Supremum & infimum of the set of all rational numbers

• rohithsen
In summary, the conversation is about finding the supremum and infimum of the set of rational numbers between √2 and √3. The suggestion is made to use the Dedekind cut approach, where √2 would be the infimum and √3 would be the supremum.
rohithsen
Hi everybody,
Please help me to find supremum & infimum of the set of rational numbers between √2 to √3

(ie) sup & inf of {x/ √2 < x < √3 , where x is rational number}

Hi rohitshen!

If you tell us what you've tried then we'll know where to help! Did you have any guess for what the sup and inf could be?

rohithsen said:
Hi everybody,
Please help me to find supremum & infimum of the set of rational numbers between √2 to √3

(ie) sup & inf of {x/ √2 < x < √3 , where x is rational number}
There is an approach to defining real numbers (Dedekind cut) where √2 and √3 would be defined as the inf and sup respectively.

1. What is the supremum of the set of all rational numbers?

The supremum, or least upper bound, of the set of all rational numbers is infinity. This means that there is no rational number that is greater than or equal to all other rational numbers in the set.

2. Is the supremum of the set of rational numbers a rational number itself?

No, the supremum is not a rational number. In fact, it is not even a real number since it is infinity. This is because the set of rational numbers is not bounded above, meaning there is no finite number that can be considered the supremum.

3. What is the infimum of the set of all rational numbers?

The infimum, or greatest lower bound, of the set of all rational numbers is negative infinity. This means that there is no rational number that is less than or equal to all other rational numbers in the set.

4. Is the infimum of the set of rational numbers a rational number itself?

No, the infimum is not a rational number. Similar to the supremum, it is not a real number since it is negative infinity. The set of rational numbers is not bounded below, meaning there is no finite number that can be considered the infimum.

5. How does the supremum and infimum of the set of rational numbers relate to the concept of completeness?

The supremum and infimum are important concepts in the study of completeness in mathematics. A set is considered complete if it has a supremum and infimum that are both part of the set. Since the set of rational numbers does not have a supremum or infimum that are rational numbers, it is not complete.

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