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   » » Wiki: Element (mathematics)
Tag Wiki 'Element (mathematics)'.

In , an element, or member, of a set is any one of the distinct objects that make up that set.

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

Sets can themselves be elements. For example, consider the set B = \{1, 2, \{3, 4\}\}. The elements of are not 1, 2, 3, and 4. Rather, there are only three elements of , namely the numbers 1 and 2, and the set \{3, 4\}.

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

Notation and terminology
The relation "is an element of", also called set membership, is denoted by the symbol " \in ". Writing

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 of  A".

p. 12 Logician strongly urged that "contains" be used for membership only and "includes" for the subset relation only.

For the relation ϵ , the converse relation ϵT may be written

A \ni x , meaning " A contains x".

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

x \notin A means that " x is not an element of  A".

The symbol ϵ was first used by Giuseppe Peano 1889 in his work italic=yes. Here he wrote on page X:

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

which means

The symbol ϵ means is. So a ϵ b is read as a is a b; ...

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

The characters for these symbols are U+2208 ('element of'), U+2209 ('not an element of'), U+220B ('contains as member') and U+220C ('does not contain as member'). The equivalent commands are "\in", "\notin", "\ni" and "\not\ni". has commands "\Element", "\NotElement", "\ReverseElement" and "\NotReverseElement".

Complement and converse
Every relation R : UV is subject to two involutions: complementation R\bar{R} and conversion RT: VU. The relation ∈ has for its domain a universal set U, and has the P( U) for its codomain or range. The complementary relation \bar{\in} \ = \ \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 \in^T \ = \ \ni swaps the domain and range with ∈. For any A in P( U), A \ni x is true when xA.

Cardinality of sets
The number of elements in a particular set is a property known as ; 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, ... }.

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 ) is infinite (proved by ).

Further reading
  • - "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).
  • - 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".

External links
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