ISO 31-11

ISO 31-11: 1992 is part of the international standard ISO 31 , which defines “ mathematical signs and symbols for use in physical sciences and technology ”. This standard was adopted in 1992, and in 2009 was replaced by a slightly supplemented standard ISO 80000-2 ^[1] .

Content

Math Symbols

Below are (not fully) the main sections of the standard ^[2] .

Mathematical Logic

Identified reading	Use	Title	Meaning and explanation	Comments
∧	p ∧ q	conjunction	p and q
∨	p ∨ q	disjunction	p or q (possibly both)
¬	¬ p	negation	invalid p ; not- p
⇒	p ⇒ q	implication	if p , then q ; from p it follows q	Sometimes written as p → q or q ⇐ p .
∀	∀ x ∈ A p ( x ) (∀ x ∈ A ) p ( x )	community quantifier	for every x from the set A, the statement p ( x )	For brevity, the specification "∈ A " is often omitted if it is clear from the context.
∃	∃ x ∈ A p ( x ) (∃ x ∈ A ) p ( x )	quantifier of existence	there exists x from the set A for which the statement p ( x ) is true	For brevity, the specification "∈ A " is often omitted if it is clear from the context. Option ∃! means that such x is unique in the set A.

Set Theory

Identified reading	Use	Meaning and explanation	Comments
∈	x ∈ A	x belongs to A ; x is an element of the set A
∉	x ∉ A	x does not belong to A ; x is not an element of the set A	The strike line can also be vertical.
∋	A ∋ x	The set A contains the element x	is equivalent to x ∈ A
∌	A ∌ x	The set A does not contain the element x	equivalent to x ∉ A
{}	{x ₁ , x ₂ , ..., x _n }	the set formed by the elements x ₁ , x ₂ , ..., x _n	also {x _i ∣ i ∈ I }, where I denotes the set of indices
{∣}	{ x ∈ A ∣ p ( x )}	the set of elements A for which the statement p ( x ) is true	Example: { x ∈ ℝ ∣ x > 5} For brevity, the specification "∈ A " is often omitted if it is clear from the context.
card	card ( A )	cardinal number of elements of the set A ; power A
∖	A ∖ B	the difference of the sets A and B ; A minus B	Many elements from A that are not in B. A ∖ B = { x ∣ x ∈ A ∧ x ∉ B } Should not be written as A - B.
∅		empty set
ℕ		many natural numbers , including zero	ℕ = {0, 1, 2, 3, ...} If zero is excluded, mark the symbol with an asterisk : ℕ ^* = {1, 2, 3, ...} The final subset: ℕ _k = {0, 1, 2, 3, ..., k - 1}
ℤ		many integers	ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...} Integer nonzero are denoted ℤ ^* = ℤ ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...}
ℚ		many rational numbers	ℚ ^* = ℚ ∖ {0}
ℝ		set of real numbers	ℝ ^* = ℝ ∖ {0}
ℂ		set of complex numbers	ℂ ^* = ℂ ∖ {0}
[,]	[ a , b ]	closed interval in ℝ from a (including) to b (including)	[ a , b ] = { x ∈ ℝ ∣ a ≤ x ≤ b }
],] (,]	] a , b ] ( a , b ]	half-open interval on the left in ℝ from a (excluding) to b (including)	] a , b ] = { x ∈ ℝ ∣ a < x ≤ b }
[, [ [,)	[ a , b [ [ a , b )	the half-open interval on the right in ℝ from a (including) to b (excluding)	[ a , b [= { x ∈ ℝ ∣ a ≤ x < b }
], [ (,)	] a , b [ ( a , b )	open interval in ℝ from a (excluding) to b (excluding)	] a , b [= { x ∈ ℝ ∣ a < x < b }
⊆	B ⊆ A	B is contained in A ; B is a subset of A	Each element of B belongs to A. Symbol variant: ⊂.
⊂	B ⊂ A	B is contained in A as a proper subset	Each element of B belongs to A , but B is not equal to A. If ⊂ stands for “contained,” then ⊊ should be used in the sense of “contained as a proper subset”.
⊈	C ⊈ A	C is not contained in A ; C is not a subset of A	Option: C ⊄ A
⊇	A ⊇ B	A contains B (as a subset)	A contains all the elements of B. Option: ⊃. B ⊆ A is equivalent to A ⊇ B.
⊃	A ⊃ B.	A contains B as its own subset .	A contains all elements of B , but A is not equal to B. If the symbol ⊃ is used, then ⊋ should be used in the sense of "contains as a proper subset".
⊉	A ⊉ C	A does not contain C (as a subset)	Option: ⊅. A ⊉ C is equivalent to C ⊈ A.
∪	A ∪ B	union of A and B	The set of elements belonging to either A or B , or both A and B. A ∪ B = { x ∣ x ∈ A ∨ x ∈ B }
⋃	$\bigcup _{i=1}^{n}A_{i}$ ${\ displaystyle \ bigcup _ {i = 1} ^ {n} A_ {i}}$	joining a family of sets	$\bigcup _{i=1}^{n}A_{i}=A_{1}\cup A_{2}\cup \ldots \cup A_{n}$ ${\ displaystyle \ bigcup _ {i = 1} ^ {n} A_ {i} = A_ {1} \ cup A_ {2} \ cup \ ldots \ cup A_ {n}}$ , the set of elements belonging to at least one of A ₁ , ..., A _n . Options: $\bigcup {}_{i=1}^{n}$ ${\ displaystyle \ bigcup {} _ {i = 1} ^ {n}}$ and $\bigcup _{i\in I}$ ${\ displaystyle \ bigcup _ {i \ in I}}$ , $\bigcup {}_{i\in I}$ ${\ displaystyle \ bigcup {} _ {i \ in I}}$ where I is the set of indices.
∩	A ∩ B	intersection of A and B	Many elements belonging to both A and B. A ∩ B = { x ∣ x ∈ A ∧ x ∈ B }
⋂	$\bigcap _{i=1}^{n}A_{i}$ ${\ displaystyle \ bigcap _ {i = 1} ^ {n} A_ {i}}$	intersection of a family of sets	$\bigcap _{i=1}^{n}A_{i}=A_{1}\cap A_{2}\cap \ldots \cap A_{n}$ ${\ displaystyle \ bigcap _ {i = 1} ^ {n} A_ {i} = A_ {1} \ cap A_ {2} \ cap \ ldots \ cap A_ {n}}$ , the set of elements belonging to each A ₁ , ..., A _n . Options: $\bigcap {}_{i=1}^{n}$ ${\ displaystyle \ bigcap {} _ {i = 1} ^ {n}}$ and $\bigcap _{i\in I}$ ${\ displaystyle \ bigcap _ {i \ in I}}$ , $\bigcap {}_{i\in I}$ ${\ displaystyle \ bigcap {} _ {i \ in I}}$ where I is the set of indices.
∁	∁ _A B	difference between A and B	The set of elements A that are not in B. The symbol A is often omitted if understood in context. Option: ∁ _A B = A ∖ B.
(,)	( a , b )	ordered pair a , b	( a , b ) = ( c , d ) if and only if a = c and b = d . Recording option: ⟨a, b⟩.
(, ...,)	( a ₁ , a ₂ , ..., a _n )	ordered n is a tuple	Recording option: ⟨a ₁ , a ₂ , ..., a n⟩ ( angle brackets ).
×	A × B	Cartesian product of the sets A and B	The set of ordered pairs ( a , b ), where a ∈ A and b ∈ B. A × B = {( a , b ) ∣ a ∈ A ∧ b ∈ B } A × A × ⋯ × A is denoted by A ⁿ , where n is the number of factors.
Δ	Δ _A	the set of pairs ( a , a ) ∈ A × A , where a ∈ A ; that is, the diagonal of the set A × A	Δ _A = {( a , a ) ∣ a ∈ A } Record Option: id _A.

Other characters

Designation		Example	Meaning and explanation	Comments
HTML	Tex	Example	Meaning and explanation	Comments
≝	${\stackrel {\mathrm {def} }{=}}$ ${\ displaystyle {\ stackrel {\ mathrm {def}} {=}}}$	a ≝ b	a is b by definition ^[2]	Record Option: a : = b
=	$=$ ${\ displaystyle =}$	a = b	a is b	Option: the symbol ≡ emphasizes that this equality is an identity.
≠	$\neq$ ${\ displaystyle \ neq}$	a ≠ b	a is not equal to b	Record Option: $a\not \equiv b$ ${\ displaystyle a \ not \ equiv b}$ indicates that a is not identically equal to b .
≙	${\stackrel {\wedge }{=}}$ ${\ displaystyle {\ stackrel {\ wedge} {=}}}$	a ≙ b	a corresponds to b	Example: on a map with a scale of 1:10 ⁶ 1 cm ≙ 10 km.
≈	$\approx$ ${\ displaystyle \ approx}$	a ≈ b	a is approximately equal to b	The symbol ≃ means "asymptotically equal."
∼ ∝	${\begin{matrix}\sim \\\propto \end{matrix}}$ ${\ displaystyle {\ begin {matrix} \ sim \\\ propto \ end {matrix}}}$	a ∼ b a ∝ b	a is proportional to b
<	$<$ ${\ displaystyle <}$	a < b	a is less than b
>	$>$ ${\ displaystyle>}$	a > b	a is greater than b
≤	$\leqslant$ ${\ displaystyle \ leqslant}$	a $\leqslant$ ${\ displaystyle \ leqslant}$ b	a is less than or equal to b	Option: ≦.
≥	$\geqslant$ ${\ displaystyle \ geqslant}$	a $\geqslant$ ${\ displaystyle \ geqslant}$ b	a is greater than or equal to b	Option: ≧.
≪	$\ll$ ${\ displaystyle \ ll}$	a ≪ b	a is much smaller than b
≫	$\gg$ ${\ displaystyle \ gg}$	a ≫ b	a is much larger than b
∞	$\infty$ ${\ displaystyle \ infty}$		infinity
() [] {} $<>$ ${\ displaystyle <>}$	${\begin{matrix}()\\{[]}\\\{\}\\\langle \rangle \end{matrix}}$ {\ displaystyle {\ begin {matrix} () \\ {[]} \\\ {\} \\\ langle \ rangle \ end {matrix}}}	${\begin{matrix}{(a+b)c}\\{[a+b]c}\\{\{a+b\}c}\\{\langle a+b\rangle c}\end{matrix}}$ ${\ displaystyle {\ begin {matrix} {(a + b) c} \\ {[a + b] c} \\ {\ {a + b \} c} \\ {\ langle a + b \ rangle c } \ end {matrix}}}$	$ac+bc$ ${\ displaystyle ac + bc}$ , parentheses $ac+bc$ ${\ displaystyle ac + bc}$ square brackets $ac+bc$ ${\ displaystyle ac + bc}$ braces $ac+bc$ ${\ displaystyle ac + bc}$ angle brackets	In algebra, the priority of different brackets $(),[],\{\},\langle \rangle$ ${\ displaystyle (), [], \ {\}, \ langle \ rangle}$ not standardized. Some sections of mathematics have special rules for use. $(),[],\{\},\langle \rangle$ ${\ displaystyle (), [], \ {\}, \ langle \ rangle}$ .
∥	$\\|$ ${\ displaystyle \ \|}$	AB ∥ CD	line AB is parallel to line CD
⊥	$\perp$ ${\ displaystyle \ perp}$	$\mathrm {AB\perp CD}$ ${\ displaystyle \ mathrm {AB \ perp CD}}$	line AB is perpendicular to line CD

Operations

Designation	Example	Meaning and explanation	Comments
+	a + b	a plus b
-	a - b	a minus b
±	a ± b	a plus or minus b
∓	a ∓ b	a minus plus b	- ( a ± b ) = - a ∓ b
...	...	...	...
⋮

Functions

Example	Meaning and explanation	Comments
$f:D\rightarrow C$ ${\ displaystyle f: D \ rightarrow C}$	the function f is defined on D and takes values in C	Used to explicitly indicate scope and value for a function.
$f\left(S\right)$ ${\ displaystyle f \ left (S \ right)}$	$\left\{f\left(x\right)\mid x\in S\right\}$ ${\ displaystyle \ left \ {f \ left (x \ right) \ mid x \ in S \ right \}}$	The set of all values of the function corresponding to the elements of the subset S of the domain of definition.
⋮

Exponential and logarithmic functions

Example	Meaning and explanation	Comments
e	base of natural logarithms	e = 2.71828 ...
e ^x	exponential function with base e
$\log _{a}x$ ${\ displaystyle \ log _ {a} x}$	base logarithm $a$ ${\ displaystyle a}$
lb x	binary logarithm (with base 2)	lb x = $\log _{2}x$ ${\ displaystyle \ log _ {2} x}$
ln x	natural logarithm (with base e)	ln x = $\log _{e}x$ ${\ displaystyle \ log _ {e} x}$
lg x	decimal logarithm (with base 10)	lg x = $\log _{1}0x$ ${\ displaystyle \ log _ {1} 0x}$
...	...	...
⋮

Circular and hyperbolic functions

Example	Meaning and explanation	Comments
$\pi$ ${\ displaystyle \ pi}$	ratio of circumference to its diameter	$\pi$ ${\ displaystyle \ pi}$ = 3.14159 ...
...	...	...
⋮

Complex numbers

Example	Meaning and explanation	Comments
i j	imaginary unit ; $i^{2}=-1$ ${\ displaystyle i ^ {2} = - 1}$	in electrical engineering instead $i$ ${\ displaystyle i}$ symbol used $j$ ${\ displaystyle j}$ .
Re z	real part z	z = x + i y , where x = Re z and y = Im z
Im z	imaginary part z	z = x + i y , where x = Re z and y = Im z
∣ z ∣	absolute value of z ; z module	Sometimes denoted by mod z
arg z	argument z ; phase z	$r=e^{i\varphi }$ ${\ displaystyle r = e ^ {i \ varphi}}$ , where r = ∣ z ∣, φ = arg z , while Re z = r cos φ , Im z = r sin φ
z *	(complex) conjugate to z	Option: a dash over z instead of an asterisk
sgn z	sgn z	sgn z = z / ∣ z ∣ = exp ( i arg z ) for z ≠ 0, sgn 0 = 0

Matrices

Example	Meaning and explanation	Comments
A	matrix A	...
...	...	...
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Coordinate systems

Coordinates	Point vector radius	Coordinate system name	Comments
x , y , z	$[xyz]=[xyz];$ ${\ displaystyle [xyz] = [xyz];}$	rectangular coordinate system (Cartesian)	x ₁ , x ₂ , x ₃ for coordinates and e ₁ , e ₂ , e ₃ for basis vectors. This symbolism is easily generalized to the multidimensional case. e _x , e _y , e _z form an orthogonal (right) basis. Base vectors in space are often denoted by i , j , k .
ρ , φ , z	$[x,y,z]=[\rho \cos(\phi ),\rho \sin(\phi ),z]$ ${\ displaystyle [x, y, z] = [\ rho \ cos (\ phi), \ rho \ sin (\ phi), z]}$	cylindrical coordinate system	e _ρ ( φ ), e _φ ( φ ), e _z form an orthogonal (right) basis. If z = 0 (two-dimensional case), then ρ and φ are polar coordinates .
r , θ , φ	$[x,y,z]=r[\sin(\theta )\cos(\phi ),\sin(\theta )\sin(\phi ),\cos(\theta )]$ {\ displaystyle [x, y, z] = r [\ sin (\ theta) \ cos (\ phi), \ sin (\ theta) \ sin (\ phi), \ cos (\ theta)]}	spherical coordinate system	e _r ( θ , φ ), e _θ ( θ , φ ), e _φ ( φ ) form an orthogonal (right) basis.

Vectors and tensors

Example	Meaning and explanation	Comments
a ${\vec {a}}$ ${\ displaystyle {\ vec {a}}}$	vector a	vectors in the literature can be shown in bold and / or italics, a dash or an arrow over a letter, etc. Any vector a can be multiplied by a scalar k , getting the vector k a .
...	...	...
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Special Functions

Example	Meaning and explanation	Comments
$J_{i}(x)$ ${\ displaystyle J_ {i} (x)}$	Bessel cylindrical functions (first kind)	...
...	...	...
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ISO 80000-2 standard

A new, supplemented standard ISO 80000-2 instead of ISO 31-11 appeared in 2009. New sections were added to it (there were 19 in total):

Standard number sets and intervals .
Elementary geometry.
Combinatorics .
Transforms

The name of the standard is changed to Quantities and units - Part 2: Mathematics .

Notes

↑ ISO 80000-2 .
↑ ¹ ² Thompson, Ambler. Guide for the Use of the International System of Units (SI) - NIST Special Publication 811, 2008 Edition - Second Printing / Ambler Thompson, Barry M Taylor. - Gaithersburg, MD, USA: National Institute of Standards and Technology , March 2008.

Links

ISO 80000-2: 2009 (neopr.) . International Organization for Standardization . Date accessed August 12, 2019.

[_3382066814f0e035-1] ISO 80000-2 .

[ISO_31-11-2] ¹ ² Thompson, Ambler. Guide for the Use of the International System of Units (SI) - NIST Special Publication 811, 2008 Edition - Second Printing / Ambler Thompson, Barry M Taylor. - Gaithersburg, MD, USA: National Institute of Standards and Technology , March 2008.