# Question about power sets and the empty set element

• Apr 26th 2010, 12:05 PM
swtdelicaterose
Question about power sets and the empty set element
Say theres a set A = {1}, set B = {2}

Then the powerset of A is P(A) = {{1}, Φ}, and powerset of B is P(B) = {{2}, Φ}

Now if I have P(A) - P(B), then this wouldn't remove the empty set element would it?

I know the empty set is a subset of all sets, and it doesn't really make sense that you could remove the empty set.

So I'm guessing P(A) - P(B) = {{1}, Φ} still.

Is this right? Or would P(A) - P(B) = {{1}}?

Thanks for the help!
• Apr 26th 2010, 12:09 PM
MoeBlee
I surmise you mean set difference. I.e., P(A) - P(B) = {x |x in P(A) but x not in P(B)}.

In this case, the empty set (0) is NOT in P(A) - P(B).

0 is in P(B) so it cannot be in P(A) - P(B).

That 0 is a subset of every set doesn't entail that 0 is a MEMBER of every set. 0 is a subset of P(A) - P(B) but 0 is not a member of P(A) - P(B).

You're right when you say P(A) - P(B) = {{1}}, since {1} is the only element of P(A) that is not an element of P(B).
• Apr 26th 2010, 12:14 PM
swtdelicaterose