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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 ...
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 (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
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 Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
"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 (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
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. in ...
strongly urged that "contains" be used for membership only, and "includes" for the subset relation only. For the relation ∈ , the
converse relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent& ...
T may be written :A \ni x , meaning "''A'' contains or includes ''x''". The
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
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 Epsilon (, ; uppercase , lowercase or lunate ; el, έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel or . In the system of Greek numerals it also has the value five. It was der ...
("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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
; 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 In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. Th ...
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 Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
, 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 universe. Acc ...
denoted ''U''. The range is the set of
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
s of ''U'' called the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
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 In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
*
Singleton (mathematics) In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...


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