Equivalence between power sets

[dndod1]

Homework Statement

Part a: Show that X [tex]\subseteq[/tex] Y and X [tex]\subseteq[/tex] Z if and only if X[tex]\subseteq[/tex] Y [tex]\cap[/tex] Z, for sets X,Y,Z. I have done this.

Part b: Use the equivalence from part a to establish the identity P(A) [tex]\cap[/tex] P(B)= P(A [tex]\cap[/tex] B), where P is the power set.

Homework Equations

The proof from part a

The Attempt at a Solution

This is as far as I can get and I'm not convinced that I am headed in the right direction.

Let P(A)= Y Let P(B)=Z Let P(A [tex]\cap[/tex] B) = X
Because we need to show =, we must show 2 propositions.
Proposition 1:
That P(A [tex]\cap[/tex] B) [tex]\subseteq[/tex] P(A) [tex]\cap[/tex] P(B)


Proposition 2:
That P(A) [tex]\cap[/tex] P(B) [tex]\subseteq[/tex] P(A intesection B)


Proposition 1: translates directly into what we had in part a.
X [tex]\subseteq[/tex] (Y [tex]\cap[/tex] Z) So no further proof needed?

Proposition 2: translates into (Y [tex]\cap[/tex] Z) [tex]\subseteq[/tex] X
Let x be an element of X
As (Y [tex]\cap[/tex] Z) [tex]\subseteq[/tex] X, x [tex]\in[/tex] (Y [tex]\cap[/tex] Z) from part a
As x [tex]\in[/tex] (Y [tex]\cap[/tex] Z), x [tex]\in[/tex] Y and x [tex]\in[/tex]Z

Here is the point where I am really lost! Did I need the "x [tex]\in[/tex] X" part at all?


Any assistance to get me on the right track would be greatly appreciated.
Many thanks!

 

[Gib Z]

Proposition 1: translates directly into what we had in part a.
X [tex]\subseteq[/tex] (Y [tex]\cap[/tex] Z) So no further proof needed?

Proposition 2: translates into (Y [tex]\cap[/tex] Z) [tex]\subseteq[/tex] X
Let x be an element of X
As (Y [tex]\cap[/tex] Z) [tex]\subseteq[/tex] X, x [tex]\in[/tex] (Y [tex]\cap[/tex] Z) from part a
As x [tex]\in[/tex] (Y [tex]\cap[/tex] Z), x [tex]\in[/tex] Y and x [tex]\in[/tex]Z

Here is the point where I am really lost! Did I need the "x [tex]\in[/tex] X" part at all?

I don't understand how you are getting things "translating" into another thing, and you should be showing all your working clearly as well. And don't rename the sets because it doesn't save that much time and hides what you are actually dealing with.

The easiest way to to showing M is a subset of N is by saying "Let x be in M" then use logical steps to show "Thus x is in N".

For Proposition 1) start with "Let x be an element of the power set of (A intersection B). By the definition of a power set, x is a subset of (A intersection B). By part a) ..."

Then do similar for the next one.

 

[dndod1]

Thanks for that. I shall give it another go. If I don't relate the sets in part b to what I showed in part a, am I not ignoring the part of the question that says "Use the equivalence from part a"?

 

[Gib Z]

Well, whether you explicitly state you used it or not, you will have to to complete the proof anyway, and since you have to explain all your reasoning, when you do use it you'll have to mention that you proved that fact in part a).