In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.
Sets
Writing
$A\; =\; \backslash \{1,\; 2,\; 3,\; 4\backslash \}$ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example
$\backslash \{1,\; 2\backslash \}$, are
of .
Sets can themselves be elements. For example, consider the set $B\; =\; \backslash \{1,\; 2,\; \backslash \{3,\; 4\backslash \}\backslash \}$. The elements of are not 1, 2, 3, and 4. Rather, there are only three elements of , namely the numbers 1 and 2, and the set $\backslash \{3,\; 4\backslash \}$.
The elements of a set can be anything. For example, $C\; =\; \backslash \{\backslash mathrm\{\backslash color\{red\}red\},\; \backslash mathrm\{\backslash color\{green\}green\},\; \backslash mathrm\{\backslash color\{blue\}blue\}\backslash \}$, is the set whose elements are the colors , and .
Notation and terminology
The relation "is an element of", also called
set membership, is denoted by the symbol "
$\backslash in$". Writing
 $x\; \backslash in\; A$
means that " x is an element of A". Equivalent expressions are " x is a member of A", " x belongs to A", " x is in A" and " x lies in A". The expressions " A includes x" and " A contains x" are also used to mean set membership, however some authors use them to mean instead " x is a subset of A".
p. 12 Logician
George Boolos strongly urged that "contains" be used for membership only and "includes" for the subset relation only.
For the relation ϵ , the converse relation ϵ^{T} may be written
 $A\; \backslash ni\; x\; ,$ meaning " A contains x".
The negation of set membership is denoted by the symbol "∉". Writing
 $x\; \backslash notin\; A$ means that " x is not an element of A".
The symbol ϵ was first used by Giuseppe Peano 1889 in his work italic=yes. Here he wrote on page X:
Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b; ...
which means
The symbol ϵ means is. So a ϵ b is read as a is a b; ...
The symbol itself is a stylized lowercase Greek letter epsilon ("ε"), the first letter of the word , which means "is".
The Unicode characters for these symbols are U+2208 ('element of'), U+2209 ('not an element of'), U+220B ('contains as member') and U+220C ('does not contain as member'). The equivalent LaTeX commands are "\in", "\notin", "\ni" and "\not\ni". Mathematica has commands "\Element", "\NotElement", "\ReverseElement" and "\NotReverseElement".
Complement and converse
Every relation
R :
U →
V is subject to two involutions: complementation
R →
$\backslash bar\{R\}$ and conversion
R^{T}:
V →
U.
The relation ∈ has for its domain a universal set
U, and has the
power set P(
U) for its codomain or range. The complementary relation
$\backslash bar\{\backslash in\}\; \backslash \; =\; \backslash \; \backslash notin$ expresses the opposite of ∈. An element
x ∈
U may have
x ∉
A, in which case
x ∈
U \
A, the complement of
A in
U.
The converse relation $\backslash in^T\; \backslash \; =\; \backslash \; \backslash ni$ swaps the domain and range with ∈. For any A in P( U), $A\; \backslash ni\; x$ is true when x ∈ A.
Cardinality of sets
The number of elements in a particular set is a property known as
cardinality; informally, this is the size of a set. In the above examples the cardinality of the set
A is 4, while the cardinality of either of the sets
B and
C is 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers = { 1, 2, 3, 4, ... }.
Examples
Using the sets defined above, namely
A = {1, 2, 3, 4 },
B = {1, 2, {3, 4}} and
C = { red, green, blue }:

2 ∈ A

{3,4} ∈ B

3,4 ∉ B

{3,4} is a member of B

Yellow ∉ C

The cardinality of D = {2, 4, 8, 10, 12} is finite and equal to 5.

The cardinality of P = {2, 3, 5, 7, 11, 13, ...} (the prime numbers) is infinite (proved by Euclid).
Further reading

 "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).

 Both the notion of set (a collection of members), membership or elementhood, the axiom of extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thorough understanding of "set element".
External links