ISO 31-11

Short description: Vector and tensor

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 It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-80000-2:2019.

Its definitions include the following:

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 ∈ [math]\displaystyle{ \mathbb{R} }[/math] | 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 can also be used.
∅		the empty set
[math]\displaystyle{ \mathbb{N} }[/math]		the set of natural numbers; the set of positive integers and zero	[math]\displaystyle{ \mathbb{N} }[/math] = {0, 1, 2, 3, ...} Exclusion of zero is denoted by an asterisk: [math]\displaystyle{ \mathbb{N} }[/math]* = {1, 2, 3, ...} [math]\displaystyle{ \mathbb{N} }[/math]k = {0, 1, 2, 3, ..., k − 1}
[math]\displaystyle{ \mathbb{Z} }[/math]		the set of integers	[math]\displaystyle{ \mathbb{Z} }[/math] = {..., −3, −2, −1, 0, 1, 2, 3, ...} [math]\displaystyle{ \mathbb{Z} }[/math]* = [math]\displaystyle{ \mathbb{Z} }[/math] ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...}
[math]\displaystyle{ \mathbb{Q} }[/math]		the set of rational numbers	[math]\displaystyle{ \mathbb{Q} }[/math]* = [math]\displaystyle{ \mathbb{Q} }[/math] ∖ {0}
[math]\displaystyle{ \mathbb{R} }[/math]		the set of real numbers	[math]\displaystyle{ \mathbb{R} }[/math]* = [math]\displaystyle{ \mathbb{R} }[/math] ∖ {0}
[math]\displaystyle{ \mathbb{C} }[/math]		the set of complex numbers	[math]\displaystyle{ \mathbb{C} }[/math]* = [math]\displaystyle{ \mathbb{C} }[/math] ∖ {0}
[,]	[a,b]	closed interval in [math]\displaystyle{ \mathbb{R} }[/math] from a (included) to b (included)	[a,b] = {x ∈ [math]\displaystyle{ \mathbb{R} }[/math] | a ≤ x ≤ b}
],] (,]	]a,b] (a,b]	left half-open interval in [math]\displaystyle{ \mathbb{R} }[/math] from a (excluded) to b (included)	]a,b] = {x ∈ [math]\displaystyle{ \mathbb{R} }[/math] | a < x ≤ b}
[,[ [,)	[a,b[ [a,b)	right half-open interval in [math]\displaystyle{ \mathbb{R} }[/math] from a (included) to b (excluded)	[a,b[ = {x ∈ [math]\displaystyle{ \mathbb{R} }[/math] | a ≤ x < b}
],[ (,)	]a,b[ (a,b)	open interval in [math]\displaystyle{ \mathbb{R} }[/math] from a (excluded) to b (excluded)	]a,b[ = {x ∈ [math]\displaystyle{ \mathbb{R} }[/math] | 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 }
⋃	[math]\displaystyle{ \bigcup_{i=1}^n A_i }[/math]	union of a collection of sets	[math]\displaystyle{ \bigcup_{i=1}^n A_i=A_1\cup A_2\cup\ldots\cup A_n }[/math], the set of elements belonging to at least one of the sets A1, ..., An. [math]\displaystyle{ \bigcup{}_{i=1}^n }[/math] and [math]\displaystyle{ \bigcup_{i\in I} }[/math], [math]\displaystyle{ \bigcup{}_{i \in I} }[/math] 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 }
⋂	[math]\displaystyle{ \bigcap_{i=1}^n A_i }[/math]	intersection of a collection of sets	[math]\displaystyle{ \bigcap_{i=1}^n A_i=A_1\cap A_2\cap\ldots\cap A_n }[/math], the set of elements belonging to all sets A1, ..., An. [math]\displaystyle{ \bigcap{}_{i=1}^n }[/math] and [math]\displaystyle{ \bigcap_{i\in I} }[/math], [math]\displaystyle{ \bigcap{}_{i \in I} }[/math] 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
≝	[math]\displaystyle{ \stackrel{\mathrm{def}}{=} }[/math]	a ≝ b	a is by definition equal to b	:= is also used
=	[math]\displaystyle{ = }[/math]	a = b	a equals b	≡ may be used to emphasize that a particular equality is an identity.
≠	[math]\displaystyle{ \ne }[/math]	a ≠ b	a is not equal to b	[math]\displaystyle{ a \not\equiv b }[/math] may be used to emphasize that a is not identically equal to b.
≙	[math]\displaystyle{ \stackrel{\wedge}{=} }[/math]	a ≙ b	a corresponds to b	On a 1:106 map: 1 cm ≙ 10 km.
≈	[math]\displaystyle{ \approx }[/math]	a ≈ b	a is approximately equal to b	The symbol ≃ is reserved for "is asymptotically equal to".
∼ ∝	[math]\displaystyle{ \begin{matrix} \sim \\ \propto \end{matrix} }[/math]	a ∼ b a ∝ b	a is proportional to b
<	[math]\displaystyle{ \lt }[/math]	a < b	a is less than b
>	[math]\displaystyle{ \gt }[/math]	a > b	a is greater than b
≤	[math]\displaystyle{ \le }[/math]	a ≤ b	a is less than or equal to b	The symbol ≦ is also used.
≥	[math]\displaystyle{ \ge }[/math]	a ≥ b	a is greater than or equal to b	The symbol ≧ is also used.
≪	[math]\displaystyle{ \ll }[/math]	a ≪ b	a is much less than b
≫	[math]\displaystyle{ \gg }[/math]	a ≫ b	a is much greater than b
∞	[math]\displaystyle{ \infty }[/math]		infinity
() [] {} ⟨⟩	[math]\displaystyle{ \begin{matrix}() \\ {[]} \\ \{\} \\ \langle \rangle \end{matrix} }[/math]	[math]\displaystyle{ \begin{matrix} {(a+b)c} \\ {[a+b]c} \\ {\{a+b\}c} \\ {\langle a+b \rangle c} \end{matrix} }[/math]	ac + bc, parentheses ac + bc, square brackets ac + bc, braces ac + bc, angle brackets	In ordinary algebra, the sequence of [math]\displaystyle{ (), [], \{\}, \langle \rangle }[/math] in order of nesting is not standardized. Special uses are made of [math]\displaystyle{ (), [], \{\}, \langle \rangle }[/math] in particular fields.
∥	[math]\displaystyle{ \| }[/math]	AB ∥ CD	the line AB is parallel to the line CD
⊥	[math]\displaystyle{ \perp }[/math]	AB ⊥ CD	the line AB is perpendicular to the line CD

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
f : D → C	function f has domain D and codomain C	Used to explicitly define the domain and codomain of a function.
f(S)	{ f(x) | x ∈ S }	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...
ex	exponential function to the base e of x
logax	logarithm to the base a of x
lb x	binary logarithm (to the base 2) of x	lb x = log2x
ln x	natural logarithm (to the base e) of x	ln x = logex
lg x	common logarithm (to the base 10) of x	lg x = log10x

Circular and hyperbolic functions

Example	Meaning and verbal equivalent	Remarks
π	ratio of the circumference of a circle to its diameter	π =~ 3.14159

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	[x y z]; [dx dy dz]	cartesian	x1, x2, x3 for the coordinates and e1, e2, e3 for the base vectors are also used. This notation easily generalizes to n-dimensional space. ex, ey, ez form an orthonormal right-handed system. For the base vectors, i, j, k are also used.
ρ, φ, z	[x, y, z] = [ρ cos(φ), ρ sin(φ), 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	[x, y, z] = r[sin(θ)cos(φ), sin(θ)sin(φ), cos(θ)]	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 [math]\displaystyle{ \vec a }[/math]	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