Set theory: proofs regarding power sets

[Stefan00]

Let X be an arbitrary set and P(X) the set of all its subsets, prove that if ∀ A,B ∈ P(X) the sets A∩B,A∪B are also ∈ P(X).

I really don't know how to get started on this proof but I tried to start with something like this:
∀ m,n ∈ A,B ⇒
m,n ∈ X ⇒

Is this the right way to start on this proof? A hint on how to get further with this would really be appreciated.

Stefan

 

[verty]

Try to use the fact that EVERY subset of X is in P(X). Can you see how that helps?

 

[Stefan00]

@verty, thanks for your reply;

So if A,B ∈ P(X) ⇒ A,B ⊂ X, and since every subset of X is in P(X), A∩B,A∪B are also in P(X)?

Stefan

 

[1MileCrash]

@verty, thanks for your reply;

So if A,B ∈ P(X) ⇒ A,B ⊂ X, and since every subset of X is in P(X), A∩B,A∪B are also in P(X)?

Stefan

Yes, though it wouldn't be a bad idea to show that this union and intersection are subsets of X, despite the triviality.

 

[Stefan00]

Thanks for you reply!

That's where it get stuck I'm afraid, I cannot link the Union and Intersection to X with the given information.

Stefan

 

[1MileCrash]

Thanks for you reply!

That's where it get stuck I'm afraid, I cannot link the Union and Intersection to X with the given information.

Stefan

Approach it like an introductory set theory proof.

Pick an element x in A intersect B. Show that this is in X.

Pick an element y in A union B. Show that this is in X.

For the first one, suppose that x is in A intersect B. Then x is in A and x is in B. Now what?

Remember that you know that A and B are subsets of X - they are in its power set!