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Elementary Set Theory Proof

  1. Sep 6, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that if P(A) [itex]\subseteq[/itex] P(B) then A [itex]\subseteq[/itex] B,
    where A and B are two sets and P symbolizes the power set (set of all subsets) of a particular set.

    2. Relevant equations



    3. The attempt at a solution
    Okay, so here goes.

    Because it's a conditional, we suppose P(A)[itex]\subseteq[/itex] P(B), and make it a "given."

    From there, we look at the goal ( A[itex]\in[/itex] B ), and let x be arbitrary such that x [itex]\in[/itex] A [itex]\rightarrow[/itex] x [itex]\in[/itex] B. Because x is arbitrary, we suppose x [itex]\in[/itex] A.

    So far, we have:

    Givens:
    P(A) is a subset of P(B), or [itex]\forall[/itex]y( y [itex]\in[/itex] P(A) [itex]\rightarrow[/itex] y [itex]\in[/itex] P(B)
    x [itex]\in[/itex] A

    Goals:
    x [itex]\in[/itex] B

    So this is where it falls apart. Looking at the given above, I see the opportunity for universal instantiation. However, in order to do that I need to know some variable that y [itex]\in[/itex] P(A), or that y [itex]\subseteq[/itex] A. I see neither. Can you help me?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 6, 2011 #2

    Dick

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    Homework Helper

    If x is an element of A then the set {x} is in P(A). Does that help?
     
  4. Sep 7, 2011 #3
    Thanks!
     
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