Union (set Theory) Basic Definition Algebraic Properties

In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else.

1 Basic definition

If A and B are sets, then the union of A and B is the set that contains all elements of A and all elements of B, but no other elements. The union of A and B is usually written "A ∪B". Formally:

For example, the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. The number 9 is not contained in the union 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.

More generally, one can take the union of several sets at once. The union of A, B, and C, for example, contains all elements of A, all elements of B, and all elements of C, and nothing else. Formally, x is an element of A ∪B ∪C iff x is in A or x is in B or x is in C.

2 Algebraic properties

Binary union (the union of just two sets at a time) is an associative operation; that is,
A ∪(B ∪C) = (A ∪B) ∪C. In fact, A ∪B ∪C is equal to both of these sets as well, so parentheses are never needed when writing only unions. Similarly, union is commutative, so you can write the sets in any order. The empty set is an identity element for the operation of union. That is, {} ∪A = A, for any set A. Thus one can think of the empty set as the union of zero sets. In terms of the definitions, these facts follow from analogous facts about logical disjunctionIn logic and mathematics, a disjunction is an "or statement". For example "John skis or Sally swims" is a disjunction. Note that in everyday language, use of the word "or" can sometimes mean "either, but not both" (eg, "would you like tea or coffee?")..

Together with intersectionIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. This article uses mathematical symbols. The intersecti and complementIn set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement . Relative complement If A and B are sets, then the relative complement of a A in B also known as the set theoretic, union makes any power setAbstract algebra Algebra Set theory In mathematics, given a set S the power set of S written P ''S or 2''S is the set of all subsets of S''. In formal language, the existence of power set of any set is presupposed by the axiom of power set. In this case S into a Boolean algebraIn mathematics and computer science, Boolean algebras or Boolean lattices are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and compleme. For example union and intersection distributes over each other, and all three operations are combined in de Morgan's laws. If you want a Boolean ringRing theory Mathematical logic In mathematics, a Boolean ring ''R is a ring for which x''2 x for all x in R that is, R consists of idempotent elements. These rings arise from (and give rise to) Boolean algebras. One example is the power set of any set X w instead of a Boolean algebra, then you can replace union with symmetric difference.

3 Infinite unions

The most general notion is the union of an arbitrary collection of sets. If M is a set whose elements are themselves sets, then x is an element of the union of M if and only if for at least one element A of M, x is an element of A. In symbols:

That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union in axiomatic set theory.

This idea subsumes the above paragraphs, in that 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. The analogy between finite unions and logical disjunction extends to one between infinite unions and existential quantification.

The notation for the general concept can vary considerably. Hardcore set theorists will simply write

while most people will instead write

The latter notation can be generalised to

which refers to the union of the collection {A_i : i is in I}. Here I is a set, and A_i is a set for every i in I. In the case that the index set I is the set of natural numbers, the notation is analogous that that of summation:

When formatting is difficult, this can also be written "A₁ ∪ A₂ ∪ A₃ ∪ ···". (This last example, a union of countably many sets, is very common in analysis; for an example see the article on σ-algebras.) Finally, let us note that whenever the symbol "∪" is placed before other symbols instead of between them, it is of a larger size.

Intersection distributes over infinitary union, in the sense that

We can also combine infinitary union with infinitary intersection to get the law

4 See also

Set theory

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