ISO 31-11

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ISO 31-11 is the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology.

Its definitions include:^[1]

1 Mathematical logic
2 Sets
3 Miscellaneous signs and symbols
4 Operations
5 Functions
6 Exponential and logarithmic functions
7 Circular and hyperbolic functions
8 Complex numbers
9 Matrices
10 Coordinate systems
11 Vectors and tensors
12 Special functions
13 Notes on display anomalies
14 References and notes
15 See also

[edit] 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 only exactly one x exists for which p(x) is true.

[edit] 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 belongs 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, ..., x_n}	set with elements x₁, x₂, ..., x_n	also {x_i : 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
∅		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, ...} ℕ_k = {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, …, A_n. $\bigcup{}_{i=1}^n$ and $\bigcup_{i\in I}$ , ⋃_i∈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, …, A_n. $\bigcap{}_{i=1}^n$ and $\bigcap_{i\in I}$ , ⋂_i∈I are also used, where I denotes a set of indices.
\	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.
∁	∁_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, …, a_n)	ordered n-tuple	⟨a₁, a₂, …, a_n⟩ 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	Δ_A	set of pairs (a, a) ∈ A × A where a ∈ A; diagonal of the set A × A	Δ_A = { (a, a) ∣ a ∈ A } id_A is also used.

[edit] Miscellaneous signs and symbols

Sign	Example	Meaning and verbal equivalent	Remarks
$\ \stackrel{\mathrm{def}}{=}\$	a $\ \overset{\underset{\mathrm{def}}{}}{=}\$ b	a is defined as b	$\ \overset{\underset{\mathrm{def}}{}}{=}\$ is used when a is by definition equal to b ^[1]
=	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 is by definition equal to b	:= is also used
≙	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.^[2]
∥	AB ∥ CD	the line AB is parallel to the line CD
$\perp$	AB ${}^\perp$ CD	the line AB is perpendicular to the line CD^[3]

[edit] 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
...	...	...	...
⋮

[edit] Functions

Example	Meaning and verbal equivalent	Remarks
f	function f	...
...	...	...
⋮

[edit] Exponential and logarithmic functions

Example	Meaning and verbal equivalent	Remarks
a^x	exponential function to the base a of x	...
e	base of natural logarithms	e = 2.718 281 8...
...	...	...
⋮

[edit] Circular and hyperbolic functions

Example	Meaning and verbal equivalent	Remarks
π	ratio of the circumference of a circle to its diameter	π = 3.141 592 6...
...	...	...
⋮

[edit] 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 = x + iy, where x = Re z and y = Im z
∣z∣	absolute value of z; modulus of z	mod z is also used
arg z	argument of z; phase of z	z = re^iφ, 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

[edit] Matrices

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

[edit] Coordinate systems

Coordinates	Position vector and its differential	Name of coordinate system	Remarks
x, y, z	...	cartesian coordinates	...
ρ, φ, z	...	cylindrical coordinates	...
r, θ, φ	...	spherical coordinates	...

[edit] Vectors and tensors

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

[edit] Special functions

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

[edit] Notes on display anomalies

Please understand Unicode is a relatively new and developing standard. If you see a ? (in Firefox) or a character does not appear as you might expect, your system may not have a font adequate to display this character. Some systems are not equipped to properly display every Unicode character: typically due to inadequate font support. Please contact your operating system or web browser vendor to request proper support for these characters. For accuracy, the article includes the proper unicode character in the text, but where glyphs may be missing on common installations, these close substitutes are provided. it is aways better to use the correct unicode character whenever possible, but failing that, substitutes may be used.

[edit] References and notes

^ ^a ^b Barry M. Taylor. Guide for the Use of the International System of Units (SI) ;Standardized mathematical signs; §10.1.2 p. 33: an abbreviated list of symbols from ISO 31-11. NIST.
^ These brace or fence characters are upper level unicode characters, fairly recently established and so may not display correctly in every browser. A close approximation of the appearance is found in the standard latin characters: ( ), [ ], { }, < >. A more accurate glyph depiction of the mathematical angle bracket characters are found in the Chinese-Japanese-Korean (CJK) punctuation category: 〈〉.
^ If the perpendicular symbol, ⟂, does not display correctly, it is similar to ⊥ (up tack: sometimes meaning orthogonal to) and it also appears similar to ⏊ (the dentistry: symbol light up and horizontal)

[edit] See also

These articles make substantial use of the Unicode repertoire of mathematical characters. The exact representation of these symbols may vary depending on the font available.

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