Unit interval


Unit interval

In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital letter I). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology.

In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: (0,1], [0,1), and (0,1) However, the notation I is most commonly reserved for the closed interval [0,1].

Contents

Properties

The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval.

In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1. Its standard orientation goes from 0 to 1.

The unit interval is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum and an infimum).

Cardinality

The size or cardinality of a set is the number of elements it contains.

The unit interval is a subset of the real numbers \mathbb{R}. However, it has the same size as the whole set: the cardinality of the continuum. Since the real numbers can be used to represent points along an infinitely long line, this implies that a line segment of length 1, which is a part of that line, has the same number of points as the whole line. Moreover, it has the same number of points as a square of area 1, as a cube of volume 1, and even as an unbounded n-dimensional Euclidean space \mathbb{R}^n (see Space filling curve).

The number of elements (either real numbers or points) in all the above-mentioned sets is uncountable, as it is strictly greater than the number of natural numbers.

Generalizations

Sometimes, the term "unit interval" is used to refer to objects that play a role in various branches of mathematics analogous to the role that [0,1] plays in homotopy theory. For example, in the theory of quivers, the (analogue of the) unit interval is the graph whose vertex set is {0,1} and which contains a single edge e whose source is 0 and whose target is 1. One can then define a notion of homotopy between quiver homomorphisms analogous to the notion of homotopy between continuous maps.

Fuzzy logic

In logic, the unit interval [0,1] can be interpreted as a generalization of the Boolean domain {0,1}, in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 − x, conjunction (AND) is replaced with multiplication (xy), and disjunction (OR) is defined via De Morgan's law.

Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.

See also

References

  • Robert G. Bartle, 1964, The Elements of Real Analysis, John Wiley & Sons.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Unit Interval — Einheit Norm Hilfsmaßeinheit Einheitenname Unit Interval Einheitenzeichen UI Das Unit Interval, abgekürzt UI, ist eine Hilfsmaßeinheit zur Bewertung von Jitter im Rahmen der digitalen Signalverarbeitung und wird unter anderem in der ITU T G.810… …   Deutsch Wikipedia

  • unit interval — noun the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one …   Wiktionary

  • Unit interval (data transmission) — The unit interval is the minimum time interval between condition changes of a data transmission signal, also known as the symbol duration time.ee also* Baud * Cycle * TelecommunicationsReferences …   Wikipedia

  • Interval-valued computation — is a special kind of theoretical models for computation. It is capable of working on “interval valued bytes”: special subsets of the unit interval. If such computers were realized, their computation power would be much greater than that of… …   Wikipedia

  • Unit — may refer to:In mathematics: * Unit vector, a vector with length equal to 1 * Unit circle, the circle with radius equal to 1, centered at the origin * Unit interval, the interval of all real numbers between 0 and 1 * Imaginary unit, i , whose… …   Wikipedia

  • Unit square — The unit square is a square with all of the side lengths equalling 1.In the real planeIn a Cartesian coordinate system with coordinates ( x , y ) the unit square is defined as the square consisting of the points where both x and y lie in the unit …   Wikipedia

  • Interval exchange transformation — In mathematics, an interval exchange transformation is a kind of dynamical system that generalises the idea of a circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several… …   Wikipedia

  • Interval International — (II) is a publicly traded company and former subsidiary of the corporation IAC/InterActiveCorp and is headquartered in Miami, Florida, USA. The CEO is Craig Nash.II is an affiliated exchange company that arranges vacation exchanges for timeshare… …   Wikipedia

  • Unit — U nit, n. [Abbrev. from unity.] 1. A single thing or person. [1913 Webster] 2. (Arith.) The least whole number; one. [1913 Webster] Units are the integral parts of any large number. I. Watts. [1913 Webster] 3. A gold coin of the reign of James I …   The Collaborative International Dictionary of English

  • Unit deme — Unit U nit, n. [Abbrev. from unity.] 1. A single thing or person. [1913 Webster] 2. (Arith.) The least whole number; one. [1913 Webster] Units are the integral parts of any large number. I. Watts. [1913 Webster] 3. A gold coin of the reign of… …   The Collaborative International Dictionary of English


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.