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Assume a set that has no elements, we consider it an empty set.

Logical Questions

Is an empty set a subset of an empty set ?

If yes, does it mean the cardinality or size of the set is 0 or 1 ?

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  • What is your definition of subset? Does it apply? Why should the cardinality be $1$? – martin.koeberl Apr 06 '17 at 02:11
  • Let $x\in\varnothing$, is it then true that $x\in\varnothing$? (If this doesn't make sense you should read https://en.wikipedia.org/wiki/Vacuous_truth) –  Apr 06 '17 at 02:13
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    Having something as a subset is different than having something as an element. The empty set does indeed have the empty set as a subset despite not having any elements. – JMoravitz Apr 06 '17 at 02:13
  • I guess I was thinking an empty set was also an element of an empty set which is not true. As you guys said, an empty set can be the subset of an empty set, but the cardinality is 0, e.g. | S | = 0 . Am I right ? – Agent 0 Apr 06 '17 at 02:16
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    There is only one empty set, and like every set, it is a subset of itself. The cardinality of the empty set is $0$ since it has no elements. However, the fact that the empty set is a subset of itself does not "mean" that its cardinality is $0$; like I said, every set, regardless of cardinality, is a subset of itself. – bof Apr 06 '17 at 02:59
  • yes only the number of elements are associated with the cardinality of a set. – Agent 0 Apr 06 '17 at 16:52

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