1 Set
1.1 Definition
In mathematics, a set is a collection of different things; the things are elements or members
of the set and are typically mathematical objects: numbers, symbols, points in space,
lines, other geometric shapes, variables, or other sets. A set may be finite or infinite.
There is a unique set with no elements, called the empty set; a set with a single element
is a singleton. Sets themselves are purely conceptual. This is an important point to
note: the set of all cows (for example) does not physically exist, even though the cows
do. The set is a ”gathering” of the cows into one conceptual unit that is not part of
physical reality. This makes it easy to see why we can have sets with an infinite number
of elements; even though we may not be able to point out infinitely many objects in
the real world, we can construct conceptual sets which an infinite number of elements.
Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo-Fraenkel
set theory, has been the standard way to provide rigorous foundations for all branches of
mathematics since the first half of the 20th century.
1.2 Basic Notation
If x is an element of a set S, one says that x belongs to S or is in S, and this is written as x 𝜖 S
[11]. The statement ”y is not in S” is written as y 
S , which can also be read as ”y is not in
S”[12][13]. For example, if ℤ is the set of the integers, one has −3 𝜖 ℤ and 1.5 
ℤ. Each set is
uniquely characterized by its elements.
This implies that there is only one set with no element, the empty set (or null set) that is denoted
∅, ∅ or {}[17][18]. A singleton is a set with exactly one element. If x is this element, the singleton
is denoted {x}. An element of a set can be itself a set. For example, the singleton {∅} is a set that
has the empty set ∅ as its unique element. Since these two sets do not have the same elements, {∅}
and ∅ are different sets.
A set is finite if there exists a natural number n such that the n first natural numbers can be put
in one to one correspondence with the elements of the set. In this case, one says the n is the
number of elements of the set. A set is infinite if such an n does not exist. The empty set is a finite
set with 0 elements.
The natural numbers form an infinite set, commonly denoted ℕ. Other examples of infinite sets
include integer numbers ℤ, rational numbers ℚ, real numbers ℝ, real vector spaces, curves and
most sorts of mathematical spaces.
1.3 Specifying a set
Extensionality implies that for specifying a set, one has either to list its elements or to provide a
property that uniquely characterizes the set elements.
1.3.1 Roster notation
Roster or enumeration notation is a notation introduced by Ernst Zermelo in 1908 that specifies a
set by listing its elements between braces, separated by commas [19][20][21][22][23]. For example,
one knows that {4, 2, 1, 3} and {blue,white,red} denote sets and not tuples because of the
enclosing braces.
Above notations {} and {x} for they empty set and for a singleton are examples of roster notation.
When specifying sets, it only matters whether each distinct element is in the set or not; this means
a set does not change if elements are repeated for arranged in a different order. For example,
[24][25][26]
When there is a clear pattern for generating all set elements, one can use ellipses for abbreviating
the notation, [27][28] such as in
for the positive integers not greater than 1000.
Ellipses allow also expanding roster notation to some infinite sets. For example, the set of all
integers can be denoted as
or
1.3.2 Set-builder notation
Set-builder notation specifies a set as being the set of all elements that satisfy some logical formula
[29][30][31]. More precisely if P(x) is a logical formula depending on a variable x, which evaluates
to true or false depending on the value of x, then
This article is a derivative work of the creative commons share alike with attribution in
[1].
References
[1] Wikipedia contributors. Set (mathematics). Wikipedia, The Free Encyclopedia; [cited
2025 Nov 22].
