N-norm and N-conorm in Neutrosophic Logic and Set, and the Neutrosophic Topologies

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1 -norm and -conorm in eutrosophic Logic and Set, and the eutrosophic Topologies Florentin Smarandache University of ew Mexico, Gallup M 870, USA smarand@unmedu Abstract: In this paper we present the -norms/-conorms in neutrosophic logic and set as extensions of T-norms/T-conorms in fuzzy logic and set Also, as an extension of the Intuitionistic Fuzzy Topology we present the eutrosophic Topologies Definition of the eutrosophic Logic/Set: Let T, I, F be real standard or non-standard subsets of ] - 0, + [, with sup T = t_sup, inf T = t_inf, sup I = i_sup, inf I = i_inf, sup F = f_sup, inf F = f_inf, and n_sup = t_sup+i_sup+f_sup, n_inf = t_inf+i_inf+f_inf Let U be a universe of discourse, and M a set included in U An element x from U is noted with respect to the set M as x(t, I, F) and belongs to M in the following way: it is t% true in the set, i% indeterminate (unknown if it is or not) in the set, and f% false, where t varies in T, i varies in I, f varies in F Statically T, I, F are subsets, but dynamically T, I, F are functions/operators depending on many known or unknown parameters In a similar way define the eutrosophic Logic: A logic in which each proposition x is T% true, I% indeterminate, and F% false, and we write it x(t,i,f), where T, I, F are defined above As a generalization of T-norm and T-conorm from the Fuzzy Logic and Set, we now introduce the -norms and -conorms for the eutrosophic Logic and Set We define a partial relation order on the neutrosophic set/logic in the following way: x(t, I, F ) y(t, I, F ) iff (if and only if) T T, I I, F F for crisp components And, in general, for subunitary set components: x(t, I, F ) y(t, I, F ) iff inf T inf T, sup T sup T, inf I inf I, sup I sup I, inf F inf F, sup F sup F

2 If we have mixed - crisp and subunitary - components, or only crisp components, we can transform any crisp component, say a with a [0,] or a ] 0, + [, into a subunitary set [a, a] So, the definitions for subunitary set components should work in any case -norms n : ( ] - 0, + [ ] - 0, + [ ] - 0, + [ ) ] - 0, + [ ] - 0, + [ ] - 0, + [ n (x(t,i,f ), y(t,i,f )) = ( n T(x,y), n I(x,y), n F(x,y)), where n T(,), n I(,), n F(,) are the truth/membership, indeterminacy, and respectively falsehood/nonmembership components n have to satisfy, for any x, y, z in the neutrosophic logic/set M of the universe of discourse U, the following axioms: a) Boundary Conditions: n (x, 0) = 0, n (x, ) = x b) Commutativity: n (x, y) = n (y, x) c) Monotonicity: If x y, then n (x, z) n (y, z) d) Associativity: n ( n (x, y), z) = n (x, n (y, z)) There are cases when not all these axioms are satisfied, for example the associativity when dealing with the neutrosophic normalization after each neutrosophic operation But, since we work with approximations, we can call these -pseudo-norms, which still give good results in practice n represent the and operator in neutrosophic logic, and respectively the intersection operator in neutrosophic set theory Let J {T, I, F} be a component Most known -norms, as in fuzzy logic and set the T-norms, are: The Algebraic Product -norm: n algebraic J(x, y) = x y The Bounded -orm: n bounded J(x, y) = max{0, x + y } The Default (min) -norm: n min J(x, y) = min{x, y} A general example of -norm would be this Let x(t, I, F ) and y(t, I, F ) be in the neutrosophic set/logic M Then: n (x, y) = (T /\T, I \/I, F \/F ) where the /\ operator, acting on two (standard or non-standard) subunitary sets, is a -norm (verifying the above -norms axioms); while the \/ operator, also acting on two (standard or non-standard) subunitary sets, is a -conorm (verifying the below -conorms axioms) For example, /\ can be the Algebraic Product T-norm/-norm, so T /\T = T T (herein we have a product of two subunitary sets using simplified notation); and \/ can be the Algebraic Product T-conorm/-conorm, so T \/T = T +T -T T (herein we have a sum, then a product, and afterwards a subtraction of two subunitary sets) Or /\ can be any T-norm/-norm, and \/ any T-conorm/-conorm from the above and below; for example the easiest way would be to consider the min for crisp components (or inf for subset components) and respectively max for crisp components (or sup for subset components) If we have crisp numbers, we can at the end neutrosophically normalize -conorms

3 c : ( ] - 0, + [ ] - 0, + [ ] - 0, + [ ) ] - 0, + [ ] - 0, + [ ] - 0, + [ c (x(t,i,f ), y(t,i,f )) = ( c T(x,y), c I(x,y), c F(x,y)), where n T(,), n I(,), n F(,) are the truth/membership, indeterminacy, and respectively falsehood/nonmembership components c have to satisfy, for any x, y, z in the neutrosophic logic/set M of universe of discourse U, the following axioms: a) Boundary Conditions: c (x, ) =, c (x, 0) = x b) Commutativity: c (x, y) = c (y, x) c) Monotonicity: if x y, then c (x, z) c (y, z) d) Associativity: c ( c (x, y), z) = c (x, c (y, z)) There are cases when not all these axioms are satisfied, for example the associativity when dealing with the neutrosophic normalization after each neutrosophic operation But, since we work with approximations, we can call these -pseudo-conorms, which still give good results in practice c represent the or operator in neutrosophic logic, and respectively the union operator in neutrosophic set theory Let J {T, I, F} be a component Most known -conorms, as in fuzzy logic and set the T-conorms, are: The Algebraic Product -conorm: c algebraic J(x, y) = x + y x y The Bounded -conorm: c bounded J(x, y) = min{, x + y} The Default (max) -conorm: c max J(x, y) = max{x, y} A general example of -conorm would be this Let x(t, I, F ) and y(t, I, F ) be in the neutrosophic set/logic M Then: n (x, y) = (T \/T, I /\I, F /\F ) Where as above - the /\ operator, acting on two (standard or non-standard) subunitary sets, is a - norm (verifying the above -norms axioms); while the \/ operator, also acting on two (standard or nonstandard) subunitary sets, is a -conorm (verifying the above -conorms axioms) For example, /\ can be the Algebraic Product T-norm/-norm, so T /\T = T T (herein we have a product of two subunitary sets); and \/ can be the Algebraic Product T-conorm/-conorm, so T \/T = T +T -T T (herein we have a sum, then a product, and afterwards a subtraction of two subunitary sets) Or /\ can be any T-norm/-norm, and \/ any T-conorm/-conorm from the above; for example the easiest way would be to consider the min for crisp components (or inf for subset components) and respectively max for crisp components (or sup for subset components) If we have crisp numbers, we can at the end neutrosophically normalize Since the min/max (or inf/sup) operators work the best for subunitary set components, let s present their definitions below They are extensions from subunitary intervals {defined in []} to any subunitary sets Analogously we can do for all neutrosophic operators defined in [] Let x(t, I, F ) and y(t, I, F ) be in the neutrosophic set/logic M eutrosophic Conjunction/Intersection: x/\y=(t /\,I /\,F /\ ),

4 where inf T /\ = min{inf T, inf T } sup T /\ = min{sup T, sup T } inf I /\ = max{inf I, inf I } sup I /\ = max{sup I, sup I } inf F /\ = max{inf F, inf F } sup F /\ = max{sup F, sup F } eutrosophic Disjunction/Union: x\/y=(t \/,I \/,F \/ ), where inf T \/ = max{inf T, inf T } sup T \/ = max{sup T, sup T } inf I \/ = min{inf I, inf I } sup I \/ = min{sup I, sup I } inf F \/ = min{inf F, inf F } sup F \/ = min{sup F, sup F } eutrosophic egation/complement: C(x) = (T C,I C,F C ), where T C = F inf I C = -sup I sup I C = -inf I F C = T Upon the above eutrosophic Conjunction/Intersection, we can define the eutrosophic Containment: We say that the neutrosophic set A is included in the neutrosophic set B of the universe of discourse U, iff for any x(t A, I A, F A ) A with x(t B, I B, F B ) B we have: inf T A inf T B ; sup T A sup T B ; inf I A inf I B ; sup I A sup I B ; inf F A inf F B ; sup F A sup F B Remarks a) The non-standard unit interval ] - 0, + [ is merely used for philosophical applications, especially when we want to make a distinction between relative truth (truth in at least one world) and absolute truth (truth in all possible worlds), and similarly for distinction between relative or absolute falsehood, and between relative or absolute indeterminacy But, for technical applications of neutrosophic logic and set, the domain of definition and range of the - norm and -conorm can be restrained to the normal standard real unit interval [0, ], which is easier to use, therefore: and n : ( [0,] [0,] [0,] ) [0,] [0,] [0,] c : ( [0,] [0,] [0,] ) [0,] [0,] [0,] b) Since in L and S the sum of the components (in the case when T, I, F are crisp numbers, not sets) is not necessary equal to (so the normalization is not required), we can keep the final result un-normalized 4

5 But, if the normalization is needed for special applications, we can normalize at the end by dividing each component by the sum all components If we work with intuitionistic logic/set (when the information is incomplete, ie the sum of the crisp components is less than, ie sub-normalized), or with paraconsistent logic/set (when the information overlaps and it is contradictory, ie the sum of crisp components is greater than, ie over-normalized), we need to define the neutrosophic measure of a proposition/set If x(t,i,f) is a L/S, and T,I,F are crisp numbers in [0,], then the neutrosophic vector norm of variable/set x is the sum of its components: vector-norm (x) = T+I+F ow, if we apply the n and c to two propositions/sets which maybe intuitionistic or paraconsistent or normalized (ie the sum of components less than, bigger than, or equal to ), x and y, what should be the neutrosophic measure of the results n (x,y) and c (x,y)? Herein again we have more possibilities: - either the product of neutrosophic measures of x and y: vector-norm ( n (x,y)) = vector-norm (x) vector-norm (y), - or their average: vector-norm ( n (x,y)) = ( vector-norm (x) + vector-norm (y))/, - or other function of the initial neutrosophic measures: vector-norm ( n (x,y)) = f( vector-norm (x), vector-norm (y)), where f(,) is a function to be determined according to each application Similarly for vector-norm ( c (x,y)) Depending on the adopted neutrosophic vector norm, after applying each neutrosophic operator the result is neutrosophically normalized We d like to mention that neutrosophically normalizing doesn t mean that the sum of the resulting crisp components should be as in fuzzy logic/set or intuitionistic fuzzy logic/set, but the sum of the components should be as above: either equal to the product of neutrosophic vector norms of the initial propositions/sets, or equal to the neutrosophic average of the initial propositions/sets vector norms, etc In conclusion, we neutrosophically normalize the resulting crisp components T`,I`,F` by multiplying each neutrosophic component T`,I`,F` with S/( T`+I`+F`), where S= vector-norm ( n (x,y)) for a -norm or S= vector-norm ( c (x,y)) for a -conorm - as defined above c) If T, I, F are subsets of [0, ] the problem of neutrosophic normalization is more difficult i) If sup(t)+sup(i)+sup(f) <, we have an intuitionistic proposition/set ii) If inf(t)+inf(i)+inf(f) >, we have a paraconsistent proposition/set iii) If there exist the crisp numbers t T, i I, and f F such that t+i+f =, then we can say that we have a plausible normalized proposition/set But in many such cases, besides the normalized particular case showed herein, we also have crisp numbers, say t T, i I, and f F such that t +i +f < (incomplete information) and t T, i I, and f F such that t +i +f > (paraconsistent information) 4 Examples of eutrosophic Operators which are -norms or -pseudonorms or, respectively -conorms or -pseudoconorms We define a binary neutrosophic conjunction (intersection) operator, which is a particular case of a -norm (neutrosophic norm, a generalization of the fuzzy T-norm): 5

6 ([ ] [ ] [ ]) [ ] [ ] [ ] c : 0, 0, 0, 0, 0, 0, c ( x, y) = TT, II+ IT + TI, FF + FI + FT + FT + FI The neutrosophic conjunction (intersection) operator x y component truth, indeterminacy, and falsehood values result from the multiplication ( T+ I+ F) ( T + I + F) since we consider in a prudent way T I F, where is a neutrosophic relationship and means weaker, ie the products TI i j will go to I, TF i j will go to F, and IiF j will go to F for all i, j {,}, i j, while of course the product T T will go to T, I I will go to I, and F F will go to F (or reciprocally we can say that F prevails in front of I which prevails in front of T, and this neutrosophic relationship is transitive): (T I F ) (T I F ) (T I F ) (T I F ) (T I F ) (T I F ) So, the truth value is TT, the indeterminacy value is I I + IT + TI and the false value is FF + FI + FT + FT + FI The norm of x y is ( T+ I+ F) ( T+ I+ F) Thus, if x and y are normalized, then x y is also normalized Of course, the reader can redefine the neutrosophic conjunction operator, depending on application, in a different way, for example in a more optimistic way, ie I T F or T prevails with respect to I, then we get: ITF c ( x, y) = ( TT + TI + TI, II, FF + FI + FT + FT + FI) Or, the reader can consider the order T F I, etc Let s also define the unary neutrosophic negation operator: n :[ 0,] [ 0,] [ 0,] [ 0,] [ 0,] [ 0,] n ( T, I, F) = ( F, I, T) by interchanging the truth T and falsehood F vector components Similarly, we now define a binary neutrosophic disjunction (or union) operator, where we consider the neutrosophic relationship F I T : FIT d : 0, 0, 0, 0, 0, 0, FIT ([ ] [ ] [ ]) [ ] [ ] [ ] = ( ) d ( x, y) TT TI T I TF T F, I F I F I I, FF We consider as neutrosophic norm of the neutrosophic variable x, where L( x) = T+ I+ F, the sum of its components: T + I + F, which in many cases is, but can also be positive < or > 6

7 Or, the reader can consider the order F T I, in a pessimistic way, ie focusing on indeterminacy I which prevails in front of the truth T, or other neutrosophic order of the neutrosophic components T,I,F depending on the application Therefore, FTI d ( x, y) = TT + TF + T F, I F + I F + I I + TI + T I, FF 4 eutrophic Composition k-law ow, we define a more general neutrosophic composition law, named k-law, in order to be able to define neutrosophic k-conjunction/intersection and neutrosophic k-disjunction/union for k variables, where k is an integer Let s consider i i i i denote T = ( T,, Tk ) I = ( I,, Ik ) F = ( F,, Fk ) We now define a neutrosophic composition law o in the following way: o : T, I, F 0, If z { T, I, F} then If zw, {,, } then o k neutrosophic variables, x ( T, I, F ), for all i {,,, k} k = i i= z z z { } [ ] o o i i j + j r= { i,, ir, jr+,, jk} {,,, k} ( i,, i ) r r C (,,, k) k r ( j,, j ) C (,,, k) r+ k z w= w z = z z w w r r k r where C (,,, k ) means the set of combinations of the elements { } k r [Similarly for C (,,, k) ] k Let s,,, k taken by r In other words, zo w is the sum of all possible products of the components of vectors z and w, such that each product has at least a z i factor and at least a w j factor, and each product has exactly k factors where each factor is a different vector component of z or of w Similarly if we multiply three vectors: k T I F = T I F F o o i i l l u+ u+ v ++ uvk,, u v= { i,, iu, ju+,, ju+ v, lu+ v+,, lk} {,,, k} u ( i,, iu) C (,,, k)(, ju+,, j u+ v) v k u v C (,,, k)(, l,, l ) C (,,, k) u j j u v k Let s see an example for k = u++ v k 7

8 (,, ) (,, ) (,, ) x T I F x T I F x T I F T T = TTT, I I = I I I, F F = FF F o o o T I= TII + ITI + IIT+ TTI + TIT+ ITT o o T F= TFF+ FTF+ FFT+ TTF+ TFT+ FTT o I F= IFF+ FIF+ FFI + IIF+ IFI + FI I To Io F = TI F+ TF I+ IT F+ IFT + FI T+ FT I For the case when indeterminacy I is not decomposed in subcomponents {as for example I = P U where P =paradox (true and false simultaneously) and U =uncertainty (true or false, not sure which one)}, the previous formulas can be easily written using only three components as: T I F = TI F o o i j r i, j, r P (,,) where P (,,) means the set of permutations of (,, ) ie {(,,),(,,),(,,),(,,,),(,,),(,,) } z w= zw w + wz z o i j jr i j r i= (, i j, r) (,,) ( jr, ) P (,,) This neurotrophic law is associative and commutative 4 eutrophic Logic and Set k-operators Let s consider the neutrophic logic crispy values of variables x, yz, (so, for k = ): L( x) = T, I, F with 0 T, I, F L( y) = T, I, F with 0 T, I, F L( z) = T, I, F with 0 T, I, F In neutrosophic logic it is not necessary to have the sum of components equals to, as in intuitionist fuzzy logic, ie Tk + Ik + Fk is not necessary, for k As a particular case, we define the tri-nary conjunction neutrosophic operator: c : 0, 0, 0, 0, 0, 0, ([ ] [ ] [ ]) [ ] [ ] [ ] ( o o o o o o ) c ( x, y, z) = T T, I I + I T, F F + F I + F T If all x, y, z are normalized, then If x, y, or y are non-normalized, then norm of w c (,, ) (,, ) x y z is also normalized c x y z = x y z, where w means 8

9 c is a --norm (neutrosophic norm, ie generalization of the fuzzy T-norm) Again, as a particular case, we define the unary negation neutrosophic operator: n : 0, 0, 0, 0, 0, 0, [ ] [ ] [ ] [ ] [ ] [ ] = = n ( x) n T, I, F F, I, T Let s consider the vectors: T I F T= T, I= I and F= F T I F F T T We note T= x T F, T= y F, T= z T, T = xy F T, etc T F T and similarly T F T F= x F, Fy = T, F = xz F, etc F F T For shorter and easier notations let s denote for the vector neutrosophic law defined previously zo w= zw and respectively z w v= zwv Then the neutrosophic tri-nary conjunction/intersection of neutrosophic variables x, y, and z is: c ( x, y, z) = ( TT, II + IT, FF + FI + FT + FIT ) = = ( TTT, III + IIT + ITI + TII + ITT + TIT + TTI, FFF + FFI + FI F+ IFF + FI I+ IFI + IIF+ + FFT+ FTF+ TFF+ FTT+ TFT+ TTF+ z is: + TI F+ TFI + IFT + ITF + FI T+ FT I) Similarly, the neutrosophic tri-nary disjunction/union of neutrosophic variables x, y, and FIT d ( x, y, z) = TT + TI + TF +, II + IF, FF = (T T T + T T I + T I T + I T T + T I I + I T I + I I T + T T F + T F T + F T T + T F F + F T F + F F T + T I F + T F I + I F T + I T F + F I T + F T I, I I I + I I F + I F I + F I I + I F F + F I F + F F I, F F F ) Surely, other neutrosophic orders can be used for tri-nary conjunctions/intersections and respectively for tri-nary disjunctions/unions among the componenets T, I, F o o 5 eutrosophic Topologies 9

10 A) General Definition of T: Let M be a non-empty set Let x(t A, I A, F A ) A with x(t B, I B, F B ) B be in the neutrosophic set/logic M, where A and B are subsets of M Then (see Section 9 about -norms / -conorms and examples): A B = {x M, x(t A \/T B, I A /\I B, F A /\F B )}, A B = {x M, x(t A /\T B, I A \/I B, F A \/F B )}, C(A) = {x M, x(f A, I A, T A )} A General eutrosophic Topology on the non-empty set M is a family η of eutrosophic Sets in M satisfying the following axioms: 0(0,0,) and (,0,0) η ; If A, B η, then A B η ; If the family {A k, k K} η, then B) An alternative version of T U A k η kk -We cal also construct a eutrosophic Topology on T = ] - 0, + [, considering the associated family of standard or non-standard subsets included in T, and the empty set, called open sets, which is closed under set union and finite intersection Let A, B be two such subsets The union is defined as: A B = A+B-A B, and the intersection as: A B = A B The complement of A, C(A) = { + }-A, which is a closed set {When a non-standard number occurs at an extremity of an internal, one can write ] instead of ( and [ instead of ) } The interval T, endowed with this topology, forms a neutrosophic topological space In this example we have used the Algebraic Product -norm/-conorm But other eutrosophic Topologies can be defined by using various -norm/-conorm operators In the above defined topologies, if all x's are paraconsistent or respectively intuitionistic, then one has a eutrosophic Paraconsistent Topology, respectively eutrosophic Intuitionistic Topology References: [] F Smarandache & J Dezert, Advances and Applications of DSmt for Information Fusion, Am Res Press, 004 [] F Smarandache, A unifying field in logics: eutrosophic Logic, eutrosophy, eutrosophic Set, eutrosophic Probability and Statistics, 998, 00, 00, 005 [] H Wang, F Smarandache, Y-Q Zhang, R Sunderraman, Interval eutrosophic Set and Logic: Theory and Applications in Computing, Hexs, 005 [4] L Zadeh, Fuzzy Sets, Information and Control, Vol 8, 8-5, 965 0

11 [5] Driankov, Dimiter; Hellendoorn, Hans; and Reinfrank, Michael, An Introduction to Fuzzy Control Springer, Berlin/Heidelberg, 99 [6] K Atanassov, D Stoyanova, Remarks on the Intuitionistic Fuzzy Sets II, otes on Intuitionistic Fuzzy Sets, Vol, o, 85 86, 995 [7] Coker, D, An Introduction to Intuitionistic Fuzzy Topological Spaces, Fuzzy Sets and Systems, Vol 88, 8-89, 997 [Published in the author s book: A Unifying Field in Logics: eutrosophic Logic eutrosophy, eutrosophic Set, eutrosophic Probability (fourth edition), Am Res Press, Rehoboth, 005]