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dndod1
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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!