Sets of subsets
Task number: 3346
Is it true that for every two sets \(X\) and \(Y\), \(2^X = 2^Y\) exactly when \(X = Y\)?
What happens if we take the union of all sets in \(2^X\)?
We will prove the implication in both directions. We can obtain the implication from right to left by merely substituting \(X\) for \(Y\) in the exponent: \(2^X = 2^Y\).
In the other direction we must first realize that \(X =\bigcup 2^X\), because the power set \(2^X\) contains \(X\) as one of its subsets and the union with the other subsets does not add any elements. Now we easily obtain \(X =\bigcup 2^X =\bigcup 2^Y = Y\).
The equivalence \(2^X = 2^Y \Longleftrightarrow X = Y\) holds.