Union (set theory) – Wikipedia

Union of three sets : The union of A, B, C, D, and E is everything except the white area. In sic hypothesis, the union ( denoted by ∪ ) of a collection of sets is the set of all elements in the collection. [ 1 ] It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of zero ( 0 { \displaystyle 0 } ) sets and it is by definition equal to the empty laid. For explanation of the symbols used in this article, refer to the table of numerical symbols .

Union of two sets [edit ]

The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. [ 2 ] In symbols ,

A ∪ B = { ten : x ∈ A or ten ∈ B } { \displaystyle A\cup B=\ { adam : x\in A { \text { or } } x\in B\ } }[3]

For exercise, if A = { 1, 3, 5, 7 } and B = { 1, 2, 4, 6, 7 } then A ∪ B = { 1, 2, 3, 4, 5, 6, 7 }. A more elaborate example ( involving two countless sets ) is :

A = {x is an even integer larger than 1}

B = {x is an odd integer larger than 1}

A ∪ B = { 2, 3, 4, 5, 6, … } { \displaystyle A\cup B=\ { 2,3,4,5,6, \dots \ } }

As another exemplar, the number 9 is not contained in the union of the set of prime numbers { 2, 3, 5, 7, 11, … } and the set of tied numbers { 2, 4, 6, 8, 10, … }, because 9 is neither flower nor even. Sets can not have duplicate elements, [ 3 ] [ 4 ] sol the coupling of the sets { 1, 2, 3 } and { 2, 3, 4 } is { 1, 2, 3, 4 }. multiple occurrences of identical elements have no effect on the cardinality of a put or its contents .

Algebraic properties [edit ]

Binary union is an associative operation ; that is, for any sets A, B, and C, { \displaystyle A, B, { \text { and } } C, } A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C. { \displaystyle A\cup ( B\cup C ) = ( A\cup B ) \cup C. } frankincense the parentheses may be omitted without ambiguity : either of the above can be written as A ∪ B ∪ C. { \displaystyle A\cup B\cup C. } besides, union is commutative, so the sets can be written in any order. [ 5 ] The empty bent is an identity element for the mathematical process of union. That is, A ∪ ∅ = A, { \displaystyle A\cup \varnothing =A, } for any set A. { \displaystyle A. } besides, the union operation is idempotent : A ∪ A = A. { \displaystyle A\cup A=A. } All these properties follow from analogous facts about legitimate disconnection. intersection distributes over union A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) { \displaystyle A\cap ( B\cup C ) = ( A\cap B ) \cup ( A\cap C ) }

A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ). { \displaystyle A\cup ( B\cap C ) = ( A\cup B ) \cap ( A\cup C ). } and coupling distributes over intersection The power set of a laid U, { \displaystyle U, } together with the operations given by union, overlap, and complementary distribution, is a boolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and complementary distribution by the formula A ∪ B = ( A hundred ∩ B carbon ) cytosine, { \displaystyle A\cup B=\left ( A^ { \text { c } } \cap B^ { \text { c } } \right ) ^ { \text { c } }, } c { \displaystyle { } ^ { \text { c } } } U. { \displaystyle U. }

Finite unions [edit ]

where the superscriptdenotes the complement in the universal put One can take the union of respective sets simultaneously. For model, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and merely if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets ; the give voice does not imply that the union laid is a finite set. [ 6 ] [ 7 ]

arbitrary unions [edit ]

The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set or course whose elements are sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A. [ 8 ] In symbols :

adam ∈ ⋃ M ⟺ ∃ A ∈ M, x ∈ A. { \displaystyle x\in \bigcup \mathbf { M } \iff \exists A\in \mathbf { M }, \ x\in A. }

This mind subsumes the preceding sections—for exemplar, A ∪ B ∪ C is the union of the collection { A, B, C }. besides, if M is the empty collection, then the coupling of M is the empty hardening .

Notations [edit ]

The notation for the general concept can vary well. For a finite union of sets S 1, S 2, S 3, …, S north { \displaystyle S_ { 1 }, S_ { 2 }, S_ { 3 }, \dots, S_ { n } } one much writes S 1 ∪ S 2 ∪ S 3 ∪ ⋯ ∪ S newton { \displaystyle S_ { 1 } \cup S_ { 2 } \cup S_ { 3 } \cup \dots \cup S_ { newton } } or ⋃ one = 1 north S iodine { \displaystyle \bigcup _ { i=1 } ^ { nitrogen } S_ { iodine } } . assorted coarse notations for arbitrary unions include ⋃ M { \displaystyle \bigcup \mathbf { M } } , ⋃ A ∈ M A { \displaystyle \bigcup _ { A\in \mathbf { M } } A } , and ⋃ i ∈ I A one { \displaystyle \bigcup _ { i\in I } A_ { one } } . The last of these notations refers to the union of the collection { A iodine : one ∈ I } { \displaystyle \left\ { A_ { iodine } : i\in I\right\ } } , where I is an index set and A iodine { \displaystyle A_ { one } } is a typeset for every one ∈ I { \displaystyle i\in I } . In the lawsuit that the exponent laid I is the set of natural numbers, one uses the notation ⋃ one = 1 ∞ A i { \displaystyle \bigcup _ { i=1 } ^ { \infty } A_ { i } } , which is analogous to that of the space sums in series. [ 8 ] When the symbol “ ∪ ” is placed before other symbols ( rather of between them ), it is normally rendered as a larger size.

notation encoding [edit ]

In Unicode, union is represented by the character U+222A ∪ UNION. In TeX, ∪ { \displaystyle \cup } is rendered from \cup .

See besides [edit ]

Notes [edit ]