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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- Aug 2nd 2011, 03:39 AMlukeheseldensupremum ,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? - Aug 2nd 2011, 04:04 AMpeceRe: supremum ,infinum ,maximum,minimum
Remember, a maximum of a set is a

**element of**such that every element of are smaller or equal to it.

So, is ? It's the first step to be a maximum.

For the rest, remember also that (and , but it's irrelevant here) can be infinite. - Aug 2nd 2011, 05:27 AMlukeheseldenRe: 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

- Aug 2nd 2011, 05:48 AMpeceRe: 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 . It's your set, right ?

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

Let's begin with the supremum (it always exists) : it's by definition (i.e. the smaller upper bound). You've seen that (i.e is an upper bound). It's a good start, now you have to show that if is a upper bound, then (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 ( ) is in . If it is : it's the maximum. If not : there is no maximum.

Now, the infinum (it always exists) : it's by definition . 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.