sub-interval
Definition
Noun: An interval that is entirely contained within another, larger interval. In mathematics, particularly in calculus and analysis, a sub-interval is a smaller segment created by partitioning a given interval.
Usage
A sub-interval is a fundamental concept used when dividing an interval into smaller, non-overlapping parts. This is common in numerical integration (like Riemann sums), probability, and time series analysis.
Examples
- The closed interval [1, 3] is a sub-interval of the larger interval [0, 5].
- To approximate the area under the curve, we partition the main interval into several equal sub-intervals.
- The probability is calculated by summing the probabilities over all sub-intervals within the sample space.
Advanced Usage
- Nested Sub-intervals: Refers to a sequence of sub-intervals where each one is contained within the previous one. This concept is used in the Nested Interval Theorem to prove the existence of a real number.
- The method of bisection generates a sequence of nested sub-intervals to find the root of the equation.
Variants and Related Words
- Interval (n): A set of real numbers with the property that any number that lies between two numbers in the set is also included in the set (e.g., [a, b], (a, b)).
- Partition (n/v): The act or result of dividing an interval into sub-intervals.
Synonyms
- Subset (of an interval)
- Segment
- Subdivision
Antonyms
- Superset (interval)
- Entire interval
Noun
-
an interval that is included in another interval
