Thread: 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?

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.

i not sure if i understand that could you state the answers and then it might be more clear to me

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.