sub-interval

Thân thiện

A student marks a sub-interval on a number line.

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

Học thuật

Noun

1. an interval that is included in another interval