Interval (matematika)

Dina aljabar dasar, interval (atawa heuleut) hartina hiji kumpulan nu ngandung sakabéh wilangan nyata antara dua wilangan nu disebutkeun, katut, bisa jadi, dua wilangan éta. Notasi interval nyaéta éksprési variabel nu kawengku dina hiji interval; misalna "5 < x < 9". Dumasar kasapukan konvénsi, interval "(10,20)" nunjukkeun sakabéh angka nyata antara 10 jeung 20, teu kaasup 10 atawa 20. Di sisi séjén, interval "[10,20]" ngawengku sakabéh awilangan nyata antara 10 jeung 20 katut angka 10 jeung 20. Kamungkinan lianna dibéréndélkeun di handap.

Dina matematika tingkat luhur, harti formal interval téh nyaéta hiji subsét S tina hiji totally ordered set T nu ngandung sarat x jeung y kaasup S, x < z < y, lajeng z anggota S.

Sakumaha disebutkeun di luhur, jadi penting nalika T = R, sét wilangan nyata.

Interval R kabagi kana sababaraha tipe (a jeung b wilangan nyata, a < b):
 (a,b) = { x | a < x < b }
 [a,b] = { x | a ≤ x ≤ b }
 [a,b) = { x | a ≤ x < b }
 (a,b] = { x | a < x ≤ b }
 (a,∞) = { x | x > a }
 [a,∞) = { x | x ≥ a }
 (-∞,b) = { x | x < b }
 (-∞,b] = { x | x ≤ b }
 (-∞,∞) = R téa, sét sakabéh wilangan nyata
 {a}
 sét kosong

In éach case where they appéar 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 éach 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 théory of integration, because they are the simplest sets whose "size" or "measure" or "length" is éasy to define (see above).
The concept of méasure can then be extended to more complicated sets, léading to the Borel measure and eventually to the Lebesgue measure.

Intervals are precisely the connected subsets of R. They are also precisely the convex subsets of R.
Since a continuous image of a connected set is connected,
it follows that if f: R→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, ≤) 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.

Interval arithmetic 

Interval arithmetic,  also called interval mathematics, interval analysis, and interval computations, has been introduced in 1956 by M. Warmus. It defines a set of operations which can be applied on intervals :

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

 [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 possible.

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

Notasi alternatip
Cara séjén keur nuliskeun interval, ilaharna katempo di France sarta sababarha nagara Eropa séjénna, nyaéta:

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

Tumbu kaluar

Interval computations research centers 

Topology
Order theory
Matematika