as you should see, S must be in the power set.. hence it should be the set {5}.. also, note that the function gives you an element of the power set..

my proof might be completely wrong because i proved it the way i know when proving something like this..

proof:

let and suppose , where is the power set of U, and defined by , is one-one..

Let .. since is one-one for , then ..

by definition, , hence and also ..

since was arbitrary and we started with or , then .

Let A=U and suppose for , .. clearly, ..

therefore, is one-one.. QED