# Element (mathematics)

In mathematics, an **element** (or **member**) of a set is any one of the distinct objects that belong to that set.

## Sets

Writing means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example , are subsets of A.

Sets can themselves be elements. For example, consider the set . The elements of B are *not* 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set .

The elements of a set can be anything. For example, is the set whose elements are the colors red, green and blue.

## Notation and terminology

The relation "is an element of", also called **set membership**, is denoted by the symbol "∈". Writing

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, although some authors use them to mean instead "*x* is a subset of *A*". Logician George Boolos strongly urged that "contains" be used for membership only, and "includes" for the subset relation only.

For the relation ∈ , the converse relation ∈^{T} may be written

- meaning "
*A*contains or includes*x*".

The negation of set membership is denoted by the symbol "∉". Writing

- means that "
*x*is not an element of*A*".

The symbol ∈ was first used by Giuseppe Peano, in his 1889 work *Arithmetices principia, nova methodo exposita*. Here he wrote on page X:

Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b; …

which means

The symbol ∈ means

is. So a ∈ b is read as ais a certainb; …

The symbol itself is a stylized lowercase Greek letter epsilon ("ϵ"), the first letter of the word ἐστί, which means "is".

Preview | ∈ | ∉ | ∋ | ∌ | ||||
---|---|---|---|---|---|---|---|---|

Unicode name | ELEMENT OF | NOT AN ELEMENT OF | CONTAINS AS MEMBER | DOES NOT CONTAIN AS MEMBER | ||||

Encodings | decimal | hex | dec | hex | dec | hex | dec | hex |

Unicode | 8712 | U+2208 | 8713 | U+2209 | 8715 | U+220B | 8716 | U+220C |

UTF-8 | 226 136 136 | E2 88 88 | 226 136 137 | E2 88 89 | 226 136 139 | E2 88 8B | 226 136 140 | E2 88 8C |

Numeric character reference | ∈ |
∈ |
∉ |
∉ |
∋ |
∋ |
∌ |
∌ |

Named character reference | ∈, ∈, ∈, ∈ | ∉, ∉, ∉ | ∋, ∋, ∋, ∋ | ∌, ∌, ∌ | ||||

LaTeX | \in | \notin | \ni | \not\ni or \notni | ||||

Wolfram Mathematica | \[Element] | \[NotElement] | \[ReverseElement] | \[NotReverseElement] |

## 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 set *B* and set *C* are both 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}, the following statements are true:

- 2 ∈
*A* - 5 ∉
*A*

- {3,4} ∈
*B* - 3 ∉
*B* - 4 ∉
*B* - yellow ∉
*C*

## Formal relation

As a relation, set membership must have a domain and a range. Conventionally the domain is called the universe denoted *U*. The range is the set of subsets of *U* called the power set of *U* and denoted P(*U*). Thus the relation is a subset of *U* x P(*U*). The converse relation is a subset of P(*U*) x *U*.