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Math Help - supremum ,infinum ,maximum,minimum

  1. #1
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    supremum ,infinum ,maximum,minimum

    i was doing a question and i got stuck again,
    {x is an element of rational numbers: x^3 ≤ pi}
    i think the answers are sup:cube root of pi,inf: no infimum ,max: cube root of pi, min:no minimum

    am i right?
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  2. #2
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    Re: supremum ,infinum ,maximum,minimum

    Remember, a maximum of a set A is a element of A such that every element of A are smaller or equal to it.

    So, is \sqrt[3]{\pi} \in \{q \in \mathbb Q / q^3 \leq \pi\} ? It's the first step to be a maximum.


    For the rest, remember also that \inf (and \sup, but it's irrelevant here) can be infinite.
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  3. #3
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    Re: supremum ,infinum ,maximum,minimum

    i not sure if i understand that could you state the answers and then it might be more clear to me
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  4. #4
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    Re: supremum ,infinum ,maximum,minimum

    Well, i'm not gonna give the cooked answers. It's your job to find them.

    But I can try to explain clearly.
    Let A = \{q \in \mathbb Q / q^3 \leq \pi\}. It's your set, right ?

    We want to find, if exist, the minimum, the infinum, the maximum and the supremum of A.

    Let's begin with the supremum (it always exists) : it's by definition \min \{M \in \mathbb R\cup\{+\infty\} / \forall x \in A, M \geq x\} (i.e. the smaller upper bound). You've seen that \forall x \in A, x \leq \sqrt[3] \pi (i.e \sqrt[3]\pi is an upper bound). It's a good start, now you have to show that if M \in \mathbb R is a upper bound, then M \geq \sqrt[3] \pi (by contradiction for example). I let you do that.

    Let's continue with the maximum : by definition, a maximum is an upper bound element of the set. So, if a maximum exists, it also is the supremum. So you just have to check if the supremum you found ( \sqrt[3] \pi) is in A. If it is : it's the maximum. If not : there is no maximum.

    Now, the infinum (it always exists) : it's by definition \max \{m \in \mathbb R\cup\{-\infty\} / \forall x \in A, m \leq x\}. So, what is it here ?

    And finally the minimum : it's a lower bound element of the set. Just like the maximum, let's check if the infinum is in the set or not and we have our answer.
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