Question of DFA closure properties?
Does closure under intersection and complementation imply closure under union?


This is immediate from De Morgan's laws.


  Does closure under intersection and union imply closure under complementation?


This one is slightly trickier. The answer is no.

Both intersection and union are what is called "monotone". That means that, if you make $X$ or&nbsp;$Y$ bigger then $X\cup Y$ and $X\cap Y$ can't get smaller. By induction, you can prove that any operator defined using only intersections and unions must also be monotone. But complement is not monotone: when you make $X$ bigger, its complement definitely gets smaller. So complement cannot be defined using only intersection and union.