Intersection

<h2 id="uniqueness">Uniqueness</h2>

<p>There can be more than one primitive object, such as points (pictured above), that form an intersection. The intersection can be viewed collectively as all of the shared objects (i.e., the intersection <a href="/facts/Operation_(mathematics)/Yplv602Q">operation</a> results in a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>, possibly empty), or as <a href="/facts/Multivalued_function/WH5ua6BL">several intersection objects</a> (<a href="/facts/Partial_function/0vJMw0Nm">possibly zero</a>).
</p>
<h2 id="in-set-theory">In set theory</h2>

<p class="note">Main article: <a href="/facts/Intersection_(set_theory)/RPfUNJh9">Intersection (set theory)</a></p>
<p>The intersection of two sets <i>A</i> and <i>B</i> is the set of elements which are in both <i>A</i> and <i>B</i>.  Formally,
</p>

A
        ∩
        B
        =
        {
        x
        :
        x
        ∈
        A
        
           and 
        
        x
        ∈
        B
        }
      
    
    {\displaystyle A\cap B=\{x:x\in A{\text{  and  }}x\in B\}}
  
.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
<p>For example, if 
  
    
      
        A
        =
        {
        1
        ,
        3
        ,
        5
        ,
        7
        }
      
    
    {\displaystyle A=\{1,3,5,7\}}
  
 and 
  
    
      
        B
        =
        {
        1
        ,
        2
        ,
        4
        ,
        6
        }
      
    
    {\displaystyle B=\{1,2,4,6\}}
  
, then 
  
    
      
        A
        ∩
        B
        =
        {
        1
        }
      
    
    {\displaystyle A\cap B=\{1\}}
  
.  A more elaborate example (involving infinite sets) is:
</p>

A
        =
        {
        x
        :
        
           x is an even integer
        
        }
      
    
    {\displaystyle A=\{x:{\text{ x is an even integer}}\}}

B
        =
        {
        x
        :
        
           x is an integer divisible by 3
        
        }
      
    
    {\displaystyle B=\{x:{\text{ x is an integer divisible by 3}}\}}

, then
        
      
    
    {\displaystyle {\text{ , then}}}

A
        ∩
        B
        =
        {
        6
        ,
        12
        ,
        18
        ,
        …
        }
      
    
    {\displaystyle A\cap B=\{6,12,18,\dots \}}

<p>As another example, the number 5 is <i>not</i> contained in the intersection of the set of <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> {2, 3, 5, 7, 11, …}  and the set of <a href="/facts/Even_number/7zq2UM4V">even numbers</a> {2, 4, 6, 8, 10, …} , because although 5 <i>is</i> a prime number, it is <i>not</i> even. In fact, the number 2 is the only number in the intersection of these two sets. In this case, the intersection has mathematical meaning: the number 2 is the only even prime number.
</p>
<h2 id="in-geometry">In geometry</h2>
<p class="note">This section is an excerpt from <a href="/facts/Intersection_(geometry)/RwH57yPq">Intersection (geometry)</a>.[<a href="https://en.wikipedia.org/w/index.php?title=Intersection_(geometry)&action=edit">edit</a>]</p>

<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, an <a href="/facts/Intersection_(geometry)/RwH57yPq">intersection</a> is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a> is the <a href="/facts/Line%25E2%2580%2593line_intersection/41R6tjCG">line–line intersection</a> between two distinct <a href="/facts/Line_(geometry)/gJqsr4Gj">lines</a>, which either is one <a href="/facts/Point_(geometry)/6YPAvX2a">point</a> (sometimes called a <i><a href="/facts/Vertex_(geometry)/ID3cea9z">vertex</a></i>) or does not exist (if the lines are <a href="/facts/Parallel_lines/16r5yq7D">parallel</a>). Other types of geometric intersection include:
</p>
<ul><li><a href="/facts/Line%25E2%2580%2593plane_intersection/N52PmX7z">Line–plane intersection</a></li>
<li><a href="/facts/Line%25E2%2580%2593sphere_intersection/rLgqJfMC">Line–sphere intersection</a></li>
<li><a href="/facts/Intersection_of_a_polyhedron_with_a_line/CfrIDlEL">Intersection of a polyhedron with a line</a></li>
<li><a href="/facts/Line_segment_intersection/RwH57yPq">Line segment intersection</a></li>
<li><a href="/facts/Intersection_curve/tpXrr2EU">Intersection curve</a></li></ul>
Determination of the intersection of <a href="/facts/Flat_(geometry)/IlMjx7RK">flats</a> – linear geometric objects embedded in a higher-<a href="/facts/Dimension/0ktmUyac">dimensional</a> space – is a simple task of <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, namely the solution of a <a href="/facts/System_of_linear_equations/fk9CFdgp">system of linear equations</a>. In general the determination of an intersection leads to <a href="/facts/Non-linear_equation/4aYItALC">non-linear equations</a>, which can be <a href="/facts/Numerical_solution/QT9cYa9z">solved numerically</a>, for example using <a href="/facts/Newton_iteration/VlRhI2FL">Newton iteration</a>. Intersection problems between a line and a <a href="/facts/Conic_section/teqgLptX">conic section</a> (circle, ellipse, parabola, etc.) or a <a href="/facts/Quadric/L3cTApd4">quadric</a> (sphere, cylinder, hyperboloid, etc.) lead to <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equations</a> that can be easily solved. Intersections between quadrics lead to <a href="/facts/Quartic_equation/SjjtXkHL">quartic equations</a> that can be solved <a href="/facts/Algebraic_equation/1QauIGxs">algebraically</a>.
<h2 id="notation">Notation</h2>
<p>Intersection is denoted by the U+2229 ∩ INTERSECTION from <a href="/facts/Unicode_Mathematical_Operators/N8sebKEI">Unicode Mathematical Operators</a>.
</p><p>The symbol U+2229 ∩ INTERSECTION was first used by <a href="/facts/Hermann_Grassmann/oHCLh2IC">Hermann Grassmann</a> in <i>Die Ausdehnungslehre von 1844</i> as general operation symbol, not specialized for intersection. From there, it was used by <a href="/facts/Giuseppe_Peano/VT2Wsmu7">Giuseppe Peano</a> (1858–1932) for intersection, in 1888 in <i>Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann</i>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>Peano also created the large symbols for general intersection and <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of more than two classes in his 1908 book <i>Formulario mathematico</i>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Constructive_solid_geometry/Dcs2fzyU">Constructive solid geometry</a>, Boolean Intersection is one of the ways of combining 2D/3D shapes</li>
<li><a href="/facts/Dimensionally_Extended_9-Intersection_Model/Ha6Mm78o">Dimensionally Extended 9-Intersection Model</a></li>
<li><a href="/facts/Meet_(lattice_theory)/5ak6ufky">Meet (lattice theory)</a></li>
<li><a href="/facts/Intersection_(set_theory)/RPfUNJh9">Intersection (set theory)</a></li>
<li><a href="/facts/Union_(set_theory)/KVVXKfWl">Union (set theory)</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Intersection.html">"Intersection"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Vereshchagin, Nikolai Konstantinovich; Shen, Alexander (2002-01-01). Basic Set Theory. American Mathematical Soc. ISBN 9780821827314. <a href="9780821827314" target="_blank">9780821827314</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Peano, Giuseppe (1888-01-01). Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann: preceduto dalle operazioni della logica deduttiva (in Italian). Torino: Fratelli Bocca. <a href="https://books.google.com/books?id=5LJi3dxLzuwC" target="_blank">https://books.google.com/books?id=5LJi3dxLzuwC</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Cajori, Florian (2007-01-01). A History of Mathematical Notations. Torino: Cosimo, Inc. ISBN 9781602067141. <a href="9781602067141" target="_blank">9781602067141</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Peano, Giuseppe (1908-01-01). Formulario mathematico, tomo V (in Italian). Torino: Edizione cremonese (Facsimile-Reprint at Rome, 1960). p. 82. OCLC 23485397. <a href="/wiki/OCLC_(identifier)" target="_blank">/wiki/OCLC_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Earliest Uses of Symbols of Set Theory and Logic <a href="http://www.math.hawaii.edu/~tom/history/set.html" target="_blank">http://www.math.hawaii.edu/~tom/history/set.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Intersection open-in-new

Intersection