# Are two identical sets each other's subsets / supersets?

Given two sets:

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

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

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

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Every set is a subset of itself. – S. Snape 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. – S. Snape 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

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 $B$.
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 $b \subset a$.