Intersection

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In

.

Uniqueness

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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,

${\displaystyle A\cap B=\{x:x\in A{\text{ and }}x\in B\}}$.^[1]

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

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

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

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

As another example, the number 5 is not contained in the intersection of the set of

In geometry

This section is an excerpt from Intersection (geometry).[edit]

In

parallel

). Other types of geometric intersection include:

• Line–plane intersection
• Line–sphere intersection
• Intersection of a polyhedron with a line
• Line segment intersection
• Intersection curve

Determination of the intersection of

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

.

This section needs expansion with: history of the symbol. You can help by adding to it. (January 2014)

The symbol U+2229 ∩ INTERSECTION 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]

See also

• Constructive solid geometry, Boolean Intersection is one of the ways of combining 2D/3D shapes
• Dimensionally Extended 9-Intersection Model
• Meet (lattice theory)
• Intersection (set theory)
• Union (set theory)

References