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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. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
This article uses mathematical symbols. The following table lists many specialized symbols commonly used in mathematics. ...
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: Venn diagram for A union B. Created by me: Paul August 02:48, Aug 24, 2004 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Venn diagram for A union B. Created by me: Paul August 02:48, Aug 24, 2004 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
- 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".) ↔ ⇔ ≡ logical symbols representing iff. ...
OR logic gate In mathematics, logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ...
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 prime) is a natural number greater than one whose only positive divisors are one and itself. ...
In mathematics, any integer (whole number) is either 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 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 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, associativity is a property that a binary operation can have. ...
In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
Algebraic properties Binary union (the union of just two sets at a time) is an associative operation; that is, In mathematics, associativity is a property that a binary operation can have. ...
- 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 the sets can be written 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 disjunction. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...
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 set. ...
0 (zero), alternatively called naught, nil, ought, or nought, is both a number and a numeral. ...
OR logic gate In mathematics, logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ...
Together with intersection and complement, union makes any power set into a Boolean algebra. For example, union and intersection distribute over each other, and all three operations are combined in De Morgan's laws. Replacing union with symmetric difference gives a Boolean ring instead of 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, all elements of B that also belong to A), but no other elements. ...
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ...
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...
In logic, De Morgans laws (or De Morgans theorem) are rules in formal logic relating pairs of dual logical operators in a systematic manner expressed in terms of negation. ...
In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ...
In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. ...
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: ↔ ⇔ ≡ logical symbols representing iff. ...
In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. ...
 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 of Zermelo_Fraenkel set theory, stating that, for any two sets, there is a set that contains exactly the elements of both. ...
Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ...
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
 while most people will instead write  The latter notation can be generalised to  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: In mathematics, an index set is another name for a function domain. ...
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ...
In mathematics, a series is the sum of a sequence of terms. ...
 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. Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ...
In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S that is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...
Intersection distributes over infinitary union, in the sense that
 We can also combine infinitary union with infinitary intersection to get the law 
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