Show that for any set X its power set PXis not emptySolution

Show that for any set X, its power set P(X)is not empty.

Solution

We have to prove that power set of any set is not empty.

Let x be any set.

X can be a null set or contains some elements

If X is a null set, then again phi, a null set is a subset of X HenceP(x) contains null set.

So even if X is a null set P(X) is not empty

If  X is not a null set, then X contains atleast one element a.

The set {a}, {} are elements of power set P(X)

Hence whether X is a null set or contains any element, P(X) cannot be empty