systems, 20, (968), pp [4] Ataassov, K. Ituitioistic fuzzy sets. Fuzzy Sets ad Systems, 20, (986), pp [5] Ataassov, K., & Gargov, G Iterval valued ituitioistic fuzzy sets. Fuzzy Sets ad Systems. Iteratioal Joural of Geeral Systems 393, 3,(998), pp [6] Wag, H., Smaradache, F., Zhag, Y.-Q. ad Suderrama, R., Iterval Neutrosophic Sets ad ogic: Theory ad Applicatios i Computig, Hexis, Phoeix, AZ, (2005). [7] A. Kharal, A Neutrosophic Multicriteria Decisio Makig Method,New Mathematics & Natural Computatio, to appear i Nov [8] S. Broumi ad F. Smaradache, Ituitioistic Neutrosophic Soft Set, Joural of Iformatio ad Computig Sciece, Eglad, K, ISSN ,Vol. 8, No. 2, (203), pp [9] S.Broumi, Geeralized Neutrosophic Soft Set, Iteratioal Joural of Computer Sciece, Egieerig ad Iformatio Techology (IJCSEIT), ISSN: , E-ISSN: , Vol.3, No.2, (203), pp [0] S. Broumi ad F. Smaradache, Cosie similarity measure of iterval valued eutrosophic sets, 204 (submitted). [] J. Ye. Sigle valued etrosophiqc miimum spaig tree ad its clusterig method De Gruyter joural of itelliget system, (203), pp [2] J. Ye, Similarity measures betwee iterval eutrosophic 254

7 sets ad their multicriteria decisio-makig method Joural of Itelliget & Fuzzy Systems, DOI: /IFS-20724,(203),. [3] M. Arora, R. Biswas,. S. Pady, Neutrosophic Relatioal Database Decompositio, Iteratioal Joural of Advaced Computer Sciece ad Applicatios, Vol. 2, No. 8, (20) [4] M. Arora ad R. Biswas, Deploymet of Neutrosophic techology to retrieve aswers for queries posed i atural laguage, i 3rdIteratioal Coferece o Computer Sciece ad Iformatio Techology ICCSIT, IEEE catalog Number CFP057E-art, Vol.3, ISBN: ,(200) [5] Asari, Biswas, Aggarwal, Proposal for Applicability of Neutrosophic Set Theory i Medical AI, Iteratioal Joural of Computer Applicatios ( ), Vol. 27 No.5, (20) 5-. [6] F. G upiáñez, "O eutrosophic topology", Kyberetes, Vol. 37, Iss: 6, (2008), pp , Doi: 0.08/ [7] S. Aggarwal, R. Biswas, A. Q. Asari, Neutrosophic Modelig ad Cotrol, /0 IEEE,( 200), pp [8] H. D. Cheg,& Y Guo. A ew eutrosophic approach to image thresholdig. New Mathematics ad Natural Computatio, 4(3), (2008), pp [20] M. Zhag,. Zhag, ad H. D. Cheg. A eutrosophic approach to image segmetatio based o watershed method. Sigal Processig 5, 90, (200), pp [2] Wag, H., Smaradache, F., Zhag, Y. Q., Suderrama, R, Sigle valued eutrosophic sets. Multispace ad Multistructure, 4 (200), pp [22] S. Broumi, F. Smaradache, Correlatio Coefficiet of Iterval Neutrosophic set, Periodical of Applied Mechaics ad Materials, Vol. 436, 203, with the title Egieerig Decisios ad Scietific Research i Aerospace, Robotics, Biomechaics, Mechaical Egieerig ad Maufacturig; Proceedigs of the Iteratioal Coferece ICMERA, Bucharest, October 203. [23] P. Majumdar, S.K. Samat, O similarity ad etropy of eutrosophic sets, Joural of Itelliget ad Fuzzy Systems, (Prit) (Olie), (203), DOI:0.3233/IFS-3080, IOS Press. [24] S.M. Che, S. M. Yeh ad P.H. Hsiao, A compariso of similarity measure of fuzzy values Fuzzy sets ad systems, 72, [25] S.M. Che, A ew approach to hadlig fuzzy decisio makig problem, IEEE trasactios o Systems, ma ad cyberetics, 8, 988., pp [26] Said Broumi, ad, Several Similarity Measures of Neutrosophic Sets, Neutrosophic Sets ad Systems, Vol. (203), pp [9] Y. Guo, &, H. D. Cheg New eutrosophic approach to image segmetatio. Patter Recogitio, 42, (2009), pp Preseted at 7th Iteratioal Coferece o Iformatio Fusio, Salamaca, Spai, 7-0 July

A New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions

A New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions A New Costructive Proof of Graham's Theorem ad More New Classes of Fuctioally Complete Fuctios Azhou Yag, Ph.D. Zhu-qi Lu, Ph.D. Abstract A -valued two-variable truth fuctio is called fuctioally complete,