# Nc: ( ]-0,1+[ × ]-0,1+[ × ]-0,1+[ )2 → ]-0,1+[ × ]-0,1+[ × ]-0,1+[

 Nc (x(T1,I1,F1), y(T2,I2,F2)) = (NcT(x,y), NcI(x,y), NcF(x,y)), 
where NnT(.,.), NnI(.,.), NnF(.,.) are the truth/membership, indeterminacy, and respectively falsehood/nonmembership components. 

Nc 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: Nc(x, 1) = 1, Nc(x, 0) = x. b) Commutativity: Nc (x, y) = Nc(y, x). c) Monotonicity: if x ≤ y, then Nc(x, z) ≤ Nc(y, z). 

d) Associativity: Nc (Nc(x, y), z) = Nc(x, Nc(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 **N-pseudo-conorms**, which still give good results in practice. 

Nc 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 N-conorms, as in fuzzy logic and set the T-conorms, are: 
- The Algebraic Product N-conorm: Nc−algebraicJ(x, y) = x + y − x · y - The Bounded N-conorm: Nc−boundedJ(x, y) = min{1, x + y} 
- The Default (max) N-conorm: Nc−maxJ(x, y) = max{x, y}. 

A general example of N-conorm would be this. 

Let x(T1, I1, F1) and y(T2, I2, F2) be in the neutrosophic set/logic M. Then: 
Nn(x, y) = (T1\/T2, I1/\I2, F1/\F2) 
Where - as above - the "/\" operator, acting on two (standard or non-standard) subunitary sets, is a Nnorm (verifying the above N-norms axioms); while the "\/" operator, also acting on two (standard or nonstandard) subunitary sets, is a N-conorm (verifying the above N-conorms axioms). 

For example, /\ can be the Algebraic Product T-norm/N-norm, so T1/\T2 = T1·T2 (herein we have a product of two subunitary sets); and \/ can be the Algebraic Product T-conorm/N-conorm, so T1\/T2 = T1+T2-T1·T2 (herein we have a sum, then a product, and afterwards a subtraction of two subunitary sets). 

 Or /\ can be any T-norm/N-norm, and \/ any T-conorm/N-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 [3]} to any subunitary sets. Analogously we can do for all neutrosophic operators defined in [3]. 

Let x(T1, I1, F1) and y(T2, I2, F2) be in the neutrosophic set/logic M. 

## Neutrosophic Conjunction/Intersection: X/\Y=(T/\,I/\,F/\),
