| Revision as of 16:06, 29 January 2014 editIncnis Mrsi (talk | contribs)Extended confirmed users, Pending changes reviewers, Rollbackers11,646 edits Incnis Mrsi moved page Intersection to Intersection (disambiguation): a broad mathematical concept is a WP:PRIMARYTOPIC | Revision as of 16:08, 29 January 2014 edit undoIncnis Mrsi (talk | contribs)Extended confirmed users, Pending changes reviewers, Rollbackers11,646 edits a WP:CONCEPTDAB stub (some borrowing from what is now Intersection (disambiguation)Next edit → | ||
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| {{about|a broad mathematical concept|the operation on sets|intersection (set theory)|intersections of planar and solid shapes|intersection (Euclidean geometry)|other uses}} | |||
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| ] (black) intersects this ] (purple) in two points]] | |||
| In ]s, '''intersection''' of two or more objects is another, usually "smaller" object. All objects are presumed to lie in certain common ] except for ] where the intersection of arbitrary sets is defined. It is one of basic concepts of ]. Intuitively, the intersection of two or more ] is a new object that lie in each of original objects. Intersection can have various ]s, but a ] is the most common in a <!-- indefinite article: there are different “plane geometries” -->]. | |||
| Definitions vary in different contexts: set theory formalizes the idea that a smaller object lies in a larger object with ], and the ] is formed of ] that belong to all intersecting sets. It is always ], but may be ]. ] defines an <!-- yes, must be indefinite article -->intersection (usually, of ]) as an object of lower ] that is ] to each of original objects. In this approach an intersection can be sometimes undefined, such as for ]. In both cases the concept of intersection relies on ]. | |||
| ] defines intersections in its own way with the ]. | |||
| <!-- In set theory, there are also intersections of ] blah-blah-blah | |||
| not sure it’s very topical --Incnis Mrsi --> | |||
| ==Uniqueness== | |||
| There can be more that one primitive objects, such as points (pictured above) that form an intersection. It can be understood ambiguously: either the intersection is all of them (i.e. the intersection ] result in a ], possibly empty), or there are are ] (]). | |||
| ==Examples in classical geometry== | |||
| {{further|Intersection (Euclidean geometry)}} | |||
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| ==Notation== | |||
| Intersection is denoted by the {{unichar|2229|intersection}} from ]. | |||
| {{expand section|history of the symbol|date=January 2014}} | |||
| ==See also== | |||
| * ], Boolean Intersection is one of the ways of combining 2D/3D shapes | |||
| * ] | |||
| ==References== | |||
| {{MathWorld|Intersection}} | |||
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| ] | |||
| {{geometry-stub}} | |||
Revision as of 16:08, 29 January 2014
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In mathematics, intersection of two or more objects is another, usually "smaller" object. All objects are presumed to lie in certain common space except for set theory where the intersection of arbitrary sets is defined. It is one of basic concepts of geometry. Intuitively, the intersection of two or more object is a new object that lie in each of original objects. 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 the intersection theory.
Uniqueness
There can be more that one primitive objects, such as points (pictured above) that form an intersection. It can be understood ambiguously: either the intersection is all of them (i.e. the intersection operation result in a set, possibly empty), or there are are several intersection objects (possibly zero).
Examples in classical geometry
Further information: Intersection (Euclidean geometry)- Line–line intersection
- Line–plane intersection
- Line–sphere intersection
- Intersection of a polyhedron with a line
- Line segment intersection
- Intersection curve
Notation
Intersection is denoted by the U+2229 ∩ INTERSECTION from Unicode Mathematical Operators.
 | This section needs expansion with: history of the symbol. You can help by adding to it. (January 2014) |
See also
- Constructive solid geometry, Boolean Intersection is one of the ways of combining 2D/3D shapes
- Meet (lattice theory)
References
Weisstein, Eric W. "Intersection". MathWorld.
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