# Why $A \cap B = \widehat{\widehat{A}\cup \widehat{B}}$ does not holds for the class of Recursively Enumerable Languages?

"The class of Recursively Enumerable Languages is closed under Union, and Intersection but they are not closed under Complement."

I know why they are not closed under Complement & why they are closed under Union & Intersection but for other classes of languages we use a reasoning for closures based on the De'Morgan Law:

$A \cap B = \widehat{\widehat{A}\cup \widehat{B}}$

That is,
If some language class is closed under any two of the three operations namely, Union, Intersection & Complement then it must be closed under the third.
So what's wrong with the class of Recursively Enumerable Languages?
are not they sets?
Or am I using implication incorrectly?

• Finite subsets of $\mathbb{N}$ are closed under intersection and union, but not under complement. Jan 9, 2016 at 9:07