(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

The proof from part a

3. 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!

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# Homework Help: Equivalence between power sets

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