# Element (mathematics)

# Element (mathematics)

In mathematics, an **element**, or **member**, of a set is any one of the distinct objects that make up that set.

Sets

`Writingmeans that the elements of the setAare the numbers 1, 2, 3 and 4. Sets of elements ofA, for example, aresubsetsofA.`

`Sets can themselves be elements. For example, consider the set. The elements ofBare`

*not*1, 2, 3, and 4. Rather, there are only three elements ofB, 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 colorsred,greenandblue.`

Notation and terminology

`Therelation"is an element of", also called`

**set membership**, is denoted by the symbol "". Writingmeans 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*".^{[1]} Logician George Boolos strongly urged that "contains" be used for membership only and "includes" for the subset relation only.^{[2]}

For the relation ∈ , the converse relation ∈T may be written

- meaning "

*A*contains

*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 1889 in his 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 ab; …

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

Character | ∈ | ∉ | ∋ | ∌ | ||||
---|---|---|---|---|---|---|---|---|

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

Encodings | decimal | hex | decimal | hex | decimal | hex | decimal | 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] |

Complement and converse

`Every relation`

*R*:*U*→*V*is subject to twoinvolutions: complementation*R*→and conversion*R*T:*V*→*U*. The relation ∈ has for its domain a universal set*U*, and has thepower set**P**(*U*) for its codomain or range. The complementary relationexpresses the opposite of ∈. An element*x*∈*U*may have*x*∉*A*, in which case*x*∈*U*\*A*, thecomplementof*A*in*U*.`Theconverse relationswaps the domain and range with ∈. For any`

*A*in**P**(*U*),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*5 ∉

*A*{3,4} ∈

*B*3 ∉

*B*4 ∉

*B*Yellow ∉

*C*

## References

*Handbook of Analysis and Its Foundations*. Academic Press. ISBN 0-12-622760-8. p. 12

*24.243 Classical Set Theory (lecture)*(Speech). Massachusetts Institute of Technology.