A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally
also ignored (unlike a list or multiset).
Members of a set are often referred to as elements and
the notation 
is used to denote that 
is an element of a set 
. The study of sets and their properties is the object of set theory.
Older words for set include aggregate and set class. Russell also uses the unfortunate term manifold to refer to a set (Russell 1903, §68).
Historically, a single horizontal overbar was used to denote a set stripped of any structure besides order, and hence to represent the order type of the set. A double overbar indicated stripping the order from the set and hence represented the cardinal number of the set. This practice was begun by set theory founder Georg Cantor (Cantor 1895, §§1 and 7).
Symbols used to operate on sets include 
(which means "AND"
or intersection), and 
(which means "OR" or
union). The symbol 
is used to denote the set containing no elements,
called the empty set.
There are a number of different notations related to the theory of sets. In the case of a finite set of elements, one often writes the collection inside curly braces, e.g.,
 |
(1)
|
for the set of natural numbers less than or equal to three. Similar notation can be used for infinite sets provided that ellipses are used to signify infiniteness, e.g.,
 |
(2)
|
for the collection of natural numbers greater than or equal to three, or
 |
(3)
|
for the set of all even numbers.
In addition to the above notation, one can use so-called set-builder notation to express sets and elements thereof. The general format for set-builder notation is
 |
(4)
|
where 
denotes an element
and 
denotes a property 
satisfied by 
. (4) can also be expanded so as to indicate
construction of a set which is a subset of some ambient
set 
, e.g.,
 |
(5)
|
The shorthand (4) does not assert that every property defines a set. In Zermelo-Fraenkel set theory, the axiom of subsets justifies the bounded form (5). Allowing every property to define a set leads to Russell's antinomy (Jech 2003, pp. 3-4).
It is worth noting that the ":" in (4) and (5) is sometimes replaced by a vertical line, e.g.,
 |
(6)
|
Also worth noting is that the sets in (1), (2), and (3) can all be rewritten in set-builder
notation as subsets of the set 
of integers, namely
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
respectively.
Other common notations related to set theory include 
,
which is used to denote the set of maps from 
to 
where 
and 
are arbitrary sets. For example, an element
of 
would be a map
from the natural numbers 
to the set 
.
Call such a function 
, then 
,
, etc., are elements
of 
, so call them 
, 
,
etc. This now looks like a sequence of elements
of 
, so sequences
are really just functions from 
to 
.
This notation is standard in mathematics and is frequently
used in symbolic dynamics to denote sequence spaces.
Let 
, 
, and 
be sets. The operations on these sets using the 
and 
binary operators
are commutative
 |
(10)
|
 |
(11)
|
 |
(12)
|
 |
(13)
|
and distributive
 |
(14)
|
 |
(15)
|
More generally, we have the infinite distributive laws
 |
(16)
|
 |
(17)
|
where 
runs through any index
set 
. The proofs
follow trivially from the definitions of union and intersection.
