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Encyclopedia > Interval (mathematics)

In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ...

Contents


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. Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In computer science and mathematics, a variable (sometimes called a pronumeral) is a symbol denoting a quantity or symbolic representation. ... For the round brackets used in punctuation, often called parentheses, see bracket. ... See parenthesis for an account of the rhetorical concept from which the name of the punctuation mark is derived. ...


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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...


As mentioned above, a particularly important case is when T=mathbb{R}, the set of real numbers. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...


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,|,aleq xleq b}
  3. [a,b)={x,|,a,leq x<b}
  4. (a,b]={x,|,a<xleq b}
  5. (a,infty)={x,|,x>a}
  6. [a,infty)={x,|,xgeq a}
  7. (-infty,b)={x,|,x<b}
  8. (-infty,b]={x,|,xleq 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. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...


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. Integration may be any of the following: In the most general sense, integration may be any bringing together of things: the integration of two or more economies, cultures, religions (usually called syncretism), etc. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a (where a < b). ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...


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}rightarrowmathbb{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. In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In analysis, the intermediate value theorem is either of two theorems of which an account is given below. ...


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]. Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...


For a partially ordered set (P, ≤) and two elements a and b of P, one defines the set

[a, b] = { x | axb }

One may choose to restrict this definition to pairs of elements with the property that ab. 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 mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...


Interval arithmetic

Interval arithmetic, also called interval mathematics, interval analysis, and interval computation, is being 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: The 1950s were the decade that spanned the years 1950 through 1959, although some sources say from 1951 through 1960. ... The outrageously crowded Woodstock festival epitomized the popular antiwar movement of the 60s. ... A round-off error is the difference between the calculated approximation of a number and its exact mathematical value. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...

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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...


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 countries (e.g., Germany, France) in secondary school: Standards are produced by many organizations, some for internal usage only, others for use by a groups of people, groups of companies, or a subsection of an industry. ... ISO 31-11 is the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology. ... Secondary school may refer to Secondary school in the United Kingdom, is the general term for the schools for children between the ages of eleven and eighteen in most areas (a few areas have schools for 13-18 year olds instead, and these are called upper schools). ...

  • ]a,b[ = { x | a < x < b }
  • [a,b] = { x | axb }
  • [a,b[ = { x | ax < b }
  • ]a,b] = { x | a < xb }

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. In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects. ...


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. The decimal separator is used to mark the boundary between the integer and the fractional parts of a decimal numeral. ...


External links


  Results from FactBites:
 
Interval (mathematics) - Wikipedia, the free encyclopedia (910 words)
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 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.
Interval arithmetic, also called interval mathematics, interval analysis, and interval computation, is being 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.
Interval (mathematics) - definition of Interval (mathematics) in Encyclopedia (780 words)
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 (1), (2), (3), (4), (10) and (11) are called bounded intervals and intervals (5), (6), (7), (8) and (9) unbounded 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.
  More results at FactBites »

 
 

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