# Subtraction of Power Sets

I have two quick questions:

With P being the power set,

P(~A) = P(U) - P(A) and
P(A-B) = P(A) - P(B)

I'm told if it's true to prove it, and if false to give a counterexample.

To be they're both false, since the null set is part of any power set, the subtraction of two power sets would get rid of the null set and the result could never be another power set. Am I wrong in assuming this?

ShayanJ
Gold Member
Seems correct to me. So just any set with two of its non-disjoint subsets will be a counter example. Try it then!

Svein
To be they're both false, since the null set is part of any power set, the subtraction of two power sets would get rid of the null set and the result could never be another power set. Am I wrong in assuming this?

Yes, you are wrong. The null set is by definition part of any set, so you cannot get rid of it.

Stephen Tashi
The null set is by definition part of any set, so you cannot get rid of it.

The phrase "part of" is ambiguous. The null set is a subset of any given set but it need not be an element of a given set. The exercise in this thread depends on whether the null set is an element of the various sets.

Svein