# Element (mathematics)

In mathematics, an **element**, or **member**, of a set is any one of the distinct objects that make up 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, 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 two involutions: complementation *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 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 swaps 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

- Eric Schechter (1997).
*Handbook of Analysis and Its Foundations*. Academic Press. ISBN 0-12-622760-8. p. 12 - George Boolos (February 4, 1992).
*24.243 Classical Set Theory (lecture)*(Speech). Massachusetts Institute of Technology.

## Further reading

- Halmos, Paul R. (1974) [1960],
*Naive Set Theory*, Undergraduate Texts in Mathematics (Hardcover ed.), NY: Springer-Verlag, ISBN 0-387-90092-6 - "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither). - Jech, Thomas (2002), "Set Theory",
*Stanford Encyclopedia of Philosophy* - Suppes, Patrick (1972) [1960],
*Axiomatic Set Theory*, NY: Dover Publications, Inc., ISBN 0-486-61630-4 - Both the notion of set (a collection of members), membership or element-hood, 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".