Intersection


Image:Circle-line intersection.svg|thumb|right|The circle intersects the line in two points. The disk intersects the line in the line segment between the two red points.
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.
Intersections can be thought of either collectively or individually, see Intersection (geometry) for an example of the latter. The definition given above exemplifies the collective view, whereby the intersection operation always results in a well-defined and unique, although possibly empty, set of mathematical objects. In contrast, the individual view focuses on the separate members of this set. Given this view, intersections need not be unique, as shown by the two points of intersection between a circle and a line pictured. Similarly, intersections need not exist as between two parallel but distinct lines in Euclidean geometry.
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 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.

In set theory

The intersection of two sets and is the set of elements which are in both and. Formally,
For example, if and, then. A more elaborate example is:
As another example, the number is not contained in the intersection of the set of prime numbers and the set of even numbers, because although is a prime number, it is not even.
Sets can have an empty intersection. For example, if and, then. Such sets are called disjoint sets and may colloquially be said to have no intersection.

Notation

Intersection is denoted by the from Unicode Mathematical Operators.
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 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.