Isomorphism as an Equivalence Relation on Sets: A Proof

[yaganon]

So it says here "Let S be a set of sets. Show that isomorphism is an equivalence relation on S."

So in order to approach this proof, can I just use the Reflexive, Symmetrical, and Transitive properties that is basically the definition of equivalence relations?

eg. suppose x, y, z are sets contained in S...

Reflexive: x~x <same as> xRx
Symmetry: x~y => y~x <same as> xRy => yRx
Transitive: x~y, y~z => x~z

This seems too elementary, but I just want to make sure.

 

[mrbohn1]

Yes - showing something is an equivalence relation always involves the same steps. Sets are isomorphic if they have the same cardinality, so you need to show that the property of "having the same number of elements" satisfies the three conditions.

 

[yaganon]

I'm also confused about the X! notation. I think X^x means the set of all selfmap functions on X. What is X! suppose to be?