Intersection


In mathematics, the intersection of two or more objects is another, usually "smaller" object. All objects are presumed to lie in a certain common space except in set theory, where the intersection of arbitrary sets is defined. The intersection is one of basic concepts of geometry. Intuitively, the intersection of two or more objects is a new object that lies in each of original objects. An intersection can have various geometric shapes, but a point is the most common in a plane geometry.
Definitions vary in different contexts: set theory formalizes the idea that a smaller object lies in a larger object with inclusion, and the intersection of sets is formed of elements that belong to all intersecting sets. It is always defined, but may be empty. Incidence geometry defines an intersection 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.
Euclidean geometry deals with the intersections of planar and solid shapes.

Uniqueness

There can be more than one primitive object, such as points, that form an intersection. The intersection can be viewed collectively as all of the shared objects, or as several intersection objects.

In set theory

The intersection of two sets A and B is the set of elements which are in both A and B. In symbols,
For example, if A = and B = then A ∩ B =. A more elaborate example is:
As another example, the number 9 is not contained in the intersection of the set of prime numbers and the set of even numbers, because 9 is neither prime nor even.

In Euclidean geometry

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.
Giuseppe Peano also created the large symbols for general intersection and union of more than two classes in 1908, at his book Formulario mathematico.