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Is ∅ equivalent to {∅}? I think they are, but I am not sure? If anyone could clarify, that would be great. Thank you!

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    $\begingroup$ "Equivalent" meaning here...? $\endgroup$ – Timbuc Sep 22 '14 at 3:39
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    $\begingroup$ I'm going to have to say no.. for one thing the cardinalities of the two sets are not equal. $\endgroup$ – graydad Sep 22 '14 at 3:40
  • $\begingroup$ See here, here and here $\endgroup$ – Joel Bosveld Sep 22 '14 at 3:46
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    $\begingroup$ I think this question has been asked at least a thousand times before. $\endgroup$ – Asaf Karagila Sep 22 '14 at 3:53
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    $\begingroup$ Absolutely not! The set $\{\emptyset\}$ contains $\emptyset$, whereas $\emptyset$ contains nothing. Edit: @JearsonNarvaezRojas beat me to the punch. $\endgroup$ – Robin Goodfellow Sep 22 '14 at 4:01
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No. $\emptyset$ has zero elements, and $\{\emptyset\}$ has one element.

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It seems that you are confused about notation. Two common ways to denote the empty set are $\emptyset$ and $\{\}$. Using alternate notation the question becomes is $\{\}$ equivalent to $\{\emptyset\}$. Note that $\{\emptyset\}$ is a set of sets. It is a set containing a single element, and that element is the empty set. On the other hand, the empty set is a set containing zero elements.

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The set of an empty set is not an empty set. $\qquad\{\{\}\}\not\equiv\{\}$

It has an element, even if that element is itself a set with no elements.

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No! they are not the same. Simple answer: The cardinality of a set containing an empty set is 1, whilst the empty set itself is 0.

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