# Unit interval > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Unit_interval > Markdown URL: https://mediated.wiki/source/Unit_interval.md > Source: https://en.wikipedia.org/wiki/Unit_interval > Source revision: 1287160926 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Closed interval [0,1] on the real number line}} {{For|the data transmission signaling interval|Unit interval (data transmission)}} [[File:Unit-interval.svg|thumb|The unit interval as a [subset](/source/subset) of the [real line](/source/real_line)]] In [mathematics](/source/mathematics), the '''unit interval''' is the [closed interval](/source/interval_(mathematics)) {{closed-closed|0,1}}, that is, the [set](/source/Set_(mathematics)) of all [real number](/source/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 {{mono|I}}). In addition to its role in [real analysis](/source/real_analysis), the unit interval is used to study [homotopy theory](/source/homotopy_theory) in the field of [topology](/source/topology). 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}}. == Properties == The unit interval is a [complete metric space](/source/complete_metric_space), [homeomorphic](/source/homeomorphism) to the [extended real number line](/source/extended_real_number_line). As a [topological space](/source/topological_space), it is [compact](/source/compact_space), [contractible](/source/contractible), [path connected](/source/connectedness) and [locally path connected](/source/Locally_connected_space). The [Hilbert cube](/source/Hilbert_cube) is obtained by taking a [topological product](/source/Product_topology) of countably many copies of the unit interval. In [mathematical analysis](/source/mathematical_analysis), the unit interval is a [one-dimensional](/source/dimension) analytical [manifold](/source/manifold) whose boundary consists of the two points 0 and 1. Its standard [orientation](/source/orientability) goes from 0 to 1. The unit interval is a [totally ordered set](/source/total_order) and a [complete lattice](/source/complete_lattice) (every subset of the unit interval has a [supremum](/source/supremum) and an [infimum](/source/infimum)). ===Cardinality=== {{Main|Cardinality of the continuum}} The ''size'' or ''[cardinality](/source/cardinality)'' of a set is the number of elements it contains. The unit interval is a [subset](/source/subset) of the [real number](/source/real_number)s \mathbb{R}. However, it has the same size as the whole set: the [cardinality of the continuum](/source/cardinality_of_the_continuum). Since the real numbers can be used to represent points along an [infinitely long line](/source/Real_line), this implies that a [line segment](/source/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](/source/area) 1, as a [cube](/source/cube) of [volume](/source/volume) 1, and even as an unbounded ''n''-dimensional [Euclidean space](/source/Euclidean_space) \mathbb{R}^n (see [Space filling curve](/source/Space_filling_curve)). The number of elements (either real numbers or points) in all the above-mentioned sets is [uncountable](/source/Uncountable_set), as it is strictly greater than the number of [natural number](/source/natural_number)s. ===Orientation=== The unit interval is a [curve](/source/curve). The open interval (0,1) is a subset of the [positive real numbers](/source/positive_real_numbers) and inherits an orientation from them. The [orientation](/source/curve_orientation) is reversed when the interval is entered from 1, such as in the integral \int_1^x \frac{dt}{t} used to define [natural logarithm](/source/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](/source/signed_area) yielding ''negative area'' over the unit interval due to reversed orientation there. == Generalizations == The interval {{closed-closed|-1,1}}, with length two, demarcated by the positive and negative units, occurs frequently, such as in the [range](/source/range_of_a_function) of the [trigonometric function](/source/trigonometric_function)s sine and cosine and the [hyperbolic function](/source/hyperbolic_function) tanh. This interval may be used for the [domain](/source/domain_of_a_function) of [inverse function](/source/inverse_function)s. For instance, when {{theta}} is restricted to {{closed-closed|−π/2, π/2}} 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 {{closed-closed|0,1}} plays in homotopy theory. For example, in the theory of [quiver](/source/quiver_(mathematics))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](/source/homotopy) between quiver [homomorphism](/source/homomorphism)s analogous to the notion of homotopy between [continuous](/source/continuous_function_(topology)) maps. == Fuzzy logic == In [logic](/source/logic), the unit interval {{closed-closed|0,1}} can be interpreted as a generalization of the [Boolean domain](/source/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](/source/negation) (NOT) is replaced with {{math|1 − ''x''}}; [conjunction](/source/Logical_conjunction) (AND) is replaced with multiplication ({{math|''xy''}}); and [disjunction](/source/Logical_disjunction) (OR) is defined, per [De Morgan's laws](/source/De_Morgan's_laws), as {{math|1 − (1 − ''x'')(1 − ''y'')}}. Interpreting these values as logical [truth value](/source/truth_value)s yields a [multi-valued logic](/source/multi-valued_logic), which forms the basis for [fuzzy logic](/source/fuzzy_logic) and [probabilistic logic](/source/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== {{wiktionary}} * [Interval notation](/source/Interval_notation) * Unit [square](/source/unit_square), [cube](/source/unit_cube), [circle](/source/unit_circle), [hyperbola](/source/unit_hyperbola) and [sphere](/source/unit_sphere) * [Unit impulse](/source/Unit_impulse) * [Unit vector](/source/Unit_vector) ==References== * Robert G. Bartle, 1964, ''The Elements of Real Analysis'', John Wiley & Sons. Category:Sets of real numbers Category:1 (number) Category:Topology --- Adapted from the Wikipedia article [Unit interval](https://en.wikipedia.org/wiki/Unit_interval) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Unit_interval?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.