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Tag Wiki 'Element (mathematics)'.
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In , 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 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.

Another possible notation for the same relation is

A \ni x

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

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".


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, ... }.


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 ) 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".


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