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Math Help - Greatest lower and leat upper bound help

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    Greatest lower and leat upper bound help

    what is the least upper bound and greatest lower bound of the set

     [-1,3] \cap (\sqrt{2} , 5] ?

     {  { x \in  {R} : e^{x} \leq 2 }   }
    Last edited by Tweety; December 5th 2013 at 01:07 AM.
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  2. #2
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    Re: Greatest lower and leat upper bound help

    Quote Originally Posted by Tweety View Post
    what is the least upper bound and greatest lower bound of the set

     [-1,3] \cap (\sqrt{2} , 5] ?

     {  { x \in  {R} : e^{x} \leq 2 }   }
    I will start be admitting that I am new to this type of notation so if my answer sounds like nonsense then you are probably correct.

    The intersection of those two intervals will be (\sqrt2,3] I guess this is a limit on the x values

    \begin{align*}e^x &\leq2\\lne^x & \leq ln2\\x&\leq ln2\\x&\leq 0.6931 approx\end{align*}

    now if x\leq ln2 in the domain (\sqrt{2} , 5] then there are no real solutions.
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    Re: Greatest lower and leat upper bound help

    Quote Originally Posted by Melody2 View Post
    I will start be admitting that I am new to this type of notation so if my answer sounds like nonsense then you are probably correct.

    The intersection of those two intervals will be (\sqrt2,3] I guess this is a limit on the x values

    \begin{align*}e^x &\leq2\\lne^x & \leq ln2\\x&\leq ln2\\x&\leq 0.6931 approx\end{align*}

    now if x\leq ln2 in the domain (\sqrt{2} , 5] then there are no real solutions.
    I'm seeing 2 sets.

    \left.[-1,3]\cap \left(\sqrt{2},5\right.\right] and

    {x : x is Real, ex <= 2}


    OP doesn't specify any relation between these two sets so I think this is 2 problems.

    (\left.[-1,3]\cap \left(\sqrt{2},5\right.\right] = \left.\left(\sqrt{2},3\right.\right] so

    the LUB of the first set is clearly 3. The GLB is clearly \sqrt{2}

    The LUB of the 2nd set = ln(2) as you noted. There is no lower bound on set 2 so there is no GLB.
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    Re: Greatest lower and leat upper bound help

    Yes, I agree that this is two separate problems.

    MelodyII is right that the first set is equivalent to the interval (\sqrt{2}, 3] so its least upper bound is 3 and greatest lower bound is \sqrt{2}.

    romset is, of course, using the fact that the logarithm is an increasing function: if b a\le b then ln(a)\le ln(b)

    So if e^x\le 2 then x\le ln(2). So there is no greatest upper bound on x and the least upper bound is ln(2).
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    Re: Greatest lower and leat upper bound help

    why is it called the least upper bound and not just the upper bound?
    and
    why is it called the greatest lower bound and not just the lower bound?
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    Re: Greatest lower and leat upper bound help

    Quote Originally Posted by Melody2 View Post
    why is it called the least upper bound and not just the upper bound?
    and why is it called the greatest lower bound and not just the lower bound?
    Definition: L is the least upper bound of a set A means that L is an upper bound of A such that if K is also an upper bound of A then L\le K.

    Let A=[0.8) then 10 is an upper bound of A but 8=\text{LUB}(A).
    Thanks from Melody2
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