# 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 *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 {1, 2}, are subsets of *A*.

Sets can themselves be elements. For example, consider the set *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 {3, 4}.

The elements of a set can be anything. For example, *C* = { red, green, blue }, 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]}

Another possible notation for the same relation is

meaning "*A* contains *x*", though it is used less often.

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

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

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

"Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b; ..."

which means

"The symbol ϵ meansis. So a ϵ 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".

The Unicode characters for these symbols are U+2208 ('element of'), U+220B ('contains as member') and U+2209 ('not an element of'). The equivalent LaTeX commands are "\in", "\ni" and "\notin". Mathematica has commands "\[Element]" and "\[NotElement]".

## 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 (this was proven by Euclid).

## 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, Cambridge, MA.

## 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".