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Thread: Some properties on supermum

  1. #1
    Apr 2008

    Some properties on supermum

    T or F:

    Suppose that A and B are bounded sets in the real numbers.

    1) sup (A U B) = max {sup A, sup B}

    2) If the elements of A and B are positive and A.B = {ab | a is in A, b is in B}, then sup(A.B) = sup(A)sup(B)

    3) The analogous problems for the greatest lower bound.

    I believe all of them are true, but just wanted to make sure before I started on the proofs. Thanks!
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  2. #2
    Senior Member
    Nov 2008

    Let $\displaystyle C$ be a bounded real set, the set of upper bounds of $\displaystyle C$ has always a minimal element, which is the supremum of $\displaystyle C.$

    Now for instance, 1):
    $\displaystyle A,B$ two bounded real sets.
    $\displaystyle a=supA$ and $\displaystyle b=supB$ and $\displaystyle c=sup(A\cup B).\ \forall t\in \mathbb{R},$

    $\displaystyle t=c$
    $\displaystyle \Leftrightarrow (\forall x\in A\cup B, x\leq c)\wedge (\forall y\in\mathbb{R}(\forall x\in A\cup B,\ y\geq x\Rightarrow y\geq c))$
    $\displaystyle \Leftrightarrow (\forall x\in A,\ x\leq c\wedge \forall x\in B, x\leq c)\wedge (\forall x \in\mathbb{R} (x\geq a\wedge x\geq b)\Rightarrow x\geq c)$
    $\displaystyle \Rightarrow (a\leq c\wedge b\leq c)\wedge (c\leq max\{a,b\})$
    $\displaystyle \Rightarrow (max\{a,b\}\leq c)\wedge (c\leq max\{a,b\})$
    $\displaystyle \Rightarrow c=max\{a,b\}$
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