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Given two sets:

$a = \{1,2,3\} $


Are they supersets of each other? Are they subsets of each other?

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Every set is a subset of itself. – Babak S. Oct 22 '12 at 6:14
@BabakSorouh and a superset of itself? – jsj Oct 22 '12 at 6:15
Yes! They are superset of itself. – Babak S. Oct 22 '12 at 6:16
Yes, subset and superset. But neither proper subset nor proper superset. – Hagen von Eitzen Oct 22 '12 at 6:16
up vote 2 down vote accepted

What is the definition of subset? Of superset? It shouldn't be difficult to prove that a set is a subset and superset of itself. Actually, two sets $A$ and $B$ are identical if and only if $A$ is both a subset and superset of $B$.

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As others have pointed out, it is a quite trivial consequence of the definition of the notion of a subset that if $a = b$ then $a \subset b$ and $b \subset a$.

But also do note that it is a bit unhappy to say 'given two sets' here when you've given two different labels to one and the same set!

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