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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an element (or member) of a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
is any one of the distinct
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...
that belong to that set.


Sets

Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of . Sets can themselves be elements. For example, consider the set B = \. 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 \. The elements of a set can be anything. For example, C = \ 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 "∈". 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, although some authors use them to mean instead "''x'' is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of ''A''". p. 12 Logician
George Boolos George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology. Life Boolos is of Greek- Jewish descent. He graduated with an A.B. ...
strongly urged that "contains" be used for membership only, and "includes" for the subset relation only. For the relation ∈ , the converse relationT may be written :A \ni x , meaning "''A'' contains or includes ''x''". The negation of set membership is denoted by the symbol "∉". Writing :x \notin A means that "''x'' is not an element of ''A''". The symbol ∈ was first used by Giuseppe Peano, in his 1889 work . Here he wrote on page X:
which means
The symbol ∈ means ''is''. So a ∈ b is read as a ''is a certain'' b; …
The symbol itself is a stylized lowercase Greek letter epsilon ("ϵ"), the first letter of the word , which means "is".


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 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 .


Examples

Using the sets defined above, namely ''A'' = , ''B'' = and ''C'' = , the following statements are true: *2 ∈ ''A'' *5 ∉ ''A'' * ∈ ''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 The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
denoted ''U''. The range is the set of
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of ''U'' called the power set of ''U'' and denoted P(''U''). Thus the relation \in is a subset of ''U'' x P(''U''). The converse relation \ni is a subset of P(''U'') x ''U''.


See also

* Identity element * Singleton (mathematics)


References


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". {{Set theory Basic concepts in set theory