Context free languages are not closed under complementation. This follows from their property of non-closure under intersection:
If CFLs were closed under complementation, then they must have also been closed under intersection, which is not the case. Now as we know, CFLs are not closed under intersection and complementation, can’t we use the following arguments to say CFLs are not closed under union, too!
In RHS, we are performing two operations viz. intersection and complementation, none of which guarantees the closure of the given CFLs. So, how does result of these two operations claims to remain a CFL? Doesn’t this prove that the result is not necessarily a CFL?
Yet, it’s known that CFLs are closed under union. Where is the flaw in the argument? Where am I wrong?