 ( 1 23 123 123 123 123 123 123 123 = ++++++ *T TT I I I I I T IT I TI I ITT TI T TT I* , ,
 *FFF FFI FI F I FF FI I I FI I I F* 1 2 3 1 23 12 3 1 2 3 123 1 23 12 3 + + + ++++
 ++ + ++++ *FFT FT F TF F FT T TFT TT F* 1 23 12 3 1 2 3 123 1 23 12 3
 ++ + + + + *TI F TF I I FT I T F FI T FT I* 12 3 1 23 1 23 12 3 123 12 3 )
NL P c x y n z T T II IT F F F I F T F IT () ( ) 12 , , ( ) , , = = + ++ + *N N zz z z z z zz z z* ( )
NL P c x n y z T T II IT F F F I F T F IT () ( ) 13 , ( ), , , = = + ++ + *N N yy y y y y yy y y* ( )
NL P c n x y z T T II IT F F F I F T F IT () ( ) 23 ( ), , , , = = + ++ + *N N xx x x x x xx x y* ( ) NL P c x n y n z T T II IT F F F I F T F IT () ( ) 1 , ( ), ( ) , , = = + ++ + *N N N yz yz yz yz yz yz yz yz yz yz* ( )
NL P c n x y n z T T II IT F F F I F T F IT () ( ) 2 ( ), , ( ) , , = = + ++ + *N N N xz xz xz xz xz xz xz xz xz xz* ( )
NL P c n x n y n z T T II IT F F F I F T F IT () ( ) 0 ( ), ( ), ( ) , , = = + ++ + = *N N N N xyz xyz xyz xyz xyz xyz xyz xyz xyz xyz* ( )
= + ++ + ( ) *FF II IF TT TI TF TIF* , , . 

## N-Ary Fuzzy Logic And Neutrosophic Logic Operators

 We can generalize for any integer n ≥ 2 . 

 The Venn Diagram has 2 2 n disjoint parts. Each part has the form 1 1 ... ... Pi i j j *kk n* + , where 
0≤ ≤k n , and of course 0 ≤−≤ *nk n* ; { } 1,..., k i i is a combination of k elements of the set 
{ } 1, 2,..., n , while { } 1,..., k n j j + the n k − elements left, i.e. { } *jkk k* +1 1 ,..., 1, 2,..., \ ,..., *j ni i* = { }{ } . 
{ } 1,..., k i i are replaced by the corresponding numbers from { } 1, 2,..., n , while { } 1,..., k n j j + are 
replaced by blanks. 
 For example, when n = 3 , 
$P_{i_{1}}i_{2}J_{3}=P13$ in $\{i_{1},i_{2}\}=\{1,3\}$,  $P_{i_{1}}j_{2}j_{3}=P1$ if $\{i_{1}\}=\{1\}$.  zy logic we have:  $P_{i_{1}}...i_{k}j_{k+1}...j_{n}=c_{F}\left(x_{i_{1}},...,x_{i_{k}},n_{F}\left(x_{j_{k+1}}\right),...,n_{F}\left(x_{j_{n}}\right)\right)$
 Hence, for fuzzy logic we have: 
 ( ) ( ) ( ) 1 1 1 1 ... ... ,..., , ,..., kk n Pi i j j c x x n x n x *kk n F i i F j F j* + + =
whence 
$$FL\big{(}P\dot{i}_{1}...\dot{i}_{k}\dot{j}_{k+1}...\dot{j}_{n}\big{)}=\left(\left(\prod_{r=1}^{k}t_{i_{r}}\right)\left(\prod_{s=k+1}^{n}\left(1-t_{j_{s}}\right)\right),\varphi\big{(}f_{1}f_{2},...,f_{n}\big{)}\right)$$  $$1\big{]}\,,$$
$${\mathrm{where~}}\varphi:\left[0,1\right]^{n}\to\left[0,1\right],$$
$$\varphi(\alpha_{1},\alpha_{2},...,\alpha_{n})=S_{1}-S_{2}+S_{3}+...+(-1)^{n+1}S_{n}=\sum_{l=1}^{n}{(-1)^{l+1}S_{l}}$$
where 