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So here is the question I'm working on

So obviously I need to prove two things, that its both injective and surjective, however I'm trying to show it is injective and am currently stuck, here is what I have

As you can see, double containment to show an arbitrary x is in both little a and little b is possible, but I have no clue how to do that.

Any ideas?

Thanks!

k9b
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Just prove that the function is an involution, that is $f\circ f = Id_{\wp(X)}$. It will follow that it's injective (otherwise it would not be invertible) and surjective (otherwise it wouldn't be the inverse of another function).

drhab
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  • Oh brainwave! Thanks a ton. Good "trick" here to prove involution + surjective instead of injective surjective. – k9b Nov 20 '14 at 16:10