In
axiomatic set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
, the axiom of union is one of the
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
. This axiom was introduced by
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...
.
The axiom states that for each set ''x'' there is a set ''y'' whose elements are precisely the elements of the elements of ''x''.
Formal statement
In the
formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of symb ...
of the Zermelo–Fraenkel axioms, the axiom reads:
:
or in words:
:
Given any
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''A'',
there is
English grammar is the set of structural rules of the English language. This includes the structure of words, phrases, clauses, sentences, and whole texts.
This article describes a generalized, present-day Standard English – a form of speech an ...
a set ''B'' such that, for any element ''c'', ''c'' is a member of ''B''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
there is a set ''D'' such that ''c'' is a member of ''D''
and
or AND may refer to:
Logic, grammar, and computing
* Conjunction (grammar), connecting two words, phrases, or clauses
* Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition
* Bitwise AND, a boolea ...
''D'' is a member of ''A''.
or, more simply:
:For any set
, there is a set
which consists of just the elements of the elements of that set
.
Relation to Pairing
The axiom of union allows one to unpack a set of sets and thus create a flatter set.
Together with the
axiom of pairing
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary se ...
, this implies that for any two sets, there is a set (called their
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
) that contains exactly the elements of the two sets.
Relation to Replacement
The axiom of replacement allows one to form many unions, such as the union of two sets.
However, in its full generality, the axiom of union is independent from the rest of the ZFC-axioms:
Replacement does not prove the existence of the union of a set of sets if the result contains an unbounded number of cardinalities.
Together with the
axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite ...
, the axiom of union implies that one can form the union of a family of sets indexed by a set.
Relation to Separation
In the context of set theories which include the axiom of separation, the axiom of union is sometimes stated in a weaker form which only produces a
superset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of the union of a set. For example, Kunen states the axiom as
:
which is equivalent to
: