Intersections and Unions of powersets confusing

[xpoferens]

Homework Statement

let A and B be finite sets

Let P(A) denote the Power Set of A (set of all subsets of A)

Homework Equations

Prove or disprove:
a) P(A inter B) = P(A) inter P(B)
b) P(A U B) = P(B) U P(B)

The Attempt at a Solution

So I tried drawing Venn diagrams and came to the conclusion that they must be true but when I asked my professor for help all he said was that Venn diagrams don't count as formal proofs and that I was wrong in any case. Now I am just totally confused. Any help will be appreciated

 

[Office_Shredder]

Just try doing it with words. You really want to prove two things for each question

a) Prove that P(a intersect B) is a subset of P(A) intersect P(B). Also prove that P(A) intersect P(B) is a subset of P(A intersect B).

So how do we do this? For the first direction: Suppose X is an element of P(A intersect B). Your goal is to conclude that X is an element of both P(A) and P(B)

 

[xpoferens]

ok so if x is an element of P(A inter B)
then a is a subset of (A inter B)
so a is a subset of A AND a is a subset of B
so a is an element of P(A) AND a is an element of P(B)
therefore a is an element of P(A) inter P(B).

Is that reasoning sound?

 

[Office_Shredder]

Yeah looks good to me.

 

[xpoferens]

Ok thanks a lot,

Now for part b, after thinking about it I can intuitively see why the equality is false but I don't know how to prove it. Could you point me in the right direction please?

 

[xpoferens]

OK I actually just thought that since it's easier to prove that something is false I used a counter example.
Consider A = {1} and B = {2}
In this case P(A) U P (B) does not contain {1, 2} but P(A U B) does. That seems logical to me..

 

[Office_Shredder]

For part a you need to do the other direction that P(A) intersect P(B) is a subset of P(A intersect B)

Part b sounds good also