-2
$\begingroup$

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

$\endgroup$
  • 3
    $\begingroup$ "Equivalent" meaning here...? $\endgroup$ – Timbuc Sep 22 '14 at 3:39
  • 1
    $\begingroup$ 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
  • 3
    $\begingroup$ I think this question has been asked at least a thousand times before. $\endgroup$ – Asaf Karagila Sep 22 '14 at 3:53
  • 1
    $\begingroup$ 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
8
$\begingroup$

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

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.