Intersection

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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. Intuitively, the intersection of objects is that which belongs to all of them. 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.

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 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.

Uniqueness[edit]

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[edit]

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 5 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 although 5 is a prime number, it is not even. In fact, the number 2 is the only number in the intersection of these two sets. In this case, the intersection has mathematical meaning: the number 2 is the only even prime number.

In Euclidean geometry[edit]

Notation[edit]

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]

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[edit]

References[edit]

  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. ISBN 9781602067141.
  4. ^ Peano, Giuseppe (1908-01-01). Formulario mathematico, tomo V (in Italian). Torino: Edizione cremonese (Facsimile-Reprint at Rome, 1960). p. 82. OCLC 23485397.
  5. ^ Earliest Uses of Symbols of Set Theory and Logic

External links[edit]