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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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

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Basic definition

The union of A and B
The union of A and B

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:

x is an element of A ∪ B if and only if
  • x is an element of A or
  • x is an element of B. ↔

(This is an inclusive "or". )

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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, the parity of an object states whether it is even or odd

Finite unions

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 if and only if x is in A or x is in B or x is in C.

Union is an associative operation, it doesn't matter in what order unions are taken. In Mathematics, associativity is a property that a Binary operation can have In mathematics a finite union means any union carried out on a finite number of sets: it doesn't imply that the union set is a finite set. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 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 Mathematics, associativity is a property that a Binary operation can have

The operations can be performed in any order, and the parentheses may be omitted without ambiguity (i. e. either of the above can be expressed equivalently as A ∪ B ∪ C). Similarly, union is commutative, so the sets can be written in any order. In Mathematics, commutativity is the ability to change the order of something without changing the end result The empty set is an identity element for the operation of union. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that 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 disjunction.

Together with intersection and complement, union makes any power set into a Boolean algebra. In 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 In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. For example, union and intersection distribute over each other, and all three operations are combined in De Morgan's laws. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed Replacing union with symmetric difference gives a Boolean ring instead of a Boolean algebra. In Mathematics, the symmetric difference of two sets is the set of elements which are in one of the sets but not in both In Mathematics, a Boolean ring R is a ring (with identity for which x 2 = x for all x in R; that

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 Predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain In symbols:

x \in \bigcup\mathbf{M} \iff \exists A{\in}\mathbf{M}, x \in A.

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. In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom of union is one of the Axioms

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. In Predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain

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

\bigcup \mathbf{M},

while most people will instead write

\bigcup_{A\in\mathbf{M}} A.

The latter notation can be generalised to

\bigcup_{i\in I} A_{i},

which refers to the union of the collection {Ai : i is in I}. Here I is a set, and Ai 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 to that of infinite series:

\bigcup_{i=1}^{\infty} A_{i}.

When formatting is difficult, this can also be written "A1 ∪ A2 ∪ A3 ∪ ···". In Mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with (This last example, a union of countably many sets, is very common in analysis; for an example see the article on σ-algebras. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty ) 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

A \cap \bigcup_{i\in I} B_{i} = \bigcup_{i\in I} (A \cap B_{i}).

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

\bigcup_{i\in I} \left(\bigcap_{j\in J} A_{i,j}\right) \subseteq \bigcap_{j\in J} \left(\bigcup_{i\in I} A_{i,j}\right).

See also

External links

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