Question

Prove that:

P(A-B) = P(A) - P(B)

Where P is a power set.

Answer 1

Like with the previous question, take an element from one side and show it's in the other.

Let \(X\in P(A-B)\), then \(X\subset A-B\), which means \(X\subset A\) and \(X\not\subset B\).

This implies \(X\in P(A)\) and \(X\not\in P(B)\), or \(X\in P(A)-P(B)\).

Thus, \(P(A-B)\subset P(A)-P(B)\).

The proof in the other direction is similar.