Intersection of the Bypass Highway and Van Giesen Street 6.jpg
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
in a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...
are not parallel, their intersection is the
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of
elements Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of ...
which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space. Intersection is one of the basic concepts of geometry. An intersection can have various
geometric shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...
s, but a
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of
flats Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...
) as an object of lower dimension that is
incident Incident may refer to: * A property of a graph in graph theory * ''Incident'' (film), a 1948 film noir * Incident (festival), a cultural festival of The National Institute of Technology in Surathkal, Karnataka, India * Incident (Scientology), a ...
to each of original objects. In this approach an intersection can be sometimes undefined, such as for parallel lines. In both cases the concept of intersection relies on logical conjunction.
Algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
defines intersections in its own way with intersection theory.


Uniqueness

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
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...
results in a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, possibly empty), or as several intersection objects ( possibly zero).


In set theory

The intersection of two sets ''A'' and ''B'' is the set of elements which are in both ''A'' and ''B''. Formally, :A \cap B = \. For example, if A = \ and B = \, then A \cap B = \. A more elaborate example (involving infinite sets) is: : ''A'' = : ''B'' = : A \cap B = \ As another example, the number 5 is ''not'' contained in the intersection of the set of prime numbers and the set of
even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
s , because although 5 ''is'' a prime number, it is ''not'' 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.


In geometry


Notation

Intersection is denoted by the from
Unicode Mathematical Operators The Unicode Standard encodes almost all standard characters used in mathematics. Unicode Technical Report #25 provides comprehensive information about the character repertoire, their properties, and guidelines for implementation. Mathematical ope ...
. The symbol was first used by Hermann Grassmann in ''Die Ausdehnungslehre von 1844'' as general operation symbol, not specialized for intersection. From there, it was used by Giuseppe Peano (1858-1932) for intersection, in 1888 in ''Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann''. Peano also created the large symbols for general intersection and union of more than two classes in his 1908 book ''Formulario mathematico''.


See also

*
Constructive solid geometry Constructive solid geometry (CSG; formerly called computational binary solid geometry) is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combi ...
, Boolean Intersection is one of the ways of combining 2D/3D shapes *
Dimensionally Extended 9-Intersection Model The Dimensionally Extended 9-Intersection Model (DE-9IM) is a topological model and a standard used to describe the spatial relations of two regions (two geometries in two-dimensions, R2), in geometry, point-set topology, geospatial topology, ...
* Meet (lattice theory)


References


External links

*{{MathWorld, Intersection zh-yue:交點 Broad-concept articles