Write range of length and relation of various range of length.

Ranges of length can be categorized in various ways based on their relationship to one another. Here are some common categories and examples:

Disjoint Ranges:

Definition: Ranges that have no overlap and are completely separate from each other.
Examples: [0, 10] and [15, 20]
Overlapping Ranges:

Definition: Ranges that share at least one common value or overlap partially.
Examples: [5, 15] and [10, 20]
Nested Ranges:

Definition: Ranges where one range is completely contained within another.
Examples: [5, 20] and [8, 15]
Adjacent Ranges:

Definition: Ranges that share an endpoint but do not overlap.
Examples: [0, 10] and [10, 20]
Continuous Ranges:

Definition: Ranges that form a continuous sequence without any gaps.
Examples: [0, 10], [11, 20], [21, 30]
Disjoint Union of Ranges:

Definition: A collection of disjoint ranges that together cover a larger span.
Example: {[0, 5], [10, 15], [20, 25]} forms a disjoint union on the range [0, 25].
Overlapping Union of Ranges:

Definition: A collection of ranges that may overlap and collectively cover a larger span.
Example: {[0, 10], [5, 15], [12, 20]} forms an overlapping union on the range [0, 20].
Empty Range:

Definition: A range with no values in it, typically represented as [a, a] where a is a constant.
Example: [5, 5] is an empty range.
Infinite Range:

Definition: A range that extends infinitely in one or both directions.
Examples: (-∞, ∞), [0, ∞), (-∞, 10]
Equal Length Ranges:

Definition: Ranges with the same length.
Example: [0, 10] and [20, 30] are equal length ranges of length 10.
These definitions and relationships can be applied in various fields such as mathematics, data analysis, programming, and more to describe and work with ranges of lengths in different contexts.