Semantic holism

ISO 31-11 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 It was superseded in 2009 by ISO 80000-2.

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 ∈ 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
ℕ		the set of natural numbers; the set of positive integers and zero	ℕ = {0, 1, 2, 3, ...} Exclusion of zero is denoted by an asterisk: ℕ* = {1, 2, 3, ...} ℕk = {0, 1, 2, 3, ..., k − 1}
ℤ		the set of integers	ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...} ℤ* = ℤ ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...}
ℚ		the set of rational numbers	ℚ* = ℚ ∖ {0}
ℝ		the set of real numbers	ℝ* = ℝ ∖ {0}
ℂ		the set of complex numbers	ℂ* = ℂ ∖ {0}
[, ]	[a, b]	closed interval in ℝ from a (included) to b (included)	[a, b] = {x ∈ ℝ ∣ a ≤ x ≤ b}
], ] (, ]	]a, b] (a, b]	left half-open interval in ℝ from a (excluded) to b (included)	]a, b] = {x ∈ ℝ ∣ a < x ≤ b}
[, [ [, )	[a, b[ [a, b)	right half-open interval in ℝ from a (included) to b (excluded)	[a, b[ = {x ∈ ℝ ∣ a ≤ x < b}
], [ (, )	]a, b[ (a, b)	open interval in ℝ from a (excluded) to b (excluded)	]a, b[ = {x ∈ 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 }
⋃	$\bigcup _{i=1}^{n}A_{i}$	union of a collection of sets	$\bigcup _{i=1}^{n}A_{i}=A_{1}\cup A_{2}\cup \ldots \cup A_{n}$, the set of elements belonging to at least one of the sets A1, …, An. $\bigcup {}_{i=1}^{n}$ and $\bigcup _{i\in I}$, $\bigcup {}_{i\in I}$ 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 }
⋂	$\bigcap _{i=1}^{n}A_{i}$	intersection of a collection of sets	$\bigcap _{i=1}^{n}A_{i}=A_{1}\cap A_{2}\cap \ldots \cap A_{n}$, the set of elements belonging to all sets A1, …, An. $\bigcap {}_{i=1}^{n}$ and $\bigcap _{i\in I}$, ⋂i∈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
≝ $\ {\stackrel {\mathrm {def} }{=}}\$	a ≝ b	a is by definition equal to b	:= is also used
=	a = b	a equals b	≡ may be used to emphasize that a particular equality is an identity.
≠	a ≠ b	a is not equal to b	$a\not \equiv b$ may be used to emphasize that a is not identically equal to b.
≙	a ≙ b	a corresponds to b	On a 1:106 map: 1 cm ≙ 10 km.
≈	a ≈ b	a is approximately equal to b	The symbol ≃ is reserved for "is asymptotically equal to".
∼ ∝	a ∼ b a ∝ b	a is proportional to b
<	a < b	a is less than b
>	a > b	a is greater than b
≤	a ≤ b	a is less than or equal to b	The symbol ≦ is also used.
≥	a ≥ b	a is greater than or equal to b	The symbol ≧ is also used.
≪	a ≪ b	a is much less than b
≫	a ≫ b	a is much greater than b
∞		infinity
( ) [ ] { } $\langle \rangle$	(a+b)c [a+b]c {a+b}c $\langle$a+b$\rangle$c	ac+bc, parentheses ac+bc, square brackets ac+bc, braces ac+bc, angle brackets	In ordinary algebra, the sequence of (), [], {}, $\langle \rangle$ in order of nesting is not standardized. Special uses are made of (), [], {}, $\langle \rangle$ in particular fields.
∥	AB ∥ CD	the line AB is parallel to the line CD
$\perp$	AB${}^{\perp }$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
$f:D\rightarrow C$	function f has domain D and codomain C	Used to explicitly define the domain and codomain of a function.
$f\left(S\right)$	$\left\{f\left(x\right)\mid x\in S\right\}$	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 (mathematical constant)|Template:Mvar]]	base of natural logarithms	Template:Mvar = 2.718 28...
Template:MvarTemplate:Mvar	exponential function to the base Template:Mvar of Template:Mvar
logTemplate:Mvar Template:Mvar	logarithm to the base Template:Mvar of Template:Mvar
lb Template:Mvar	binary logarithm (to the base 2) of Template:Mvar	lb Template:Mvar = log2 Template:Mvar
ln Template:Mvar	natural logarithm (to the base Template:Mvar) of Template:Mvar	ln Template:Mvar = logTemplate:Mvar Template:Mvar
lg Template:Mvar	common logarithm (to the base 10) of Template:Mvar	lg Template:Mvar = log10 Template:Mvar
...	...	...
⋮

Circular and hyperbolic functions

Example	Meaning and verbal equivalent	Remarks
Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.	ratio of the circumference of a circle to its diameter	Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park. = 3.141 59...
...	...	...
⋮

Complex numbers

Example	Meaning and verbal equivalent	Remarks
Template:Mvar   Template:Mvar	imaginary unit; Template:Mvar² = −1	In electrotechnology, Template:Mvar is generally used.
Re Template:Mvar	real part of Template:Mvar	Template:Mvar = Template:Mvar + Template:MvarTemplate:Mvar, where Template:Mvar = Re Template:Mvar and Template:Mvar = Im Template:Mvar
Im Template:Mvar	imaginary part of Template:Mvar
∣Template:Mvar∣	absolute value of Template:Mvar; modulus of Template:Mvar	mod Template:Mvar is also used
arg Template:Mvar	argument of Template:Mvar; phase of Template:Mvar	Template:Mvar = Template:MvarTemplate:MvarTemplate:MvarTemplate:Mvar, where Template:Mvar = ∣Template:Mvar∣ and Template:Mvar = arg Template:Mvar, i.e. Re Template:Mvar = Template:Mvar cos Template:Mvar and Im Template:Mvar = Template:Mvar sin Template:Mvar
Template:Mvar*	(complex) conjugate of Template:Mvar	sometimes a bar above Template:Mvar is used instead of Template:Mvar*
sgn Template:Mvar	signum Template:Mvar	sgn Template:Mvar = Template:Mvar / ∣Template:Mvar∣ = exp(Template:Mvar arg Template:Mvar) for Template:Mvar ≠ 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] = [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-mensional space. ex, ey, ex 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 ${\vec {a}}$	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
Template:MvarTemplate:Mvar(Template:Mvar)	cylindrical Bessel functions (of the first kind)	...
...	...	...
⋮

See also

• Mathematical symbols
• Mathematical notation

References and notes

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