“Intersection”的意思、由来-开放百科全书

词条

Intersection

释义

Uniqueness
Intersection of two sets
Examples in classical geometry
Notation
See also
References

{{about|a broad mathematical concept|the operation on sets|Intersection (set theory)|intersections of planar and solid shapes|Intersection (Euclidean geometry)|the point where two or more roads intersect|Intersection (road)|other uses}}{{refimprove|date=January 2014}}

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.

Uniqueness

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).

Intersection of two 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 A ∩ B = {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.

Examples in classical 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 {{unichar|2229|intersection}} from Unicode Mathematical Operators.

References

1. ^{{Cite book|url=https://books.google.com/books?id=LBvpfEMhurwC|title=Basic Set Theory|last=Vereshchagin|first=Nikolai Konstantinovich|last2=Shen|first2=Alexander|date=2002-01-01|publisher=American Mathematical Soc.|isbn=9780821827314|language=en}}

1 : Geometry

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

Uniqueness

Intersection of two sets

Examples in classical geometry

Notation

See also

References