Intersection

Last updated January 26, 2024

In 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 in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of elements 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 shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower dimension that is incident 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 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 results in a set, 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=\{x:x\in A{\text{  and  }}x\in B\}

.^[1]

For example, if $A=\{1,3,5,7\}$ and $B=\{1,2,4,6\}$ , then $A\cap B=\{1\}$ . A more elaborate example (involving infinite sets) is:

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

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

{\text{ , then}}

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

As another example, the number $5$ is not contained in the intersection of the set of prime numbers ${2, 3, 5, 7, 11, \dots}$ and the set of even numbers ${2, 4, 6, 8, 10, \dots}$ , 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

The red dot represents the point at which the two lines intersect. Schnittpunkt-2g.svg — The red dot represents the point at which the two lines intersect.

In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a vertex ) or does not exist (if the lines are parallel). Other types of geometric intersection include:

Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically.

Notation

Intersection is denoted by the U+2229∩INTERSECTION from Unicode Mathematical Operators.

The symbol U+2229∩ 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.^[2]^[3]

Peano also created the large symbols for general intersection and union of more than two classes in his 1908 book Formulario mathematico.^[4]^[5]

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<span class="mw-page-title-main">Intersection curve</span> Curve that is common to two geometric objects

In geometry, an intersection curve is a curve that is common to two geometric objects. In the simplest case, the intersection of two non-parallel planes in Euclidean 3-space is a line. In general, an intersection curve consists of the common points of two transversally intersecting surfaces, meaning that at any common point the surface normals are not parallel. This restriction excludes cases where the surfaces are touching or have surface parts in common.

References

↑ Vereshchagin, Nikolai Konstantinovich; Shen, Alexander (2002-01-01). Basic Set Theory. American Mathematical Soc. ISBN 9780821827314.
↑ Peano, Giuseppe (1888-01-01). Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann: preceduto dalle operazioni della logica deduttiva (in Italian). Torino: Fratelli Bocca.
↑ Cajori, Florian (2007-01-01). A History of Mathematical Notations. Torino: Cosimo, Inc. ISBN 9781602067141.
↑ Peano, Giuseppe (1908-01-01). Formulario mathematico, tomo V (in Italian). Torino: Edizione cremonese (Facsimile-Reprint at Rome, 1960). p. 82. OCLC 23485397.
↑ Earliest Uses of Symbols of Set Theory and Logic

External links

Weisstein, Eric W. "Intersection". MathWorld .

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[:0-1] Vereshchagin, Nikolai Konstantinovich; Shen, Alexander (2002-01-01). Basic Set Theory. American Mathematical Soc. ISBN 9780821827314.

[:1-2] Peano, Giuseppe (1888-01-01). Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann: preceduto dalle operazioni della logica deduttiva (in Italian). Torino: Fratelli Bocca.

[:2-3] Cajori, Florian (2007-01-01). A History of Mathematical Notations. Torino: Cosimo, Inc. ISBN 9781602067141.

[:3-4] Peano, Giuseppe (1908-01-01). Formulario mathematico, tomo V (in Italian). Torino: Edizione cremonese (Facsimile-Reprint at Rome, 1960). p. 82. OCLC 23485397.

[5] Earliest Uses of Symbols of Set Theory and Logic

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