Interval (mathematics)

In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers.

Algebra
In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. Interval notation is the notation in which permitted values for a variable are expressed as ranging over a certain interval; "5 < x < 9" is an example of the application of interval notation. In conventional interval notation, parentheses ( ) indicate exclusion while square brackets ( [] ) indicate inclusion. For example, the interval "(10,20)" indicates the set of all real numbers between 10 and 20 but does not include 10 or 20, the first and last numbers of the interval, respectively. On the other hand, the interval "[10,20]" includes both every number between 10 and 20 as well as 10 and 20. Other possibilities are listed below.

Higher mathematics
In higher mathematics, a formal definition is the following: An interval is a subset $$S$$ of a totally ordered set $$T$$ with the property that whenever $$x$$ and $$y$$ are in $$S$$ and $$xa\}$$
 * 6) $$[a,\infty)=\{x\,|\,x\geq a\}$$
 * 7) $$(-\infty,b)=\{x\,|\,x<b\}$$
 * 8) $$(-\infty,b]=\{x\,|\,x\leq b\}$$
 * 9) $$(-\infty,\infty)=\mathbb{R}$$ itself, the set of all real numbers
 * 10) $$\{a\}$$
 * 11) $$\varnothing$$ the empty set

In each case where they appear above, a and b are known as endpoints of the interval. Note that a square bracket [ or ] indicates that the endpoint is included in the interval, while a round bracket ( or ) indicates that it is not. For more information about the notation used above, see Naive set theory.

Intervals of type (1), (5), (7), (9) and (11) are called open intervals (because they are open sets) and intervals (2), (6), (8), (9), (10) and (11) closed intervals (because they are closed sets). Intervals (3) and (4) are sometimes called half-closed (or, not surprisingly, half-open) intervals. Notice that intervals (9) and (11) are both open and closed, which is not the same thing as being half-open and half-closed.

Intervals (1), (2), (3), (4), (10) and (11) are called bounded intervals and intervals (5), (6), (7), (8) and (9) unbounded intervals. Interval (10) is also known as a singleton.

The length of the bounded intervals (1), (2), (3), (4) is b-a in each case. The total length of a sequence of intervals is the sum of the lengths of the intervals. No allowance is made for the intersection of the intervals. For instance, the total length of the sequence {(1,2),(1.5,2.5)} is 1+1=2, despite the fact that the union of the sequence is an interval of length 1.5.

Intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define (see above). The concept of measure can then be extended to more complicated sets, leading to the Borel measure and eventually to the Lebesgue measure.

Intervals are precisely the connected subsets of $$\mathbb{R}$$. They are also precisely the convex subsets of $$\mathbb{R}$$. Since a continuous image of a connected set is connected, it follows that if $$f:\mathbb{R}\rightarrow\mathbb{R}$$ is a continuous function and I is an interval, then its image $$f(I)$$ is also an interval. This is one formulation of the intermediate value theorem.

Intervals in partial orders
In order theory, one usually considers partially ordered sets. However, the above notations and definitions can immediately be applied to this general case as well. Of special interest in this general setting are intervals of the form [a,b].

For a partially ordered set (P, &le;) and two elements a and b of P, one defines the set
 * [a, b] = { x | a &le; x &le; b }

One may choose to restrict this definition to pairs of elements with the property that a &le; b. Alternatively, the intervals without this condition will just coincide with the empty set, which in the former case would not be considered as an interval.

Interval arithmetic
Interval arithmetic, also called interval mathematics, interval analysis, and interval computation, was introduced by mathematicians in the 1950s and 1960s as an approach to putting bounds on rounding errors in mathematical computation. Where classical arithmetic defines operations on individual numbers, interval arithmetic defines a set of operations on intervals:


 * T &middot; S = { x | there is some y in T, and some z in S, such that x = y &middot; z }.

The basic operations of interval arithmetic are, for two intervals [a,b] and [c,d] that are subsets of the real line (-&infin;, &infin;),
 * [a,b] + [c,d] = [a+c, b+d]
 * [a,b] - [c,d] = [a-d, b-c]
 * [a,b] * [c,d] = [min (ac, ad, bc, bd), max (ac, ad, bc, bd)]
 * [a,b] / [c,d] = [min (a/c, a/d, b/c, b/d), max (a/c, a/d, b/c, b/d)]

Division by an interval containing zero is not defined under the basic interval arithmetic.

The addition and multiplication operations are commutative, associative and sub-distributive: the set X ( Y + Z ) is a subset of XY + XZ.

Relational operations
Relational operations on intervals can be defined in tri-state logic {true, false, uncertain}:


 * T &middot; S is true if for any x in T, and any y in S, x &middot; y is true
 * T &middot; S is false if for any x in T, and any y in S, x &middot; y is false
 * otherwise T &middot; S is uncertain

Often intervals are considered as estimations of some individual numbers. In that case for both arithmetic and relational interval operations the following is true: if x in T and y in S, then the result of T &middot; S contains x &middot; y.

Alternative notation
An alternative way of writing intervals, commonly seen in France and some other European countries, is shown below:


 * ] a,b [ = { x | a < x < b }
 * [a,b] = { x | a &le; x &le; b }
 * [ a,b [ = { x | a &le; x < b }
 * ] a,b ] = { x | a < x &le; b }

Another notation in use in some countries (e.g. Czech Republic) uses parentheses for an open interval and angle brackets for an closed interval, with numbers separated by semicolon.