Unit interval

 ( Sets Of Real Numbers ) 

In mathematics, the unit interval is the closed interval , that is, the set of all 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 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: , , and . However, the notation is most commonly reserved for the closed interval .

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

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

The unit interval is a total order and a complete lattice (every subset of the unit interval has a supremum and an infimum).

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

The unit interval is a subset of the $\backslash mathbb\{R\}$. 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 Real 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 $\backslash mathbb\{R\}^n$ (see Space filling curve).

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

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Orientation The unit interval is a curve. The open interval (0,1) is a subset of the positive real numbers and inherits an orientation from them. The orientation is reversed when the interval is entered from 1, such as in the integral $\backslash int\_1^x\; \backslash frac\{dt\}\{t\}$ used to define natural logarithm for x in the interval, thus yielding negative values for logarithm of such x. In fact, this integral is evaluated as a signed area yielding negative area over the unit interval due to reversed orientation there.

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Generalizations The interval , with length two, demarcated by the positive and negative units, occurs frequently, such as in the range of the trigonometric functions sine and cosine and the hyperbolic function tanh. This interval may be used for the domain of . For instance, when is restricted to then $\backslash sin\backslash theta$ 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 quivers, the (analogue of the) unit interval is the graph whose vertex set is $\backslash \{0,1\backslash \}$ 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 analogous to the notion of homotopy between continuous maps.

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Fuzzy logic In logic, the unit interval 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 ; conjunction (AND) is replaced with multiplication (); and disjunction (OR) is defined, per De Morgan's laws, as .

Interpreting these values as logical 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.

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See also

• Interval notation
• Unit unit square, unit cube, unit circle, unit hyperbola and unit sphere
• Unit impulse
• Unit vector

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

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