From en.wikipedia.org:
[Concept in mathematics] [a broad mathematical concept] [date=January 2014]
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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, (individual) 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 <!--
indefinite article: there are different “plane geometries” -->plane geometry.
Incidence geometry defines an <!-- yes, must be indefinite article
-->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.<!-- In set
theory, there are also intersections of classes blah-blah-blah not sure it’s
very topical --Incnis Mrsi -->
** In set theory
[Intersection (set theory)]
The intersection of two sets [_A_] and [_B_] is the set of elements which are
in both [_A_] and [_B_]. Formally,
<math> A \cap B = \{ x: x \in A \text{ and } x \in B\} </math> . [1]
For example, if <math>A = \{1, 3, 5, 7\}</math> and <math>B = \{1, 2, 4,
6\}</math>, then <math>A \cap B = \{1\}</math>. A more elaborate example
(involving infinite sets) is:
<math> A = \{ x:\text{ x is an even integer} \} </math>
<math> B = \{ x:\text{ x is an integer divisible by 3} \} </math> <math> \text{ , then} </math>
<math> A \cap B = \{6, 12, 18, \dots\} </math>
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.
Sets can have an empty intersection. For example, if <math>A = \{Amy,\,
Aaron, \, Arjun\}</math> and <math>B = \{Barack,\, Bas,\, Betty \}</math>,
then <math>A \cap B = \emptyset</math>. Such sets are called disjoint sets
and may colloquially be said to have no intersection.
** In geometry
[Intersection (geometry)]
** Notation
Intersection is denoted by the [2229] from Unicode Mathematical Operators.
The symbol [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
- 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)
** References
References:
[1]. [url=https://books.google.com/books?id=LBvpfEMhurwC]
[2]. [url=https://books.google.com/books?id=5LJi3dxLzuwC]
[3]. [url=https://books.google.com/books?id=bT5suOONXlgC]
[4]. [title=Formulario mathematico, tomo V]
[5]. [http://www.math.hawaii.edu/~tom/history/set.html]
** External links
- [Intersection]
粵語: 交點 Category:Broad-concept articles