# Is ∅ equivalent to {∅}?

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

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

No. $\emptyset$ has zero elements, and $\{\emptyset\}$ has one element.

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.

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.

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.