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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.
The intersection (red) of two disks (white and red with black boundaries).
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, 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 (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. Euclidean geometry deals with the intersections of planar and solid shapes.

UniquenessEdit

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 theoryEdit

 
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 A and B is the set of elements which are in both A and B. In symbols,

 .[1]

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

A = {x is an even integer}
B = {x is an integer divisible by 3}
 

As another example, the number 9 is not contained in the intersection of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of even numbers {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.

In Euclidean geometryEdit

NotationEdit

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].

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

See alsoEdit

ReferencesEdit

  1. ^ Vereshchagin, Nikolai Konstantinovich; Shen, Alexander (2002-01-01). Basic Set Theory. American Mathematical Soc. ISBN 9780821827314.
  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.
  3. ^ Cajori, Florian (2007-01-01). A History of Mathematical Notations. Torino: Cosimo, Inc.
  4. ^ Peano, Giuseppe (1908-01-01). Formulario mathematico, tomo V (in Italian). Torino: Edizione cremonese (Facsimile-Reprint at Rome, 1960). p. 82.
  5. ^ Earliest Uses of Symbols of Set Theory and Logic

External linksEdit