Element (mathematics)

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Short description: Any one of the distinct objects that make up a set in set theory

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.

Notation and terminology

The relation "is an element of", also called set membership, is denoted by the symbol "∈". 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 ∈ , the converse relationT 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 "∉". Writing

[math]\displaystyle{ x \notin A }[/math] means that "x is not an element of A".

The symbol ∈ was first used by Giuseppe Peano, in his 1889 work Arithmetices principia, nova methodo exposita.[4] 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 b; …

The symbol itself is a stylized lowercase Greek letter epsilon ("ϵ"), the first letter of the word ἐστί, which means "is".[4]

Character information
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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 0 U+220B 8716 0 0 0 U+220C
UTF-8 226 136 136 0 0 0 E2 88 88 00 00 00 226 136 137 0 0 0 E2 88 89 00 00 00 226 136 139 0 0 0 E2 88 8B 00 00 00 226 136 140 0 0 0 E2 88 8C 00 00 00
Numeric character reference ∈��� ∈��� ∉��� ∉��� ∋��� ∋��� ∌��� ∌���
Named character reference ∈ ∉ ∋
LaTeX \in \notin \ni \not\ni or \notni
Wolfram Mathematica \[Element] \[NotElement] \[ReverseElement] \[NotReverseElement]

Cardinality of sets

Main page: Cardinality

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 ∈ A
  • 5 ∉ A
  • {3,4} ∈ B
  • 3 ∉ B
  • 4 ∉ B
  • yellow ∉ 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

  1. Weisstein, Eric W.. "Element" (in en). https://mathworld.wolfram.com/Element.html. 
  2. Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8.  p. 12
  3. George Boolos (February 4, 1992). 24.243 Classical Set Theory (lecture) (Speech). Massachusetts Institute of Technology.
  4. 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): 367–372. doi:10.1305/ndjfl/1093891001. 
  5. "Sets - Elements | Brilliant Math & Science Wiki" (in en-us). https://brilliant.org/wiki/sets-elements/. 

Further reading