Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Intersection (set theory)

From Wikipedia, the free encyclopedia
Set of elements common to all of some sets
For broader coverage of this topic, seeIntersection (mathematics).
Intersection
The intersection of two setsA{\displaystyle A} andB,{\displaystyle B,} represented by circles.A∩B{\displaystyle A\cap B} is in red.
TypeSet operation
FieldSet theory
StatementThe intersection ofA{\displaystyle A} andB{\displaystyle B} is the setA∩B{\displaystyle A\cap B} of elements that lie in both setA{\displaystyle A} and setB{\displaystyle B}.
Symbolic statementA∩B={x:x∈A and x∈B}{\displaystyle A\cap B=\{x:x\in A{\text{ and }}x\in B\}}

Inset theory, theintersection of twosetsA{\displaystyle A} andB,{\displaystyle B,} denoted byA∩B,{\displaystyle A\cap B,}[1] is the set containing all elements ofA{\displaystyle A} that also belong toB{\displaystyle B} or equivalently, all elements ofB{\displaystyle B} that also belong toA.{\displaystyle A.}[2] The notion of intersection as analgebraic operationwith sets as operands has been generalized fromgeometry, where it isencountered in the case of geometric sets ofpoints, such as individual points, lines (infiniteuncountable sets of points), planes, etc.

Notation and terminology

[edit]

Intersection is written using the symbol "∩{\displaystyle \cap }" between the terms; that is, ininfix notation. For example:{1,2,3}∩{2,3,4}={2,3}{\displaystyle \{1,2,3\}\cap \{2,3,4\}=\{2,3\}}{1,2,3}∩{4,5,6}=∅{\displaystyle \{1,2,3\}\cap \{4,5,6\}=\varnothing }Z∩N=N{\displaystyle \mathbb {Z} \cap \mathbb {N} =\mathbb {N} }{x∈R:x2=1}∩N={1}{\displaystyle \{x\in \mathbb {R} :x^{2}=1\}\cap \mathbb {N} =\{1\}}The intersection of more than two sets (generalized intersection) can be written as:⋂i=1nAi{\displaystyle \bigcap _{i=1}^{n}A_{i}}which is similar tocapital-sigma notation.

For an explanation of the symbols used in this article, refer to thetable of mathematical symbols.

Definition

[edit]
Intersection of three sets:
 A∩B∩C{\displaystyle ~A\cap B\cap C}
Intersections of the unaccented modernGreek,Latin, andCyrillic scripts, considering only the shapes of the letters and ignoring their pronunciation
Example of an intersection with sets

The intersection of two setsA{\displaystyle A} andB,{\displaystyle B,} denoted byA∩B{\displaystyle A\cap B},[3] is the set of all objects that are members of both the setsA{\displaystyle A} andB.{\displaystyle B.}In symbols:A∩B={x:x∈A and x∈B}.{\displaystyle A\cap B=\{x:x\in A{\text{ and }}x\in B\}.}

That is,x{\displaystyle x} is an element of the intersectionA∩B{\displaystyle A\cap B}if and only ifx{\displaystyle x} is both an element ofA{\displaystyle A} and an element ofB.{\displaystyle B.}[3]

For example:

  • The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
  • The number 9 isnot in the intersection of the set ofprime numbers {2, 3, 5, 7, 11, ...} and the set ofodd numbers {1, 3, 5, 7, 9, 11, ...}, because 9 is not prime.
  • The intersection of two geometric sets of points such as two lines is asingleton set of one point for distinct non-parallel lines in the same plane.

Intersecting and disjoint sets

[edit]

We say thatA{\displaystyle A} intersects (meets)B{\displaystyle B} if there exists somex{\displaystyle x} that is an element of bothA{\displaystyle A} andB,{\displaystyle B,} in which case we also say thatA{\displaystyle A} intersects (meets)B{\displaystyle B}atx{\displaystyle x}. Equivalently,A{\displaystyle A} intersectsB{\displaystyle B} if their intersectionA∩B{\displaystyle A\cap B} is aninhabited set, meaning that there exists somex{\displaystyle x} such thatx∈A∩B.{\displaystyle x\in A\cap B.}

We say thatA{\displaystyle A} andB{\displaystyle B} aredisjoint ifA{\displaystyle A} does not intersectB.{\displaystyle B.} In plain language, they have no elements in common.A{\displaystyle A} andB{\displaystyle B} are disjoint if their intersection isempty, denotedA∩B=∅.{\displaystyle A\cap B=\varnothing .}

For example:

Algebraic properties

[edit]
See also:List of set identities and relations andAlgebra of sets

Binary intersection is anassociative operation; that is, for any setsA,B,{\displaystyle A,B,} andC,{\displaystyle C,} one has

A∩(B∩C)=(A∩B)∩C.{\displaystyle A\cap (B\cap C)=(A\cap B)\cap C.}Thus the parentheses may be omitted without ambiguity: either of the above can be written asA∩B∩C{\displaystyle A\cap B\cap C}. Intersection is alsocommutative. That is, for anyA{\displaystyle A} andB,{\displaystyle B,} one hasA∩B=B∩A.{\displaystyle A\cap B=B\cap A.}The intersection of any set with theempty set results in the empty set; that is, that for any setA{\displaystyle A},A∩∅=∅{\displaystyle A\cap \varnothing =\varnothing }Also, the intersection operation isidempotent; that is, any setA{\displaystyle A} satisfies thatA∩A=A{\displaystyle A\cap A=A}. All these properties follow from analogous facts aboutlogical conjunction.

Intersectiondistributes overunion and union distributes over intersection. That is, for any setsA,B,{\displaystyle A,B,} andC,{\displaystyle C,} one hasA∩(B∪C)=(A∩B)∪(A∩C)A∪(B∩C)=(A∪B)∩(A∪C){\displaystyle {\begin{aligned}A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\\A\cup (B\cap C)=(A\cup B)\cap (A\cup C)\end{aligned}}}Inside a universeU,{\displaystyle U,} one may define thecomplementAc{\displaystyle A^{c}} ofA{\displaystyle A} to be the set of all elements ofU{\displaystyle U} not inA.{\displaystyle A.} Furthermore, the intersection ofA{\displaystyle A} andB{\displaystyle B} may be written as the complement of theunion of their complements, derived easily fromDe Morgan's laws:A∩B=(Ac∪Bc)c{\displaystyle A\cap B=\left(A^{c}\cup B^{c}\right)^{c}}

Arbitrary intersections

[edit]
Further information:Iterated binary operation

The most general notion is the intersection of an arbitrarynonempty collection of sets.IfM{\displaystyle M} is anonempty set whose elements are themselves sets, thenx{\displaystyle x} is an element of theintersection ofM{\displaystyle M} if and only iffor every elementA{\displaystyle A} ofM,{\displaystyle M,}x{\displaystyle x} is an element ofA.{\displaystyle A.}In symbols:(x∈⋂A∈MA)⇔(∀A∈M, x∈A).{\displaystyle \left(x\in \bigcap _{A\in M}A\right)\Leftrightarrow \left(\forall A\in M,\ x\in A\right).}

The notation for this last concept can vary considerably.Set theorists will sometimes write "⋂M{\displaystyle \bigcap M}", while others will instead write "⋂A∈MA{\displaystyle {\bigcap }_{A\in M}A}".The latter notation can be generalized to "⋂i∈IAi{\displaystyle {\bigcap }_{i\in I}A_{i}}", which refers to the intersection of the collection{Ai:i∈I}.{\displaystyle \left\{A_{i}:i\in I\right\}.}HereI{\displaystyle I} is a nonempty set, andAi{\displaystyle A_{i}} is a set for everyi∈I.{\displaystyle i\in I.}

In the case that theindex setI{\displaystyle I} is the set ofnatural numbers, notation analogous to that of aninfinite product may be seen:⋂i=1∞Ai.{\displaystyle \bigcap _{i=1}^{\infty }A_{i}.}

When formatting is difficult, this can also be written "A1∩A2∩A3∩⋯{\displaystyle A_{1}\cap A_{2}\cap A_{3}\cap \cdots }". This last example, an intersection of countably many sets, is actually very common; for an example, see the article onσ-algebras.

Nullary intersection

[edit]
Conjunctions of the arguments in parentheses

The conjunction of no argument is thetautology (compare:empty product); accordingly the intersection of no set is theuniverse.

In the previous section, we excluded the case whereM{\displaystyle M} was theempty set (∅{\displaystyle \varnothing }). The reason is as follows: The intersection of the collectionM{\displaystyle M} is defined as the set (seeset-builder notation)⋂A∈MA={x: for all A∈M,x∈A}.{\displaystyle \bigcap _{A\in M}A=\{x:{\text{ for all }}A\in M,x\in A\}.}IfM{\displaystyle M} is empty, there are no setsA{\displaystyle A} inM,{\displaystyle M,} so the question becomes "whichx{\displaystyle x}'s satisfy the stated condition?" The answer seems to beevery possiblex{\displaystyle x}. WhenM{\displaystyle M} is empty, the condition given above is an example of avacuous truth. So the intersection of the empty family should be theuniversal set (theidentity element for the operation of intersection),[4]but in standard (ZF) set theory, the universal set does not exist.

However, when restricted to the context of subsets of a given fixed setX{\displaystyle X}, the notion of the intersection of an empty collection of subsets ofX{\displaystyle X} is well-defined. In that case, ifM{\displaystyle M} is empty, its intersection is⋂M=⋂∅={x∈X:x∈A for all A∈∅}{\displaystyle \bigcap M=\bigcap \varnothing =\{x\in X:x\in A{\text{ for all }}A\in \varnothing \}}. Since allx∈X{\displaystyle x\in X} vacuously satisfy the required condition, the intersection of the empty collection of subsets ofX{\displaystyle X} is all ofX.{\displaystyle X.} In formulas,⋂∅=X.{\displaystyle \bigcap \varnothing =X.} This matches the intuition that as collections of subsets become smaller, their respective intersections become larger; in the extreme case, the empty collection has an intersection equal to the whole underlying set.

Also, intype theoryx{\displaystyle x} is of a prescribed typeτ,{\displaystyle \tau ,} so the intersection is understood to be of typeset τ{\displaystyle \mathrm {set} \ \tau } (the type of sets whose elements are inτ{\displaystyle \tau }), and we can define⋂A∈∅A{\displaystyle \bigcap _{A\in \emptyset }A} to be the universal set ofset τ{\displaystyle \mathrm {set} \ \tau } (the set whose elements are exactly all terms of typeτ{\displaystyle \tau }).

See also

[edit]

References

[edit]
  1. ^"Intersection of Sets".web.mnstate.edu. Archived fromthe original on 2020-08-04. Retrieved2020-09-04.
  2. ^"Stats: Probability Rules". People.richland.edu. Retrieved2012-05-08.
  3. ^ab"Set Operations | Union | Intersection | Complement | Difference | Mutually Exclusive | Partitions | De Morgan's Law | Distributive Law | Cartesian Product".www.probabilitycourse.com. Retrieved2020-09-04.
  4. ^Megginson, Robert E. (1998). "Chapter 1".An introduction to Banach space theory.Graduate Texts in Mathematics. Vol. 183. New York: Springer-Verlag. pp. xx+596.ISBN 0-387-98431-3.

Further reading

[edit]
  • Devlin, K. J. (1993).The Joy of Sets: Fundamentals of Contemporary Set Theory (Second ed.). New York, NY: Springer-Verlag.ISBN 3-540-94094-4.
  • Munkres, James R. (2000). "Set Theory and Logic".Topology (Second ed.). Upper Saddle River: Prentice Hall.ISBN 0-13-181629-2.
  • Rosen, Kenneth (2007). "Basic Structures: Sets, Functions, Sequences, and Sums".Discrete Mathematics and Its Applications (Sixth ed.). Boston: McGraw-Hill.ISBN 978-0-07-322972-0.

External links

[edit]
Wikimedia Commons has media related toIntersection (set theory).
Overview
Venn diagram of set intersection
Axioms
Operations
  • Concepts
  • Methods
Set types
Theories
Set theorists
General
Theorems
(list),
paradoxes
Logics
Traditional
Propositional
Predicate
Set theory
Types
ofsets
Maps,
cardinality
Theories
Formal
systems
(list),
language,
syntax
Example
axiomatic
systems
(list)
Proof theory
Model theory
Computability
theory
Related
Authority control databasesEdit this at Wikidata
Retrieved from "https://en.wikipedia.org/w/index.php?title=Intersection_(set_theory)&oldid=1323836858"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2026 Movatter.jp