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Math Help - collection of subsets

  1. #1
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    collection of subsets

    Let B be the collection of subsets of R of the form [a,b) for a<b, nulset, and the whole set R.

    Show that if A and C are elements of B then A intersect B is in B.
    and
    Show that if {A_i| 0 <= i <= n} is a finite collection of subsets of B then the collection of intersections (from i=1 to n) of A_i is in B.

    I'm not really sure how to go about this. Any help would be great. Thanks!!
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  2. #2
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    Just verify the different possible cases

    If A or C is \emptyset, then their intersection is the \emptyset and is in B.

    If A or C is \mathbb{R}, then their intersection is the other set and is in B.

    If A=[a,b[ and C=[c,d[, then A\cap C=[\max (a,c),\min (b,d)[ if b<c, and A\cap C=\emptyset else.

    An induction is enough to generalize that result for your second question.
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  3. #3
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    Quote Originally Posted by EricaMae View Post
    Let B be the collection of subsets of R of the form [a,b) for a<b, nulset, and the whole set R. Show that if A and C are elements of B then A intersect B is in B.
    You must mean C where you have B above.
    A=[a,b)\;\&\; C=[c,d). Then let \alpha=\max\{a,c\}\;\&\; \gamma=\min\{b,d\}.
    If \gamma \leq \alpha\text{ then } A\cap C = \emptyset \in B.
    Else A \cap C = [\alpha , \gamma) \in B.
    You do the general case.
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