Element character

∈
Mathematical signs
arithmetic
Plus sign	+
Minus sign	- , ./.
Mark	⋅ , ×
Divided sign	: , ÷ , /
Plus minus sign	± , ∓
Comparison sign	< , ≤ , = , ≥ , >
Root sign	√
Percent sign	%
Analysis
Sum symbol	Σ
Product mark	Π
Difference sign , Nabla	∆ , ∇
Prime	′
Partial differential	∂
Integral sign	∫
Concatenation characters	∘
Infinity symbol	∞
geometry
Angle sign	∠ , ∡ , ∢ , ∟
Vertical , parallel	⊥ , ∥
Triangle , square	△ , □
Diameter sign	⌀
Set theory
Union , cut	∪ , ∩
Difference , complement	∖ , ∁
Element character	∈
Subset , superset	⊂ , ⊆ , ⊇ , ⊃
Empty set	∅
logic
Follow arrow	⇒ , ⇔ , ⇐
Universal quantifier	∀
Existential quantifier	∃
Conjunction , disjunction	∧ , ∨
Negation sign	¬

The element character (∈) is a mathematical symbol used to indicate that an object is an element of a set . It goes back to Giuseppe Peano and was created through stylization from the Greek lowercase letter epsilon . A number of modifications exist for the element sign; it is often used in a crossed-out form (∉) or upside down (∋).

history

The founder of set theory Georg Cantor did not yet use an abbreviation for the expression a is an element of b . The element sign goes back to the Italian mathematician Giuseppe Peano, who first used it in the form of a Greek lowercase letter ϵ (epsilon) in 1889 in a work on the Peano axioms written in Latin :

"Signum ϵ significat est . Ita a ϵ b legitur a est quoddam b "

“The sign ϵ means is . So a ϵ b is read as a is a b "

- Giuseppe Peano : Arithmetices principia nova methodo exposita , 1889, p. X

The epsilon ϵ, which Peano wrote from 1890 in the form ε , is the initial of the Greek word ἐστί (esti) with the meaning is . In the form ε and the verbalization that is common today , the element sign was used in 1907 by Ernst Zermelo in his work on Zermelo set theory . In its original form ϵ, the element symbol spread from 1910 onwards via the Principia Mathematica by Bertrand Russell and Alfred North Whitehead . In the course of time it was then stylized to ∈.

use

If an object is an element of a set , this fact is noted ${\displaystyle x}$ ${\displaystyle M}$

${\displaystyle x\in M}$

and says "x is an element of M".

Occasionally it makes sense to reverse the order and then note it down

${\displaystyle M\ni x}$

and says "M contains as element x".

If there is no element of the set , one writes accordingly ${\displaystyle x}$ ${\displaystyle M}$

${\displaystyle x\notin M}$   or   . ${\displaystyle M\not \ni x}$

Formally, the element sign stands for a relation , the so-called element relation.

Coding

Element character

The element character can be found in the Unicode block mathematical operators and is coded as follows in computer systems.

Coding in Unicode , HTML and LaTeX

character	Unicode		designation	HTML			Latex
	position	designation		hexadecimal	decimal	named
∈	U+2208	element of	Element of	& # x2208;	& # 8712;	?	\in
∉	U+2209	not an element of	no element of	& # x2209;	& # 8713;	& notin;	\notin
∊	U+220A	small element of	small element of	& # x220A;	& # 8714;
∋	U+220B	contains as member	contains as an element	& # x220B;	& # 8715;	?	\ni
∌	U+220C	does not contain as a member	does not contain as an element	& # x220C;	& # 8716;		\not\ni
∍	U+220D	small contains as member	contains small as an element	& # x220D;	& # 8717;
⟒	U+27D2	element of opening upwards	Element opened from the top	& # x27D2;	& # 10194;
⫙	U+2AD9	element of opening downwards	Element opened from the bottom	& # x2AD9;	& # 10969;

epsilon

The Greek lowercase letter epsilon is also occasionally used as an element character.

Coding in Unicode , HTML and LaTeX

character	Unicode		designation	HTML			Latex
	position	designation		hexadecimal	decimal	named
ε	U+03B5	greek small letter epsilon	greek lowercase letter epsilon	& # x03B5;	& # 949;	ε	\varepsilon
ϵ	U+03F5	greek lunate epsilon symbol	greek crescent-shaped epsilon symbol	& # x03F5;	& # 1013;		\epsilon
϶	U+03F6	greek reversed lunate epsilon symbol	greek inverted crescent-shaped epsilon symbol	& # x03F6;	& # 1014;

Modifications

The following modifications of the element symbol also exist.

Coding in Unicode and HTML

character	Unicode		designation	HTML
	position	designation		hexadecimal	decimal
⋲	U+22F2	element of with long horizontal stroke	Element of with a long horizontal bar	& # x22F2;	& # 8946;
⋳	U+22F3	element of with vertical bar at end of horizontal stroke	Element of with a vertical bar at the end of the horizontal bar	& # x22F3;	& # 8947;
⋴	U+22F4	small element of with vertical bar at end of horizontal stroke	small element of with a vertical bar at the end of the horizontal line	& # x22F4;	& # 8948;
⋵	U+22F5	element of with dot above	Element from with point above	& # x22F5;	& # 8949;
⋶	U+22F6	element of with overbar	Element of with overline	& # x22F6;	& # 8950;
⋷	U+22F7	small element of with overbar	small element of with overline	& # x22F7;	& # 8951;
⋸	U+22F8	element of with underbar	Element of with underscore	& # x22F8;	& # 8952;
⋹	U+22F9	element of with two horizontal strokes	Element of with two horizontal bars	& # x22F9;	& # 8953;
⋺	U+22FA	contains with long horizontal stroke	contains with a long horizontal line	& # x22FA;	& # 8954;
⋻	U+22FB	contains with vertical bar at end of horizontal stroke	includes with a vertical bar at the end of the horizontal bar	& # x22FB;	& # 8955;
⋼	U+22FC	small contains with vertical bar at end of horizontal stroke	small contains with a vertical bar at the end of the horizontal bar	& # x22FC;	& # 8956;
⋽	U+22FD	contains with overbar	contains with overline	& # x22FD;	& # 8957;
⋾	U+22FE	small contains with overbar	small contains with overline	& # x22FE;	& # 8958;

See also

• Element icon
• Epsilon induction

literature

• Oliver Deiser: Introduction to set theory . Springer, 2009, ISBN 3-642-01445-3 .

Individual evidence

1. ↑ ^{a b} Oliver Deiser: Introduction to set theory . Springer, 2009, ISBN 978-3-642-01444-4 , pp. 21 .
2. ↑ See https://archive.org/details/arithmeticespri00peangoog for a link to the original work. File: First usage of the symbol ∈.png contains a picture of the corresponding text passage.
3. ^ Giuseppe Peano: Démonstration de l'intégrabilité des equations différentielles ordinaires . In: Mathematical Annals . tape 37 , 1890, p. 183 .
4. ↑ Ernst Zermelo: Investigations on the basics of set theory . In: Mathematical Annals . tape 65 , 1908, pp. 262 .
5. ^ Bertrand Russell, Alfred North Whitehead: Principia Mathematica . Volume 1. Cambridge University Press, 1910, pp. 26 .