ISO 31-11

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ISO 31-11:1992 was the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology. It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-80000-2:2019.^[1]

Its definitions include the following:^[2]

Mathematical logic[]

Sign	Example	Name	Meaning and verbal equivalent	Remarks
∧	p ∧ q	conjunction sign	p and q
∨	p ∨ q	disjunction sign	p or q (or both)
¬	¬ p	negation sign	negation of p; not p; non p
⇒	p ⇒ q	implication sign	if p then q; p implies q	Can also be written as q ⇐ p. Sometimes → is used.
∀	∀x∈A p(x) (∀x∈A) p(x)	universal quantifier	for every x belonging to A, the proposition p(x) is true	The "∈A" can be dropped where A is clear from context.
∃	∃x∈A p(x) (∃x∈A) p(x)	existential quantifier	there exists an x belonging to A for which the proposition p(x) is true	The "∈A" can be dropped where A is clear from context. ∃! is used where exactly one x exists for which p(x) is true.

Sets[]

Sign	Example	Meaning and verbal equivalent	Remarks
∈	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 negation stroke can also be vertical.
∋	A ∋ x	the set A contains x (as an element)	same meaning as x ∈ A
∌	A ∌ x	the set A does not contain x (as an element)	same meaning as x ∉ A
{ }	{x1, x2, ..., xn}	set with elements x1, x2, ..., xn	also {xi ∣ i ∈ I}, where I denotes a set of indices
{ ∣ }	{x ∈ A ∣ p(x)}	set of those elements of A for which the proposition p(x) is true	Example: {x ∈ ${\displaystyle \mathbb {R} }$ ∣ x > 5} The ∈A can be dropped where this set is clear from the context.
card	card(A)	number of elements in A; cardinal of A
∖	A ∖ B	difference between A and B; A minus B	The set of elements which belong to A but not to B. A ∖ B = { x ∣ x ∈ A ∧ x ∉ B } A − B should not be used.
∅		the empty set
${\displaystyle \mathbb {N} }$		the set of natural numbers; the set of positive integers and zero	${\displaystyle \mathbb {N} }$ = {0, 1, 2, 3, ...} Exclusion of zero is denoted by an asterisk: ${\displaystyle \mathbb {N} }$* = {1, 2, 3, ...} ${\displaystyle \mathbb {N} }$k = {0, 1, 2, 3, ..., k − 1}
${\displaystyle \mathbb {Z} }$		the set of integers	${\displaystyle \mathbb {Z} }$ = {..., −3, −2, −1, 0, 1, 2, 3, ...} ${\displaystyle \mathbb {Z} }$* = ${\displaystyle \mathbb {Z} }$ ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...}
${\displaystyle \mathbb {Q} }$		the set of rational numbers	${\displaystyle \mathbb {Q} }$* = ${\displaystyle \mathbb {Q} }$ ∖ {0}
${\displaystyle \mathbb {R} }$		the set of real numbers	${\displaystyle \mathbb {R} }$* = ${\displaystyle \mathbb {R} }$ ∖ {0}
${\displaystyle \mathbb {C} }$		the set of complex numbers	${\displaystyle \mathbb {C} }$* = ${\displaystyle \mathbb {C} }$ ∖ {0}
[,]	[a,b]	closed interval in ${\displaystyle \mathbb {R} }$ from a (included) to b (included)	[a,b] = {x ∈ ${\displaystyle \mathbb {R} }$ ∣ a ≤ x ≤ b}
],] (,]	]a,b] (a,b]	left half-open interval in ${\displaystyle \mathbb {R} }$ from a (excluded) to b (included)	]a,b] = {x ∈ ${\displaystyle \mathbb {R} }$ ∣ a < x ≤ b}
[,[ [,)	[a,b[ [a,b)	right half-open interval in ${\displaystyle \mathbb {R} }$ from a (included) to b (excluded)	[a,b[ = {x ∈ ${\displaystyle \mathbb {R} }$ ∣ a ≤ x < b}
],[ (,)	]a,b[ (a,b)	open interval in ${\displaystyle \mathbb {R} }$ from a (excluded) to b (excluded)	]a,b[ = {x ∈ ${\displaystyle \mathbb {R} }$ ∣ a < x < b}
⊆	B ⊆ A	B is included in A; B is a subset of A	Every element of B belongs to A. ⊂ is also used.
⊂	B ⊂ A	B is properly included in A; B is a proper subset of A	Every element of B belongs to A, but B is not equal to A. If ⊂ is used for "included", then ⊊ should be used for "properly included".
⊈	C ⊈ A	C is not included in A; C is not a subset of A	⊄ is also used.
⊇	A ⊇ B	A includes B (as subset)	A contains every element of B. ⊃ is also used. B ⊆ A means the same as A ⊇ B.
⊃	A ⊃ B.	A includes B properly.	A contains every element of B, but A is not equal to B. If ⊃ is used for "includes", then ⊋ should be used for "includes properly".
⊉	A ⊉ C	A does not include C (as subset)	⊅ is also used. A ⊉ C means the same as C ⊈ A.
∪	A ∪ B	union of A and B	The set of elements which belong to A or to B or to both A and B. A ∪ B = { x ∣ x ∈ A ∨ x ∈ B }
⋃	${\displaystyle \bigcup _{i=1}^{n}A_{i}}$	union of a collection of sets	${\displaystyle \bigcup _{i=1}^{n}A_{i}=A_{1}\cup A_{2}\cup \ldots \cup A_{n}}$, the set of elements belonging to at least one of the sets A1, ..., An. ${\displaystyle \bigcup {}_{i=1}^{n}}$ and ${\displaystyle \bigcup _{i\in I}}$, ${\displaystyle \bigcup {}_{i\in I}}$ are also used, where I denotes a set of indices.
∩	A ∩ B	intersection of A and B	The set of elements which belong to both A and B. A ∩ B = { x ∣ x ∈ A ∧ x ∈ B }
⋂	${\displaystyle \bigcap _{i=1}^{n}A_{i}}$	intersection of a collection of sets	${\displaystyle \bigcap _{i=1}^{n}A_{i}=A_{1}\cap A_{2}\cap \ldots \cap A_{n}}$, the set of elements belonging to all sets A1, ..., An. ${\displaystyle \bigcap {}_{i=1}^{n}}$ and ${\displaystyle \bigcap _{i\in I}}$, ${\displaystyle \bigcap {}_{i\in I}}$ are also used, where I denotes a set of indices.
∁	∁AB	complement of subset B of A	The set of those elements of A which do not belong to the subset B. The symbol A is often omitted if the set A is clear from context. Also ∁AB = A ∖ B.
(,)	(a, b)	ordered pair a, b; couple a, b	(a, b) = (c, d) if and only if a = c and b = d. ⟨a, b⟩ is also used.
(,...,)	(a1, a2, ..., an)	ordered n-tuple	⟨a1, a2, ..., an⟩ is also used.
×	A × B	cartesian product of A and B	The set of ordered pairs (a, b) such that a ∈ A and b ∈ B. A × B = { (a, b) ∣ a ∈ A ∧ b ∈ B } A × A × ⋯ × A is denoted by An, where n is the number of factors in the product.
Δ	ΔA	set of pairs (a, a) ∈ A × A where a ∈ A; diagonal of the set A × A	ΔA = { (a, a) ∣ a ∈ A } idA is also used.

Miscellaneous signs and symbols[]

Sign		Example	Meaning and verbal equivalent	Remarks
HTML	TeX
≝	${\displaystyle {\stackrel {\mathrm {def} }{=}}}$	a ≝ b	a is by definition equal to b [2]	:= is also used
=	${\displaystyle =}$	a = b	a equals b	≡ may be used to emphasize that a particular equality is an identity.
≠	${\displaystyle \neq }$	a ≠ b	a is not equal to b	${\displaystyle a\not \equiv b}$ may be used to emphasize that a is not identically equal to b.
≙	${\displaystyle {\stackrel {\wedge }{=}}}$	a ≙ b	a corresponds to b	On a 1:106 map: 1 cm ≙ 10 km.
≈	${\displaystyle \approx }$	a ≈ b	a is approximately equal to b	The symbol ≃ is reserved for "is asymptotically equal to".
∼ ∝	${\displaystyle {\begin{matrix}\sim \\\propto \end{matrix}}}$	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
≤	${\displaystyle \leq }$	a ≤ b	a is less than or equal to b	The symbol ≦ is also used.
≥	${\displaystyle \geq }$	a ≥ b	a is greater than or equal to b	The symbol ≧ is also used.
≪	${\displaystyle \ll }$	a ≪ b	a is much less than b
≫	${\displaystyle \gg }$	a ≫ b	a is much greater than b
∞	${\displaystyle \infty }$		infinity
() [] {} ⟨⟩	${\displaystyle {\begin{matrix}()\\{[]}\\\{\}\\\langle \rangle \end{matrix}}}$	${\displaystyle {\begin{matrix}{(a+b)c}\\{[a+b]c}\\{\{a+b\}c}\\{\langle a+b\rangle c}\end{matrix}}}$	${\displaystyle ac+bc}$, parentheses ${\displaystyle ac+bc}$, square brackets ${\displaystyle ac+bc}$, braces ${\displaystyle ac+bc}$, angle brackets	In ordinary algebra, the sequence of ${\displaystyle (),[],\{\},\langle \rangle }$ in order of nesting is not standardized. Special uses are made of ${\displaystyle (),[],\{\},\langle \rangle }$ in particular fields.
∥	${\displaystyle \|}$	AB ∥ CD	the line AB is parallel to the line CD
⊥	${\displaystyle \perp }$	${\displaystyle \mathrm {AB\perp CD} }$	the line AB is perpendicular to the line CD[3]

Operations[]

Sign	Example	Meaning and verbal equivalent	Remarks
+	a + b	a plus b
−	a − b	a minus b
±	a ± b	a plus or minus b
∓	a ∓ b	a minus or plus b	−(a ± b) = −a ∓ b

Functions[]

Example	Meaning and verbal equivalent	Remarks
${\displaystyle f:D\rightarrow C}$	function f has domain D and codomain C	Used to explicitly define the domain and codomain of a function.
${\displaystyle f\left(S\right)}$	${\displaystyle \left\{f\left(x\right)\mid x\in S\right\}}$	Set of all possible outputs in the codomain when given inputs from S, a subset of the domain of f.

Exponential and logarithmic functions[]

Example	Meaning and verbal equivalent	Remarks
e	base of natural logarithms	e = 2.718 28...
e${\displaystyle x}$	exponential function to the base e of ${\displaystyle x}$
log${\displaystyle a}$${\displaystyle x}$	logarithm to the base ${\displaystyle a}$ of ${\displaystyle x}$
lb ${\displaystyle x}$	binary logarithm (to the base 2) of ${\displaystyle x}$	lb ${\displaystyle x}$ = log2${\displaystyle x}$
ln ${\displaystyle x}$	natural logarithm (to the base e) of ${\displaystyle x}$	ln ${\displaystyle x}$ = loge${\displaystyle x}$
lg ${\displaystyle x}$	common logarithm (to the base 10) of ${\displaystyle x}$	lg ${\displaystyle x}$ = log10${\displaystyle x}$

Circular and hyperbolic functions[]

Example	Meaning and verbal equivalent	Remarks
π	ratio of the circumference of a circle to its diameter	π = 3.141 59...

Complex numbers[]

Example	Meaning and verbal equivalent	Remarks
i,   j	imaginary unit; i2 = −1	In electrotechnology, j is generally used.
Re z	real part of z	z = x + iy, where x = Re z and y = Im z
Im z	imaginary part of z
∣z∣	absolute value of z; modulus of z	mod z is also used
arg z	argument of z; phase of z	z = reiφ, where r = ∣z∣ and φ = arg z, i.e. Re z = r cos φ and Im z = r sin φ
z*	(complex) conjugate of z	sometimes a bar above z is used instead of z*
sgn z	signum z	sgn z = z / ∣z∣ = exp(i arg z) for z ≠ 0, sgn 0 = 0

Matrices[]

Example	Meaning and verbal equivalent	Remarks
A	matrix A	...

Coordinate systems[]

Coordinates	Position vector and its differential	Name of coordinate system	Remarks
x, y, z	${\displaystyle [xyz]=[xyz];}$ ${\displaystyle [dxdydz];}$	cartesian	x1, x2, x3 for the coordinates and e1, e2, e3 for the base vectors are also used. This notation easily generalizes to n-mensional space. ex, ey, ez form an orthonormal right-handed system. For the base vectors, i, j, k are also used.
ρ, φ, z	${\displaystyle [x,y,z]=[\rho \cos(\phi ),\rho \sin(\phi ),z]}$	cylindrical	eρ(φ), eφ(φ), ez form an orthonormal right-handed system. lf z= 0, then ρ and φ are the polar coordinates.
r, θ, φ	${\displaystyle [x,y,z]=r[\sin(\theta )\cos(\phi ),\sin(\theta )\sin(\phi ),\cos(\theta )]}$	spherical	er(θ,φ), eθ(θ,φ),eφ(φ) form an orthonormal right-handed system.

Vectors and tensors[]

Example	Meaning and verbal equivalent	Remarks
a ${\displaystyle {\vec {a}}}$	vector a	Instead of italic boldface, vectors can also be indicated by an arrow above the letter symbol. Any vector a can be multiplied by a scalar k, i.e. ka.

Special functions[]

Example	Meaning and verbal equivalent	Remarks
Jl(x)	cylindrical Bessel functions (of the first kind)	...

See also[]

• Mathematical symbols
• Mathematical notation

References and notes[]