mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the unit interval is the closed interval , that is, the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of all

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...

in the field of

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

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

Properties

The unit interval is a complete metric space, homeomorphic to the

extended real number line In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...

. As a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

, 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 Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in ...

, the unit interval is a one-dimensional analytical

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

'' of a set is the number of elements it contains. The unit interval is a

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of the

\mathbb

. However, it has the same size as the whole set: the

cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \math ...

. Since the real numbers can be used to represent points along an infinitely long line, this implies that a

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...

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 Area is the quantity that expresses the extent of a Region (mathematics), region on the plane (geometry), plane or on a curved surface (mathematics), surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar ...

1, as a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

1, and even as an unbounded ''n''-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

\mathbb^n

(see Space filling curve). The number of elements (either real numbers or points) in all the above-mentioned sets is

uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...

, as it is strictly greater than the number of

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

Generalizations

The interval , with length two, demarcated by the positive and negative units, occurs frequently, such as in the range of the

trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

s sine and cosine and the

hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...

tanh. This interval may be used for the domain of

inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X ...

s. For instance, when is restricted to then

\sin\theta

is in this interval and arcsine is defined there. 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 plays in homotopy theory. For example, in the theory of

quiver A quiver is a container for holding arrows, bolts, ammo, projectiles, darts, or javelins. It can be carried on an archer's body, the bow, or the ground, depending on the type of shooting and the archer's personal preference. Quivers were tr ...

s, the (analogue of the) unit interval is the graph whose vertex set is

\

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

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...

, the unit interval can be interpreted as a generalization of the Boolean domain , 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 ; conjunction (AND) is replaced with multiplication (); and disjunction (OR) is defined, per De Morgan's laws, as . Interpreting these values as logical

truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some prog ...

s 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.

References

* Robert G. Bartle, 1964, ''The Elements of Real Analysis'', John Wiley & Sons. Sets of real numbers 1 (number) Topology

Properties

Cardinality

Generalizations

Fuzzy logic

See also

References