
$$\begin{array}{l}{{c_{{}_{N}}^{T F}:\left([0,1]\times[0,1]\times[0,1]\right)^{-}\rightarrow[0,1]\times[0,1]\times[0,1]}}\\ {{c_{{}_{N}}^{T F}\left(x,y\right)=\left(T_{1}T_{2},I_{1}I_{2}+I_{1}T_{2}+T_{1}I_{2},F_{1}F_{2}+F_{1}I_{2}+F_{1}T_{2}+F_{2}T_{1}+F_{2}I_{1}\right).}}\end{array}$$

The neutrosophic conjunction (intersection) operator N x ∧ y component truth, indeterminacy, and falsehood values result from the multiplication 
( )( ) *TIF TI F* 11 1 2 2 2 ++ ⋅ ++
since we consider in a prudent way TIF ≺ ≺ , where "≺" is a **neutrosophic relationship** and means "weaker", i.e. the products *T Ii j* will go to I , *T Fi j* will go to F , and i j I F will go to F for all i, j ∈{1,2}, i ≠ j, while of course the product T1T2 will go to T, I1I2 will go to I, and F1F2 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): 

$\begin{array}{ccccccccc}(\text{T}_{1}&\text{I}_{1}&\text{F}_{1})&(\text{T}_{1}&\text{I}_{1}&\text{F}_{1})&(\text{T}_{1}&\text{I}_{1}\\ \text{T}_{2}&\text{I}_{2}&\text{F}_{2})&(\text{T}_{2}&\text{I}_{2}&\text{F}_{2})&(\text{T}_{2}&\text{I}_{2}\\ \end{array}$

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

So, the truth value is TT1 2 , the indeterminacy value is 12 12 12 I *I IT TI* + + and the false value is FF FI FT FT F I 1 2 12 12 21 21 ++++ . The norm of x ∧ Ny is ( )( ) *TI FT I F* 11 1 2 2 2 ++ ⋅ ++ . Thus, if x and y are normalized, then N 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, i.e. I ≺ ≺T F or T prevails with respect to I , then we get: 
( ) 12 12 21 12 1 2 12 12 21 21 (, ) , , N
ITF *c x y TT TI T I I I FF FI FT FT F I* = ++ ++ + + . 

Or, the reader can consider the order TFI ≺ ≺ , etc. 

the the unary neutrosophic negation operator:  $n_{N}:[0,1]\times[0,1]\times[0,1]\to[0,1]\times[0,1]\times[0,1]$  $n_{N}\left(T,I,F\right)=\left(F,I,T\right)$  **with $T$ and $\text{fakelhood}F$** vector components
 Let's also define the unary **neutrosophic negation** operator: 
 : 0,1 0,1 0,1 0,1 0,1 0,1 [ ][ ][ ] [ ][ ][ ] N n ××→ ××
 ( )( ) ,, ,, N *n TIF FIT* = 
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 : 

$$d_{s}^{F T}:\left([0,1]\times[0,1]\times[0,1]\right)^{2}\rightarrow[0,1]\times[0,1]\times[0,1]$$ $$d_{s}^{F T}(x,y)=\left(T_{1}T_{2}+T_{1}I_{2}+T_{2}I_{1}+T_{1}F_{2}+T_{2}F_{1},I_{1}F_{2}+I_{2}F_{1}+I_{1}I_{2},F_{1}F_{2}\right)$$

 We consider as neutrosophic norm of the neutrosophic variable x , where 11 1 *NL x T I F* ( ) =++ , the sum of its components: TIF 11 1 + + , which in many cases is 1, but can also be positive <1 or >1. 