# Intervals

An interval is a compact connected subset of the real axis R. It is thus completely defined by two numbers, namely, its lower endpoint and upper endpoint (sometimes called left endpoint and right endpoint respectively), so that `[a, b]` denotes the interval `{x ∈ R|a ≤ x ≤ b}`. The set of all real intervals is denoted by IR. In mathematical notation, taking the lower and upper endpoints of an interval is usually denoted by
`inf[a, b] = a, sup[a, b] = b.`

In the discussion below, intervals and interval objects are denoted by boldface letters, while underscores and overscores designate the lower and upper endpoints of the interval `x = [x,x]`

Every interval is uniquely determined by its midpoint,

and radius,

the latter being equivalent to the width wid `a = a-a`. Intervals of the form `[a, a]` that have equal lower and upper endpoints, that is, intervals of zero width, are called degenerate or point or thin, and they coincide with usual real numbers so that it can be implied `R ⊂ IR`. On the contrary, the intervals with nonzero width are called thick intervals.

Since intervals are sets, set-theoretical relations and operations between them are applicable, for example, inclusion, intersection, and so on. In particular, a point `t ∈ R` is a member of the interval a (written as `t ∈ a`) if `a ≤ t ≤ a`. Also, the inclusion is defined as `a ⊆ b` if and only if `a ≥ b` and `a ≤ b`

Intervals and interval objects (vectors, matrices, etc.) are a convenient tool to represent the so-called bounded uncertainty and ambiguity, when only the lower and upper bounds of the possible variation of some value are known. In this sense, intervals provide an alternative to probabilistic and fuzzy approaches for describing quantitative uncertainty.

Arithmetic operations, such as addition, subtraction, multiplication and division, can be extended to intervals according to the fundamental principle

a*b: = {a*b|a ∈ a,b ∈ b}, *∈ {+, -, ·,/}, | (1)

which makes it possible to define the so-called classical interval arithmetic. Note that the empty interval [] is often incorporated into the computer interval arithmetic structures.

Parent topic: Interval Linear Systems