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| {{For|the data transmission signaling interval|Unit interval (data transmission)}}
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| [[File:Unit-interval.svg|thumb|The unit interval as a subset of the [[real line]]]]
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| In [[mathematics]], the '''unit interval''' is the [[interval (mathematics)|closed interval]] {{closed-closed|0,1}}, that is, the set of all [[real number]]s that are greater than or equal to 0 and less than or equal to 1. It is often denoted ''{{math|I}}'' (capital letter <big><tt>I</tt></big>).
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| In addition to its role in [[real analysis]], the unit interval is used to study [[homotopy theory]] in the field of [[topology]].
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| In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: {{open-closed|0,1}}, {{closed-open|0,1}}, and {{open-open|0,1}}. However, the notation ''{{math|I}}'' is most commonly reserved for the closed interval {{closed-closed|0,1}}.
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| == Properties ==
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| The unit interval is a [[complete metric space]], [[homeomorphism|homeomorphic]] to the [[extended real number line]]. As a [[topological space]] it is [[compact space|compact]], [[contractible]], [[connectedness|path connected]] and [[Locally connected space|locally path connected]]. The [[Hilbert cube]] is obtained by taking a topological product of countably many copies of the unit interval.
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| In mathematical analysis, the unit interval is a [[dimension|one-dimensional]] analytical [[manifold]] whose boundary consists of the two points 0 and 1. Its standard [[orientability|orientation]] goes from 0 to 1.
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| The unit interval is a [[total order|totally ordered set]] and a [[complete lattice]] (every subset of the unit interval has a [[supremum]] and an [[infimum]]).
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| ===Cardinality===
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| {{Main|Cardinality of the continuum}}
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| The size or '''cardinality''' of a [[Set (mathematics)|set]] is the number of elements it contains.
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| The unit interval is a [[subset]] of the [[real number]]s <math>\mathbb{R}</math>. 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|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]] <math>\mathbb{R}^n</math> (see [[Space filling curve]]).
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| The number of elements (either real numbers or points) in all the above-mentioned sets is [[Uncountable set|uncountable]], as it is strictly greater than the number of [[natural number]]s.
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| == Generalizations ==
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| The interval [−1,1], with length two, demarcated by the positive and negative units, occurs frequently, such as in the [[range (mathematics)|range]] of the [[trigonometric function]]s sine and cosine and the [[hyperbolic function]] tanh. This interval may be used for the [[domain of a function|domain]] of [[inverse function]]s. For instance, when θ is restricted to [−π/2, π/2] then sin(θ) is in this interval and arcsine is defined there.
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| 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 [[quiver (mathematics)|quiver]]s, 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 [[homomorphism]]s analogous to the notion of homotopy between [[continuous function (topology)|continuous]] maps.
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| == Fuzzy logic ==
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| 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 <math>1-x,</math> conjunction (AND) is replaced with multiplication (<math>xy</math>), and disjunction (OR) is defined via [[De Morgan's law]].
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| Interpreting these values as logical [[truth value]]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.
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| ==See also==
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| * [[Interval notation]]
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| * Unit [[unit square|square]], [[unit cube|cube]], [[unit circle|circle]], [[unit hyperbola|hyperbola]] and [[unit sphere|sphere]]
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| * [[Unit vector]]
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| ==References== | |
| * Robert G. Bartle, 1964, ''The Elements of Real Analysis'', John Wiley & Sons.
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| [[Category:Sets of real numbers]]
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| [[Category:One]]
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| [[Category:Topology]]
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