# Do two empty sets have any elements in common?

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: $A=\{\;\}$, $B=\{\;\}$.

• What do you mean by "two" empty sets?
– user223391
Commented Sep 26, 2016 at 0:22
• They share none and all of their elements :). Commented Sep 26, 2016 at 0:40
• There is only one empty set. Commented Sep 26, 2016 at 0:54
• That's just two names for the empty set. Commented Sep 26, 2016 at 7:54
• $A=\{\}$, $B=\{\}$, and $\varnothing=\{\}$. Two empty sets do not have any element in common. They do not have any element with any property whatsoever, as you observe.
– MPW
Commented Sep 26, 2016 at 12:26

You are right. In particular, $\emptyset$ is not a common element, but rather a common subset. That is: $\emptyset$ has no elements, but is indeed a subset of itself (and of every other set!).

• I chose this answer because it was most clear for me to understand. Thank you for all answers. Commented Sep 26, 2016 at 0:43

Your argument looks good, $\varnothing\cap \varnothing=\varnothing$. In fact $\varnothing \cap X=\varnothing$ for any set $X$ !

• ∅∩X=∅ does this mean that there is nothing in common? Thanks for your reply. Commented Sep 26, 2016 at 0:26
• @pigletwithcurls Yes, that is what it means. Commented Sep 26, 2016 at 0:36

I think that two empty sets do not have any element in common since they do not have any elements in the first place.

Exactly.

Should I count $\varnothing$ as a common element?

No, because $\varnothing$ is not an element of neither of the sets. It only is an element of their power set (i.e. a common subset), but the empty set is not an element of the empty set itself and therefore not a common element of the intersection between two empty sets.

$A$ 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.