The question I'm working on says
"determine for any sets A and B, it is true thatP(A − B) = P(A) − P(B). If it is true prove it, if it is not, give a counterexample"
Slightly unsure about this "for any sets" statement. Do they mean for any specific set? Or for all? If it is true for any specific set then how can a counter example prove falsehood? Maybe I'm answering my own question here.
I'm also going to provide a proof here to see if my logic is making any sense to you folks.
Proof: P(A − B) = P(A) − P(B)
let X ∈ P(A-B) ⇒ X ∈ A and X ∉ B ⇒ X ∈ P(A) and X ∉ P(B) ⇒ X ∈ P(A) - P(B)
let X ∈ P(A) - P(B) ⇒ X ∈ P(A) and X ∉ P(B) ⇒ X ∈ A and X ∉ B ⇒ X ∈ (A - B) ⇒ X ∈ P(A - B)
so we have P(A − B) = P(A) − P(B)
For a counterexample, take $A=\{x,y\}, B=\{y\}$. Then $A-B=\{x\}$, so $$\mathcal P(A-B)=\{\emptyset,\{x\}\}$$ And $$\mathcal P(A)-\mathcal P(B)=\{\emptyset,\{x\},\{y\},\{x,y\}\}-\{\emptyset,\{y\}\}=\text{what?}$$ As you can see, the two expressions are different.
In fact, for a quick general proof that the statement is false, you just have to note that for any two sets $A$ and $B$, $\emptyset$ is in $\mathcal P(A-B)$ but not in $\mathcal P(A)-\mathcal P(B)$.