Intersection

File:Venn0001.svg

The intersection (red) of two disks (white and red with black boundaries).

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The circle (black) intersects the line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points.

File:Example of a non pairwise disjoint family of sets.svg

The intersection of D and E is shown in grayish purple. The intersection of A with any of B, C, D, or E is the empty set.

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. It simply means the overlapping area of two or more objects or geometries.

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 the original objects. In this approach an intersection can be sometimes undefined, such as for parallel lines. In both the cases the concept of intersection relies on logical conjunction. Algebraic geometry defines intersections in its own way with intersection theory.

UniquenessTemplate:Clarify

Template:Unreferenced Section 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

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Considering a road to correspond to the set of all its locations, a road intersection (cyan) of two roads (green, blue) corresponds to the intersection of their sets.

The intersection of two sets Template:Math and Template:Math is the set of elements which are in both Template:Math and Template:Math. Formally,

$A\cap B=\{x:x\in A\text{ and }x\in B\}$.

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 Template:Math is not contained in the intersection of the set of prime numbers Template:Math and the set of even numbers Template:Math, because although Template:Math is a prime number, it is not even. In fact, the number Template:Math is the only number in the intersection of these two sets. In this case, the intersection has mathematical meaning: the number Template:Math is the only even prime number.

Notation

Intersection is denoted by the Template:Unichar from Unicode Mathematical Operators.

The symbol Template:Unichar 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, 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)

External links

• Template:MathWorld