A lower bound of a set of numbers is a number such that the number is less than or equal to all numbers in the set.^{[1]} An upper bound of a set of numbers is a number such that the number is greater than or equal to all numbers in the set.
The least upper bound of a set of numbers is the smallest number that is greater than or equal to all numbers in the set. The least upper bound is also called the supremum. A set is said to be bounded from above if it has a least upper bound.
The greatest lower bound of a set of numbers is the largest number that is less than or equal to all numbers in the set. The greatest lower bound is called the infimum. A set is said to be bounded from below if it has a greatest lower bound.
A set is bounded if it has an upper bound and a lower bound. A set is unbounded if it does not have an upper bound or does not have a lower bound.
Set | Greatest Lower Bound | Least Upper Bound | Is Bounded? |
---|---|---|---|
All real numbers {x: x∈ℝ} | None | None | No |
All positive integers {x: x∈^{+}ℤ} | 1 | None | No |
{x: x∈ℝ, -1 < x < 5} | -1 | 5 | Yes |
{x: x∈ℝ, -1 ≤ x ≤ 5} | -1 | 5 | Yes |
When one says a function is bounded, one is referring to the range of the function. If the range of the function has a greatest lower bound, then one says that the function has a greatest lower bound.
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