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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an element (or member) of a set is any one of the distinct
objects Object may refer to: General meanings * Object (philosophy) An object is a philosophy, philosophical term often used in contrast to the term ''Subject (philosophy), subject''. A subject is an observer and an object is a thing observed. For mo ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
of ''A''". p. 12 Logician
George Boolos George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Met ...
strongly urged that "contains" be used for membership only, and "includes" for the subset relation only. For the relation ∈ , the
converse relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
T may be written :A \ni x , meaning "''A'' contains or includes ''x''". The
negation In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...

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'' b; …
The symbol itself is a stylized lowercase Greek letter
epsilon Epsilon (, ; ; uppercase Letter case is the distinction between the letters Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A symbol is a mark, sign, or word th ...

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
; 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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''


See also

*
Identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
*
Singleton (mathematics) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...


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{{Commons This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed. Mathematical concepts ...