Results 1 to 2 of 2

Math Help - set theory sup of a union

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    39

    set theory sup of a union

    I have an assignment quesiton :

    " let A_i be a subset of the reals and i is element of I = (1,...,n)

    now let A = UNION (i is element of I) A_i

    show that sup(A) = sup_iEI(sup(A))"

    I hope that is understandable...

    so taking the limit of both sides I have limA = lim U A_i, I'm not sure what the limit of the union of a set is. Am I on the right track?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1
    Let I = \left\{ {1,2, \cdots ,n} \right\}, I take this to be your question.
    \sup (A) = \sup \left( {\bigcup\limits_{i \in I} {A_i } } \right) = \mathop {\sup }\limits_{i \in I} \left[ {\sup \left( {A_i } \right)} \right].
    For notation: \alpha _i  = \sup \left( {A_i } \right)\,,\,\alpha  = \sup \left\{ {\alpha _1 ,\alpha _2  \cdots ,\alpha _n } \right\}\,\& \,\alpha ' = \sup (A)
    \left( {\forall x \in A} \right)\left( {\exists m \in I} \right)\left[ {x \in A_m  \Rightarrow \,x \leqslant \alpha _m  \leqslant \alpha } \right] but that means that \alpha ' \leqslant \alpha (WHY?)

    Now suppose that \alpha ' < \alpha and find a contradiction to prove that <br />
\alpha ' = \alpha.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. conclude that the closure of a union is the union of the closures.
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 13th 2011, 07:50 PM
  2. Union of a set
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 3rd 2009, 02:00 PM
  3. union
    Posted in the Calculus Forum
    Replies: 5
    Last Post: September 7th 2008, 06:22 AM
  4. A union B union C
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 27th 2008, 06:26 PM
  5. Axiom of Union (Set Theory)
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: February 8th 2008, 09:49 PM

Search Tags


/mathhelpforum @mathhelpforum