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 only 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 phrase does not imply that the union set is a finite set. Arbitrary unions The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set or class 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. In symbols: This idea subsumes the preceding sections—for example, A ∪ B ∪ C is the union of the collection {A, B, C}. Also, if M is the empty collection, then the union of M is the empty set. Notations 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 , where I is an index set and is a set for every . In the case that the index set I is the set of natural numbers, one uses the notation , which is analogous to that of the infinite sums in series. When the symbol "∪" is placed before other symbols (instead of between them), it is usually rendered as a larger size. Notation encoding In Unicode, union is represented by the character . In TeX, is rendered from \cup and is rendered from \bigcup. See also − the union of sets of strings Notes External links Infinite Union and Intersection at ProvenMath De Morgan's laws formally proven from the axioms of set theory. Boolean algebra Basic concepts in set theory Operations on sets Set theory