Union of two sets: Union of three sets: The union of A, B, C, D, and E is everything except the white area.
Inset theory, theunion (denoted by ∪) of a collection ofsets is the set of allelements in the collection.[1] It is one of the fundamental operations through which sets can be combined and related to each other. Anullary union refers to a union ofzero () sets and it is by definition equal to theempty set.
As another example, the number 9 isnot contained in the union of the set ofprime numbers {2, 3, 5, 7, 11, ...} and the set ofeven numbers {2, 4, 6, 8, 10, ...}, because 9 is neither prime nor even.
Sets cannot have duplicate elements,[3][4] so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}.
One can take the union of several sets simultaneously. For example, the union of three setsA,B, andC contains all elements ofA, all elements ofB, and all elements ofC, and nothing else. Thus,x is an element ofA ∪B ∪C if and only ifx is in at least one ofA,B, andC.
Afinite union is the union of a finite number of sets; the phrase does not imply that the union set is afinite set.[5][6]
The notation for the general concept can vary considerably. For a finite union of sets one often writes or. Various common notations for arbitrary unions include,, and. The last of these notations refers to the union of the collection, whereI is anindex set and is a set for every. In the case that the index setI is the set ofnatural numbers, one uses the notation, which is analogous to that of theinfinite sums in series.[7]
When the symbol "∪" is placed before other symbols (instead of between them), it is usually rendered as a larger size.
The most general notion is the union of an arbitrary collection of sets, sometimes called aninfinitary union. IfM is a set orclass whose elements are sets, thenx is an element of the union ofMif and only if there isat least one elementA ofM such thatx is an element ofA.[7] In symbols:
This idea subsumes the preceding sections—for example,A ∪B ∪C is the union of the collection {A,B,C}. Also, ifM is the empty collection, then the union ofM is the empty set.
InZermelo–Fraenkel set theory (ZFC) and other set theories, the ability to take the arbitrary union of any sets is granted by theaxiom of union, which states that, given any set of sets, there exists a set, whose elements are exactly those of the elements of. Sometimes this axiom is less specific, where there exists a which contains the elements of the elements of, but may be larger. For example if then it may be that since contains 1 and 2. This can be fixed by using theaxiom of specification to get the subset of whose elements are exactly those of the elements of. Then one can use theaxiom of extensionality to show that this set is unique. For readability, define the binarypredicate meaning " is the union of" or "" as:
Then, one can prove the statement "for all, there is a unique, such that is the union of":
After the union operator has been defined, the binary union can be defined by showing there exists a unique set using theaxiom of pairing, and defining. Then, finite unions can be defined inductively as:
Binary union is anassociative operation; that is, for any sets,Thus, the parentheses may be omitted without ambiguity: either of the above can be written as. Also, union iscommutative, so the sets can be written in any order.[9]Theempty set is anidentity element for the operation of union. That is,, for any set. Also, the union operation is idempotent:. All these properties follow from analogous facts aboutlogical disjunction.
Intersection distributes over union and union distributes over intersection[2]Thepower set of a set, together with the operations given by union,intersection, andcomplementation, is aBoolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formulawhere the superscript denotes the complement in theuniversal set. Alternatively, intersection can be expressed in terms of union and complementation in a similar way:. These two expressions together are calledDe Morgan's laws.[10][11][12]
The english wordunion comes from the term inmiddle French meaning "coming together", which comes from thepost-classical Latinunionem, "oneness".[13] The original term for union in set theory wasVereinigung (in german), which was introduced in 1895 byGeorg Cantor.[14] The english use ofunion of two sets in mathematics began to be used by at least 1912, used byJames Pierpont.[15][16] The symbol used for union in mathematics was introduced byGiuseppe Peano in hisArithmetices principia in 1889, along with the notations for intersection, set membership, and subsets.[17]
