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: $A=\{\;\}$, $B=\{\;\}$.
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5$\begingroup$ What do you mean by "two" empty sets? $\endgroup$– user223391Commented Sep 26, 2016 at 0:22
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10$\begingroup$ They share none and all of their elements :). $\endgroup$– YoTengoUnLCDCommented Sep 26, 2016 at 0:40
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1$\begingroup$ There is only one empty set. $\endgroup$– Rene SchipperusCommented Sep 26, 2016 at 0:54
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1$\begingroup$ That's just two names for the empty set. $\endgroup$– BakuriuCommented Sep 26, 2016 at 7:54
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$\begingroup$ $A=\{\}$, $B=\{\}$, 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. $\endgroup$– MPWCommented Sep 26, 2016 at 12:26
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4 Answers
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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!).
answered Sep 26, 2016 at 0:33
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$\begingroup$ I chose this answer because it was most clear for me to understand. Thank you for all answers. $\endgroup$ Commented Sep 26, 2016 at 0:43
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Your argument looks good, $\varnothing\cap \varnothing=\varnothing$. In fact $\varnothing \cap X=\varnothing$ for any set $X$ !
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$\begingroup$ ∅∩X=∅ does this mean that there is nothing in common? Thanks for your reply. $\endgroup$ Commented Sep 26, 2016 at 0:26
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1$\begingroup$ @pigletwithcurls Yes, that is what it means. $\endgroup$ Commented Sep 26, 2016 at 0:36
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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.
answered Sep 26, 2016 at 0:35
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$A$ 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.
answered Sep 26, 2016 at 17:21