# Subtraction of Power Sets

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1. Jan 11, 2015

### spinnaker

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?

2. Jan 12, 2015

### ShayanJ

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

3. Jan 28, 2015

### Svein

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

4. Jan 28, 2015

### Stephen Tashi

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.

5. Jan 28, 2015

### Svein

Sorry. I haven't done this actively in the last 50 years...