TITLE: Do two empty sets have any elements in common?
QUESTION [6 upvotes]: I think that two empty sets do not have any element in common since they do not have any elements in the first place. 
Should I count $\emptyset$ as a common element?
Edited: two empty sets as: $A=\{\;\}$, $B=\{\;\}$.

REPLY [2 votes]: $A$ and $B$ do not have any elements in common.
$$
  \not\exists x: x\in A \wedge x\in B
$$
However, $A$ does have all its elements in common with $B$ and vice versa!
$$
  \forall x \in A: x\in B
$$
This is of course a vacuous truth.