# Element (mathematics)

__: Any one of the distinct objects that make up a set in set theory__

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

## Sets

Writing [math]\displaystyle{ A = \{1, 2, 3, 4\} }[/math] means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example [math]\displaystyle{ \{1, 2\} }[/math], are subsets of A.

Sets can themselves be elements. For example, consider the set [math]\displaystyle{ B = \{1, 2, \{3, 4\}\} }[/math]. 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 [math]\displaystyle{ \{3, 4\} }[/math].

The elements of a set can be anything. For example, [math]\displaystyle{ C = \{\mathrm{\color{Red}red}, \mathrm{\color{green}green}, \mathrm{\color{blue}blue}\} }[/math] is the set whose elements are the colors red, green and blue.

In logical terms, (*x* โ *y*) โ (โ*x*[P* _{x}* =

*y*] :

*x*โ ๐

*y*).

## Notation and terminology

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

- [math]\displaystyle{ x \in A }[/math]

means that "*x* is an element of *A*".^{[1]} 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*".^{[2]} Logician George Boolos strongly urged that "contains" be used for membership only, and "includes" for the subset relation only.^{[3]}

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

- [math]\displaystyle{ A \ni x }[/math]

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

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

- [math]\displaystyle{ x \notin A }[/math]

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

The symbol โ was first used by Giuseppe Peano, in his 1889 work *Arithmetices principia, nova methodo exposita*.^{[4]} Here he wrote on page X:

Signum โ significat est. Ita a โ b legitur a est quoddam b; โฆ

which means

The symbol โ means

is. So a โ b is read as ais a certainb; โฆ

The symbol itself is a stylized lowercase Greek letter epsilon ("ฯต"), the first letter of the word แผฯฯฮฏ, which means "is".^{[4]}

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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 0 0 0 | U+2208 | 8713 0 0 0 | U+2209 | 8715 0 0 | U+Lua error: Internal error: The interpreter exited with status 1. |
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UTF-8 | ||||||||

Numeric character reference | ||||||||

Named character reference | Expression error: Unexpected < operator. |
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| ||||

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.^{[5]} 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 โ
*A* - 5 โ
*A*

- {3, 4} โ
*B* - 3 โ
*B* - 4 โ
*B* - yellow โ
*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 [math]\displaystyle{ \in }[/math] is a subset of *U* x P(*U*). The converse relation [math]\displaystyle{ \ni }[/math] is a subset of P(*U*) x *U*.

## See also

## References

- โ Weisstein, Eric W.. "Element" (in en). https://mathworld.wolfram.com/Element.html.
- โ Eric Schechter (1997).
*Handbook of Analysis and Its Foundations*. Academic Press. ISBN 0-12-622760-8. p. 12 - โ
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^{4.0}^{4.1}Kennedy, H. C. (July 1973). "What Russell learned from Peano".*Notre Dame Journal of Formal Logic*(Duke University Press)**14**(3): 367โ372. doi:10.1305/ndjfl/1093891001. - โ "Sets - Elements | Brilliant Math & Science Wiki" (in en-us). https://brilliant.org/wiki/sets-elements/.

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## Further reading

- Halmos, Paul R. (1974),
*Naive Set Theory*, Undergraduate Texts in Mathematics (Hardcover ed.), NY: Springer-Verlag, ISBN 0-387-90092-6, https://archive.org/details/naivesettheory0000halm_r4g0 - "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*, Metaphysics Research Lab, Stanford University, http://plato.stanford.edu/entries/set-theory/ - Suppes, Patrick (1972),
*Axiomatic Set Theory*, NY: Dover Publications, Inc., ISBN 0-486-61630-4, https://archive.org/details/axiomaticsettheo00supp_0 - 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".

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