# set theory

Set theory is a branch of mathematics created by Georg Cantor at the end of the nineteenth century. Initially controversial, set theory has
come to play the role of a foundational theory in modern mathematics, in
the sense of a theory invoked to justify assumptions made in mathematics
concerning the existence of mathematical objects (such as numbers or functions)
and their properties. Formal versions of set theory also have a foundational
role to play as specifying a theoretical ideal of mathematical rigor in
proofs. Cantor's basic discovery was that if we define two sets *A* and *B* to have the same number of members (the same **cardinality**)
then there is a way of pairing off members of *A* exhaustively with
members of *B*. The appearance around the turn of the century of the
so-called set-theoretical paradoxes, such as Russell's
paradox, prompted the formulation in 1908 by Ernst Zermelo of an axiomatic
theory of sets. The axioms for set theory
now most often studied and used are those called the **Zermelo-Fraenkel
axioms**, usually together with the axiom
of choice. The Zermelo-Fraenkel axioms are commonly abbreviated to ZF,
or ZFC if the axiom of choice is included. An important feature of ZFC is
that every object that it deals with is a set. In particular, every element
of a set is itself a set. Other familiar mathematical objects, such as numbers,
must be subsequently defined in terms of sets.

## Some basic set theory

A set is a collection of objects or quantities symbolized by a capital letter. Thus

*S* = {2, 4, 6, 8}

means that *S* is the set of even numbers less than 10. A member
of a set is called an element: symbolically, 2 ∈ *S* means
that 2 is an element of *S*; 3 ∉ means that 3 is not an element
of *S*.

An order set is equivalent to a sequence.
A set may be infinite, finite, or empty: the set of all even numbers is
infinite, that of all those less than 10 is finite, and that of all those
less than 1 is empty. This empty set,
or **null set**, symbolized by {}, should not be confused with
{0}, which is a set with one member, zero.

If there is a one-to-one correspondence between the elements of two sets, then they are said to be equivalent, and if the sets have identical elements they are equal:

*S*_{1} = {*a*, *b*, *c*, *d*}

*S*_{2} = {*e*, *f*, *g*, *h*}

*S*_{3} = {*d*, *c*, *b*, *a*}
shows three equivalent sets. Moreover, *S*_{1} = *S*_{2}.
Two equivalent sets are writter *S*_{1} ↔ *S*_{2}.

If some elements of one set are also elements of another, then those elements
are called the intersection of the
two sets, symbolized *S*_{1} ∩ *S*_{2}.
The sets of all elements that are members of at least one of the two sets
is their union, written *S*_{1} ∪ *S*_{2}. A set whose members are all members of another
set is termed a subset. Thus, if

*S*_{1} = {*a*, *b*, *c*, *d*, *e*}

*S*_{2} = {*b*, *d*, *f*, *g*}

then *S*_{1} ∩ *S*_{2} = {*b*, *d*} and *S*_{1} ∪ *S*_{2} {*a*, *b*, *c*, *d*, *e*, *f*, *g*}

Moreover, *S*_{1} and *S*_{2} are subsets
of *S*_{1} ∪ *S*_{2}, written *S*_{1} ⊂ *S*_{1} ∪ *S*_{2} and *S*_{s} ⊂ *S*_{1} ∪ *S*_{2}. The set of
all elements of all the sets in a particular discussion is the universal
set, or domain, symbolized by *U*.
The domain may contain elements in addition to all those under discussion.

Set theory is of importance throughout mathematics. In analytical
geometry, for example, a curve may be
considered as a set of points, or point set. For two functions, *f(x)* and *g(x)*, represented by the sets *S _{f}* and

*S*,

_{g}*S*∩

_{f}*S*gives those points at which the curves intersect (see intersection). Sets can be represented pictorially by Venn diagrams.

_{g}