# Infimum and supremum of empty set

• strobeda
In summary, inf∅ is defined as ∞ and sup∅ as -∞ in order to avoid conflicts with the calculation of inf and sup. This allows for a smoother calculation process and avoids potential errors.
strobeda
Hello,

I can't wrap my mind around this:

inf∅= ∞
sup∅= - ∞

inf∅ is artificially defined to be ∞ so that inf will work well. Suppose we had defined inf∅ = 998 and we had a set with one element, 999. Then we want inf({999}) = 999. But since ∅ is also a subset of {999}, we would have inf({999}) = inf∅ = 998. The only way to avoid this problem is to make inf∅ greater than any possible number. So inf∅ = ∞. Similarly we have to define sup∅ smaller than any possible number. So sup∅ = -∞.

In a sense, this is just getting ∅ out of the way of the calculation of inf and sup.

Indeed, it gets ∅ out of the way!

Thank you very much, FactChecker!

## 1. What is the infimum of the empty set?

The infimum of the empty set is undefined. This is because the infimum is defined as the greatest lower bound of a set, but since the empty set has no elements, there is no lower bound to compare to.

## 2. Is the infimum of the empty set equal to zero?

No, the infimum of the empty set is not equal to zero. As mentioned before, the infimum of the empty set is undefined because there is no lower bound to compare to.

## 3. What is the supremum of the empty set?

The supremum of the empty set is also undefined for the same reason as the infimum. The supremum is defined as the least upper bound of a set, but the empty set has no elements to compare to.

## 4. Can the infimum or supremum of the empty set be any real number?

No, the infimum and supremum of the empty set cannot be any real number. As stated before, they are both undefined because there are no elements to compare to in the empty set. Therefore, they cannot be assigned any value.

## 5. Why is it important to understand the infimum and supremum of the empty set?

Understanding the infimum and supremum of the empty set is important in mathematical analysis and set theory. It helps in defining the concepts of lower and upper bounds and understanding the behavior of sets with no elements. It also aids in proving theorems and solving problems related to sets and their properties.

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