Interval (mathematics)

In elementary algebra, an interval is a set that contains every real number between two indicated numbers and may contain 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 every number between $10$ and $20$ as well as $10$ and $20$. Other possibilities are listed below.

Because an interval is also a neighborhood of every real number x lying inside the interval, the term interval is sometimes loosely used as a synonym for neighborhood when discussing multi-dimensional metric spaces, such as the complex plane.

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 $x<z<y$ then $z$ is in $S$.

As mentioned above, a particularly important case is when $T=\mathbb {R}$, the set of real numbers.

Intervals of $\mathbb {R}$ are of the following eleven different types (where $a$ and $b$ are real numbers, with $a<b$):

1. $(a,b)=\{x\,|\,a<x<b\}$
2. $[a,b]=\{x\,|\,a\leq x\leq b\}$
3. $[a,b)=\{x\,|\,a\,\leq x<b\}$
4. $(a,b]=\{x\,|\,a<x\leq b\}$
5. $(a,\infty )=\{x\,|\,x>a\}$
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. $\varnothing$, the empty set
11. $\{a\}\,$, singleton

The last two are called degenerate intervals. In each case where they appear above, a and b are known as endpoints of the interval.

If one considers the extended reals, seven new cases arise:

1. $[-\infty ,b]=\{x\,|\,x\leq b\}\cup \{-\infty \}$
2. $[-\infty ,b)=\{x\,|\,x<b\}\cup \{-\infty \}$
3. $[a,\infty ]=\{x\,|\,x\geq a\}\cup \{\infty \}$
4. $(a,\infty ]=\{x\,|\,x>a\}\cup \{\infty \}$
5. $(-\infty ,\infty ]=\mathbb {R} \cup \{\infty \}$
6. $[-\infty ,\infty )=\mathbb {R} \cup \{-\infty \}$
7. $[-\infty ,\infty ]={\overline {\mathbb {R} }}$

Intervals using the round brackets ( or ) as in the general interval (a,b) or specific examples (-1,3) and (2,4) are called open intervals and the endpoints are not included in the set. Intervals using the square brackets [ or ] as in the general interval [a,b] or specific examples [-1,3] and [2,4] are called closed intervals and the endpoints are included in the set. Intervals using both square and round brackets [ and ) or ( and ] as in the general intervals (a,b] and [a,b) or specific examples [-1,3) and (2,4] are called half-closed intervals or half-open intervals.

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 order theory

In order theory, the concept of an interval can be extended to {{#invoke:preorder|interval|those in a preordered set}}. For example, given a partially ordered set (P, ≤) and two elements a and b of P, one defines the set

[a, b] = { x | a ≤ x ≤ b }

One may choose to restrict this definition to pairs of elements with the property that a ≤ 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.

In a subset S of an ordered set P, where the order on S is the restriction of that on P, an interval [a, b] is the intersection of S and the interval [a, b] in P, and similarly for open intervals. In a subset S of R there is an empty open interval (a, b) with a < b if and only if S is not dense in R.

Interval arithmetic

Interval arithmetic, also called interval mathematics, interval analysis, and interval computation, has been developed by mathematicians since the 1950s and 1960s as an approach to putting bounds on rounding errors in mathematical computation and thus obtaining very reliable results. Where classical arithmetic defines operations on individual numbers, interval arithmetic defines a set of operations on intervals:

T · S = { x | there is some y in T, and some z in S, such that x = y · z }.

The basic operations of interval arithmetic are, for two intervals [a, b] and [c, d] that are subsets of the real line (−∞, ∞),

• [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 · S is true if for any x in T, and any y in S, x · y is true
• T · S is false if for any x in T, and any y in S, x · y is false
• otherwise T · 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 · S contains x · y.

Alternative notation

International standard ISO 31-11 also defines another notation for intervals, which is the one commonly taught in many European and South American countries (e.g., Germany, France, Brazil) in secondary school:

• ]a,b[ = { x | a < x < b }
• [a,b] = { x | a ≤ x ≤ b }
• [a,b[ = { x | a ≤ x < b }
• ]a,b] = { x | a < x ≤ b }

This notation is somewhat easier to remember (inwards pointing bracket for inclusion, outwards-pointing bracket for exclusion). Another advantage is that this notation does not overlap with the tuple notation, which is equally commonly used in set theory.

Where numbers are written with a decimal comma, the endpoints in the interval notation may also be separated by a semicolon instead of a comma, to avoid ambiguity.

See also

• inequality

External links

• An American Scientist article provides an introduction.
• Interval Notation Basics
• Interval computations website
• Interval computations research centers
• Sun Studio compilers - implement interval arithmetic