Element (mathematics)

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

Sets[edit]

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

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

Notation and terminology[edit]

First usage of the symbol ∈ in the work Arithmetices principia, nova methodo exposita by Giuseppe Peano.

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 subset of A".^[1] Logician George Boolos strongly urged that "contains" be used for membership only and "includes" for the subset relation only.^[2]

For the relation ∈ , the converse relation ∈^T may be written

A\ni x,

meaning "A contains 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 1889 in his work Arithmetices principia, nova methodo exposita. Here he wrote on page X:

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

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 ("ϵ"), the first letter of the word ἐστί, which means "is".

The Unicode 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 LaTeX commands are "\in", "\notin", "\ni" and "\not\ni". Mathematica has commands "\[Element]", "\[NotElement]", "\[ReverseElement]" and "\[NotReverseElement]".

Character	∈		∉		∋		∌
Unicode name	ELEMENT OF		NOT AN ELEMENT OF		CONTAINS AS MEMBER		DOES NOT CONTAIN AS MEMBER
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	8712	U+2208	8713	U+2209	8715	U+220B	8716	U+220C
UTF-8	226 136 136	E2 88 88	226 136 137	E2 88 89	226 136 139	E2 88 8B	226 136 140	E2 88 8C
Numeric character reference	∈	∈	∉	∉	∋	∋	∌	∌
Named character reference	∈		∉		&ni;
LaTeX	\in		\notin		\ni		\not\ni
Wolfram Mathematica	\[Element]		\[NotElement]		\[ReverseElement]		\[NotReverseElement]

Complement and converse[edit]

Every relation R : U → V is subject to two involutions: complementation R → ${\bar {R}}$ and conversion R^T:V → U. The relation ∈ has for its domain a universal set U, and has the power set P(U) for its codomain or range. The complementary relation ${\bar {\in }}\ =\ \notin$ expresses the opposite of ∈. An element x ∈ U may have x ∉ A, in which case x ∈ U \ 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 x ∈ A.

Cardinality of sets[edit]

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 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[edit]

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 prime numbers) is infinite (proved by Euclid).

References[edit]

^ Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8. p. 12
^ George Boolos (February 4, 1992). 24.243 Classical Set Theory (lecture) (Speech). Massachusetts Institute of Technology.

External links[edit]

Weisstein, Eric W. "Element". MathWorld.

[schech-1] Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8. p. 12

[boolos-2] George Boolos (February 4, 1992). 24.243 Classical Set Theory (lecture) (Speech). Massachusetts Institute of Technology.

[1]

[2]

v t e Mathematical logic
General	Formal language Formation rule Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Rule of inference Relation Theorem Logical consequence Type theory Symbol Syntax Theory
Systems	Formal system Deductive system Axiomatic system Hilbert style systems Natural deduction Sequent calculus
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Propositional calculus and Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic
Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus
Naive set theory	Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image Map Function Relation Ordered pair
Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory
Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity
Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax
Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

v t e Set theory
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Concepts Methods	Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram
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Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine

Element (mathematics)

Contents

Sets[edit]

Notation and terminology[edit]

Complement and converse[edit]

Cardinality of sets[edit]

Examples[edit]

References[edit]

Further reading[edit]

External links[edit]

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