lower bound
Definition
Noun (Mathematics): A lower bound is a number that is less than or equal to every element in a given set of numbers. It establishes a minimum limit for the values within the set.
Usage
The term is used to describe a numerical limit or threshold for a collection of numbers. A set can have many lower bounds. The greatest of all the lower bounds is called the greatest lower bound or infimum.
Examples
- In the set {5, 8, 12, 20}, the numbers 1, 4, and 5 are all lower bounds. The number 5 is the greatest lower bound.
- To prove convergence, we first need to find a lower bound for the sequence.
- Zero is a lower bound for the set of all positive real numbers.
Advanced Usage
- Tight/Sharp Lower Bound: A lower bound that is very close to or equal to the actual smallest possible value in the set or sequence. For example, in the set {2, 4, 6}, the number 2 is a tight lower bound.
- The concept is fundamental in analysis for defining the infimum of a set.
Variants and Related Words
- Bound (n): A more general term for either a lower limit or an upper limit.
- Upper Bound (n): The complementary concept; a number greater than or equal to every element in a set.
- Infimum (n): The greatest lower bound of a set.
- Minimum (n): The smallest element in a set. A minimum is always a lower bound, but a lower bound is not necessarily a minimum (if the bound is not actually an element of the set).
Synonyms
- Minimum limit
- Floor (in certain contexts, e.g., "a price floor")
Antonyms
- Upper bound
Noun
-
(mathematics) a number equal to or less than any other number in a given set
