

![0_image_0.png](0_image_0.png)

$$S_{n}=\alpha_{1}\!\cdot\!\alpha_{2}\!\cdot\!\cdots\!\cdot\!\alpha_{n}$$

And for neutrosophic logic we have: 

$$\begin{array}{l}{{\stackrel{..}{\ldots}}}\\ {{\stackrel{..}{\ldots}}}\end{array}$$  logic we have:  $$\begin{array}{l}{\stackrel{..}{\ldots}}\\ {\stackrel{..}{\ldots}}\end{array}$$  $$\begin{array}{l}{\stackrel{..}{\ldots}}\\ {\stackrel{..}{\ldots}}\end{array}$$  $$\begin{array}{l}{\stackrel{..}{\ldots}}\\ {\stackrel{..}{\ldots}}\end{array}$$  $$\begin{array}{l}{\stackrel{..}{\ldots}}\\ {\stackrel{..}{\ldots}}\end{array}$$  $$\begin{array}{l}{\stackrel{..}{\ldots}}\\ {\stackrel{..}{\ldots}}\end{array}$$  \[\begin{array}{l}{\stackrel{..}{\ldots}}\\ {\stackrel{..}{\ldots}}\end{array}  
whence: 
$$\mathrm{where}$$
$$N L\left(P i_{1}...i_{k}j_{k+1}...j_{n}\right)=\left(T_{12\dots n},I_{12\dots n},F_{12\dots n}\right),$$
1 1 12... ... ...1 1 j jj j r s n n k k k n n x xx x i j r sk TTT T F + + = =+ ⎛ ⎞ = =⋅ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎝ ⎠ ∏ ∏ .  1 12... ... j jn k n xx I II IT + = + ,  1 1 1 11 1 1 12... ... ... ... ... ... ... ... j j j j j j j jj j j j j j n n n nn n n k k k kk k k F F F F I F T F IT n x x x x x x x xx x x x x x + + + ++ + + = ++ +

## Conclusion:

 A generalization of Knuth's Boolean binary operations is presented in this paper, i.e. we present n-ary Fuzzy Logic Operators and Neutrosophic Logic Operators based on Smarandache's codification of the Venn Diagram and on a defined vector neutrosophic law which helps in calculating fuzzy and neutrosophic operators. Better neutrosophic operators than in [2] are proposed herein. 

## References:

15
[1] F. Smarandache & J. Dezert, Advances and Applications of DSmt for Information Fusion, Am. Res. Press, 2004. 

[2] F. Smarandache, A unifying field in logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, 1998, 2001, 2003. 

[3] Knuth, Art of Computer Programming, The, Volumes 1-3 Boxed Set (2nd Edition), Addison-Wesley Professional, 1998. 

[4] Zadeh, Fuzzy Sets, Information and Control, Vol. 8, 338-353, 1965. [5] Driankov, Dimiter; Hellendoorn, Hans; and Reinfrank, Michael: *An Introduction* to Fuzzy Control. Springer, Berlin/Heidelberg, 1993. 

[6] Atanassov, K., Stoyanova, D., Remarks on the intuitionistic fuzzy sets. II, Notes on Intuitionistic Fuzzy Sets, Vol. 1, No. 2, 85 - 86, 1995. 