# 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 ${\displaystyle A=\{1,2,3,4\}}$ means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example ${\displaystyle \{1,2\}}$, are subsets of A.

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

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

## Notation and terminology

First usage of the symbol ∈ in the work Arithmetices principia, nova methodo exposita by Giuseppe Peano.

The relation "is an element of", also called set membership, is denoted by the symbol "${\displaystyle \in }$". Writing

${\displaystyle x\in A}$

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".[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 relationT may be written

${\displaystyle A\ni x,}$ meaning "A contains x".

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

${\displaystyle x\notin A}$ 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 a is a b; …

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 &#8712; &#x2208; &#8713; &#x2209; &#8715; &#x220B; &#8716; &#x220C;
Named character reference &isin; &notin; &ni;
LaTeX \in \notin \ni \not\ni
Wolfram Mathematica \[Element] \[NotElement] \[ReverseElement] \[NotReverseElement]

### Complement and converse

Every relation R : UV is subject to two involutions: complementation R${\displaystyle {\bar {R}}}$ and conversion RT: VU. 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 ${\displaystyle {\bar {\in }}={\mathord {\notin }}}$ expresses the opposite of ∈. An element xU may have xA, in which case xU \ A, the complement of A in U.

The converse relation ${\displaystyle \in ^{\textsf {T}}={\mathord {\ni }}}$ swaps the domain and range with ∈. For any A in P(U), ${\displaystyle A\ni x}$ is true when xA.

## 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).

## References

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