SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Unit interval</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Unit interval</span></span>

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">For the data transmission signaling interval, see <a href="Unit_interval_(data_transmission)" title="Unit interval (data transmission)">Unit interval (data transmission)</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>unit interval</b> is the <a href="Interval_(mathematics)" title="Interval (mathematics)">closed interval</a> <span class="texhtml">[0,1]</span>, that is, the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of all <a href="Real_number" title="Real number">real numbers</a> that are greater than or equal to 0 and less than or equal to 1. It is often denoted <i><span class="texhtml">I</span></i> (capital letter <big><style data-mw-deduplicate="TemplateStyles:r886049734">

/* start https://en.wikipedia.org/ */

.mw-parser-output .monospaced{font-family:monospace,monospace}

/* end https://en.wikipedia.org/ */

</style><span class="monospaced">I</span></big>). In addition to its role in <a href="Real_analysis" title="Real analysis">real analysis</a>, the unit interval is used to study <a href="Homotopy_theory" title="Homotopy theory">homotopy theory</a> in the field of <a href="Topology" title="Topology">topology</a>.

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

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>The unit interval is a <a href="Complete_metric_space" title="Complete metric space">complete metric space</a>, <a href="Homeomorphism" title="Homeomorphism">homeomorphic</a> to the <a href="Extended_real_number_line" title="Extended real number line">extended real number line</a>. As a <a href="Topological_space" title="Topological space">topological space</a>, it is <a href="Compact_space" title="Compact space">compact</a>, <a href="Contractible" class="mw-redirect" title="Contractible">contractible</a>, <a href="Connectedness" title="Connectedness">path connected</a> and <a href="Locally_connected_space" title="Locally connected space">locally path connected</a>. The <a href="Hilbert_cube" title="Hilbert cube">Hilbert cube</a> is obtained by taking a <a href="Product_topology" title="Product topology">topological product</a> of countably many copies of the unit interval.

</p><p>In <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the unit interval is a <a href="Dimension" title="Dimension">one-dimensional</a> analytical <a href="Manifold" title="Manifold">manifold</a> whose boundary consists of the two points 0 and 1. Its standard <a href="Orientability" title="Orientability">orientation</a> goes from 0 to 1.

</p><p>The unit interval is a <a href="Total_order" title="Total order">totally ordered set</a> and a <a href="Complete_lattice" title="Complete lattice">complete lattice</a> (every subset of the unit interval has a <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a> and an <a href="Infimum" class="mw-redirect" title="Infimum">infimum</a>).

</p>

<div class="mw-heading mw-heading3"><h3 id="Cardinality">Cardinality</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cardinality_of_the_continuum" title="Cardinality of the continuum">Cardinality of the continuum</a></div>

<p>The <i>size</i> or <i><a href="Cardinality" title="Cardinality">cardinality</a></i> of a set is the number of elements it contains.

</p><p>The unit interval is a <a href="Subset" title="Subset">subset</a> of the <a href="Real_number" title="Real number">real numbers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>. However, it has the same size as the whole set: the <a href="Cardinality_of_the_continuum" title="Cardinality of the continuum">cardinality of the continuum</a>. Since the real numbers can be used to represent points along an <a href="Real_line" class="mw-redirect" title="Real line">infinitely long line</a>, this implies that a <a href="Line_segment" title="Line segment">line segment</a> 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 <a href="Area" title="Area">area</a> 1, as a <a href="Cube" title="Cube">cube</a> of <a href="Volume" title="Volume">volume</a> 1, and even as an unbounded <i>n</i>-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> (see <a href="Space_filling_curve" class="mw-redirect" title="Space filling curve">Space filling curve</a>).

</p><p>The number of elements (either real numbers or points) in all the above-mentioned sets is <a href="Uncountable_set" title="Uncountable set">uncountable</a>, as it is strictly greater than the number of <a href="Natural_number" title="Natural number">natural numbers</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Orientation">Orientation</h3></div>

<p>The unit interval is a <a href="Curve" title="Curve">curve</a>. The open interval (0,1) is a subset of the <a href="Positive_real_numbers" title="Positive real numbers">positive real numbers</a> and inherits an orientation from them. The <a href="Curve_orientation" title="Curve orientation">orientation</a> is reversed when the interval is entered from 1, such as in the integral <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{1}^{x}{\frac {dt}{t}}}">

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{1}^{x}{\frac {dt}{t}}}</annotation>

</semantics>

</math></span><img src="./2884b87fe3cd17f338bc72cf9dd88e05dec8e64f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.915ex; height:5.843ex;" alt="{\displaystyle \int _{1}^{x}{\frac {dt}{t}}}" loading="lazy"></span> used to define <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a> for <i>x</i> in the interval, thus yielding negative values for logarithm of such <i>x</i>. In fact, this integral is evaluated as a <a href="Signed_area" title="Signed area">signed area</a> yielding <i>negative area</i> over the unit interval due to reversed orientation there.

</p>

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

<p>The interval <span class="texhtml">[-1,1]</span>, with length two, demarcated by the positive and negative units, occurs frequently, such as in the <a href="Range_of_a_function" title="Range of a function">range</a> of the <a href="Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> sine and cosine and the <a href="Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic function</a> tanh. This interval may be used for the <a href="Domain_of_a_function" title="Domain of a function">domain</a> of <a href="Inverse_function" title="Inverse function">inverse functions</a>. For instance, when 𝜃 is restricted to <span class="texhtml">[−π/2, π/2]</span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \theta }</annotation>

</semantics>

</math></span><img src="./efa733f6703578b0c3af870a3170b4ab0dd99c00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.333ex; height:2.176ex;" alt="{\displaystyle \sin \theta }" loading="lazy"></span> is in this interval and arcsine is defined there.

</p><p>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 <span class="texhtml">[0,1]</span> plays in homotopy theory. For example, in the theory of <a href="Quiver_(mathematics)" title="Quiver (mathematics)">quivers</a>, the (analogue of the) unit interval is the graph whose vertex set is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{0,1\}}</annotation>

</semantics>

</math></span><img src="./28de5781698336d21c9c560fb1cbb3fb406923eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{0,1\}}" loading="lazy"></span> and which contains a single edge <i>e</i> whose source is 0 and whose target is 1. One can then define a notion of <a href="Homotopy" title="Homotopy">homotopy</a> between quiver <a href="Homomorphism" title="Homomorphism">homomorphisms</a> analogous to the notion of homotopy between <a href="Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> maps.

</p>

<div class="mw-heading mw-heading2"><h2 id="Fuzzy_logic">Fuzzy logic</h2></div>

<p>In <a href="Logic" title="Logic">logic</a>, the unit interval <span class="texhtml">[0,1]</span> can be interpreted as a generalization of the <a href="Boolean_domain" title="Boolean domain">Boolean domain</a> {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, <a href="Negation" title="Negation">negation</a> (NOT) is replaced with <span class="texhtml">1 − <i>x</i></span>; <a href="Logical_conjunction" title="Logical conjunction">conjunction</a> (AND) is replaced with multiplication (<span class="texhtml"><i>xy</i></span>); and <a href="Logical_disjunction" title="Logical disjunction">disjunction</a> (OR) is defined, per <a href="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a>, as <span class="texhtml">1 − (1 − <i>x</i>)(1 − <i>y</i>)</span>.

</p><p>Interpreting these values as logical <a href="Truth_value" title="Truth value">truth values</a> yields a <a href="Multi-valued_logic" class="mw-redirect" title="Multi-valued logic">multi-valued logic</a>, which forms the basis for <a href="Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a> and <a href="Probabilistic_logic" title="Probabilistic logic">probabilistic logic</a>. 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.

</p>

/* start https://en.wikipedia.org/ */

.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .infobox .side-box{font-size:100%}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1237033735">

/* start https://en.wikipedia.org/ */

@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}

/* end https://en.wikipedia.org/ */

</style><div class="side-box side-box-right sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">

/* start https://en.wikipedia.org/ */

.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}

/* end https://en.wikipedia.org/ */

</style>

<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/unit_interval" class="extiw external" title="wiktionary:Special:Search/unit interval">unit interval</a></b></i> in Wiktionary, the free dictionary.</div></div>

</div>

<ul><li><a href="Interval_notation" class="mw-redirect" title="Interval notation">Interval notation</a></li>

<li>Unit <a href="Unit_square" title="Unit square">square</a>, <a href="Unit_cube" title="Unit cube">cube</a>, <a href="Unit_circle" title="Unit circle">circle</a>, <a href="Unit_hyperbola" title="Unit hyperbola">hyperbola</a> and <a href="Unit_sphere" title="Unit sphere">sphere</a></li>

<li><a href="Unit_impulse" class="mw-redirect" title="Unit impulse">Unit impulse</a></li>

<li><a href="Unit_vector" title="Unit vector">Unit vector</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>Robert G. Bartle, 1964, <i>The Elements of Real Analysis</i>, John Wiley & Sons.</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-24" href="https://en.wikipedia.org/wiki/?title=Unit_interval&oldid=1287160926">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>