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This question already has an answer here:

Are empty-sets equal to themselves? Is there one universally acepted empty-sets? Or many? And to stipulate on this, are there any axioms by which we can follow such a premise if one is put forward or am I to assert three potential outcomes? That empty sets are equal to themselves, are not equal, or are equal given certain parameters

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marked as duplicate by Daniel W. Farlow, Nate Eldredge, user147263, Joel Reyes Noche, John Gowers Mar 16 '15 at 0:44

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    $\begingroup$ There is only one empty set. And of course it is equal to itself. $\endgroup$ – TonyK Mar 15 '15 at 21:01
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    $\begingroup$ What does that have to do with the title? (Talk about a bait and switch...) $\endgroup$ – Asaf Karagila Mar 15 '15 at 21:07
  • $\begingroup$ If you're saying that positing the existence of multiple empty sets can get you out of Russell's paradox, maybe you should explain how that works. ${}\qquad{}$ $\endgroup$ – Michael Hardy Mar 15 '15 at 21:14
  • $\begingroup$ If there was only one empty set and it was equal to itself then the empty set of any one thing would be equal to the set of all empty sets, a problem? $\endgroup$ – Edward Baker Mar 15 '15 at 21:23
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    $\begingroup$ But the set of all empty sets wouldn't be empty, as there is an empty set. So the empty set does not equal the set of empty sets. Also, I don't know what you mean by “the empty set of any one thing”. $\endgroup$ – Joffysloffy Mar 15 '15 at 21:31
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By the Axiom of Extensionality two sets are equal if they contain the same elements. Since any two empty sets contain the same elements, they are equal. Namely, suppose $x$ and $y$ are empty, then the sentence $$\forall z(z\in x\leftrightarrow z\in y)$$ is true (the implications are vacuously true, because there are no elements in $x$ and $y$). Hence $x=y$.

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Sets $A$ and $B$ are equal if every element $a$ of $A$ is also an element of $B$ and every element $b$ of $B$ is also an element of $A$.

If $A$ is empty then the first half of this statement is true, no matter what $B$ is. If $B$ is also empty then the second half of the statement is true. So if each is empty then they are equal to each other. So any empty set is equal to any other empty set.

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