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Help indexed family sets proof!

  1. Sep 10, 2011 #1
    1. The problem statement, all variables and given/known data
    Ai and Bi are indexed families of sets. Prove that Ui (Ai [itex]\bigcap[/itex] Bi) [itex]\subseteq[/itex] (UiAi) [itex]\bigcap[/itex] (UiBi).

    2. Relevant equations

    3. The attempt at a solution
    Suppose arbitrary x. Let x [itex]\in[/itex]
    {x l [itex]\forall[/itex]i[itex]\in[/itex]I(x[itex]\in[/itex]Ai[itex]\bigcap[/itex]Bi)
    This means x [itex]\in[/itex]{x l [itex]\forall[/itex]i[itex]\in[/itex]I(x[itex]\in[/itex]Ai)[itex]\wedge[/itex][itex]\forall[/itex]i[itex]\in[/itex]I(x[itex]\in[/itex]Bi).
    1. The problem statement, all variables and given/known data
    This means x [itex]\in[/itex][itex]\neg[/itex][itex]\exists[/itex]i[itex]\in[/itex]I(x[itex]\notin[/itex]Ai)[itex]\wedge[/itex][itex]\neg\exists[/itex]i[itex]\in[/itex]I(x[itex]\in[/itex][itex]\notin[/itex]Bi)}
    Which is equivalent to: x[itex]\in[/itex]{x l [itex]\exists[/itex]i[itex]\in[/itex]I(x[itex]\in[/itex]Ai)[itex]\wedge[/itex][itex]\exists[/itex]i[itex]\in[/itex]I(x[itex]\in[/itex]Bi)}
    Therefore, x[itex]\in[/itex]{x l ([itex]\bigcup[/itex]i[itex]\in[/itex]IAi)[itex]\bigcap[/itex](Ui[itex]\in[/itex]IBi)}
    Therefore, is equiv. to Ui[itex]\in[/itex]I(Ai[itex]\bigcap[/itex]Bi), then x[itex]\in[/itex](Ui[itex]\in[/itex]IAi)[itex]\bigcap[/itex](Ii[itex]\in[/itex]IBi).
    Therefore Ui[itex]\in[/itex]I(Ai[itex]\bigcap[/itex]Bi)[itex]\subseteq[/itex](Ui[itex]\in[/itex]IAi)[itex]\bigcap[/itex](Ii[itex]\in[/itex]IBi).
  2. jcsd
  3. Sep 11, 2011 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    Are you trying to being by taking an arbitrary x in [itex] \bigcup_{i=1}^\infty (A_i \cap B_i) [/itex] ? If so, why are you applying the "[itex] \forall [/itex]" quantifier to the index?

    I suggest beginning this way:

    Let x be an arbitrary element of [itex] \bigcup_{i=1}^\infty (A_i \cap B_i) [/itex]

    For such an x, we know that there exists an index j such that [itex] x \in A_j \cap B_j [/itex]. This follows from the definition and properties of a union of sets.

    This implies [itex] x \in A_j [/itex] and [itex] x \in B_j [/itex] by definition of an intersection of two sets.
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