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Math Help - prove or disprove: sup of two sets

  1. #1
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    prove or disprove: sup of two sets

    Hi,
    i need to prove or disprove the following:

    Sup(A\cup B)=max\left \{ SupA, SupB \right \}

    thanks in advanced!

    *edit:
    both sets are bounded from above.
    Last edited by Stormey; January 21st 2013 at 04:51 AM.
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  2. #2
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    Re: prove or disprove: sup of two sets

    Prove (1) \sup(A\cup B)\le\max(\sup A,\sup B) and (2) \sup(A\cup B)\ge\max(\sup A,\sup B).

    For (1), show that \max(\sup A,\sup B) is an upper bound of A\cup B.

    For (2), use the fact that x\ge y and x\ge z imply x\ge\max(y,z).
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  3. #3
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    Re: prove or disprove: sup of two sets

    I've found the following to be useful:
    supremums of larger sets are larger and infimums of larger sets are smaller

    Formally, (assuming the sets are bounded):
    1. If C\subseteq D then sup(C)\leq sup(D)
    2. If C\subseteq D then inf(C)\geq inf(D)

    Proof of 1 - Let u=sup(D). Let x\in C, then x\in D and so x\leq u; i.e. u is an upper bound of C. So sup(C)\leq u. The proof for infs is exactly similar.

    For your original question, since A\subseteq A\cup B, sup(A)\leq sup(A\cup B), and similarly for B.
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  4. #4
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    Re: prove or disprove: sup of two sets

    Thanks guys.
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