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.
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.