ISO 31-11

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 It was superseded in 2009 by ISO 80000-2.[1]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, ..., 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
∖	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, ...} 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 $A 1, ..., A n$ . $\bigcup {}_{i=1}^{n}$ 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 $A 1, ..., A n$ . $\bigcap {}_{i=1}^{n}$ and $\bigcap _{i\in I}$ , $\bigcap {}_{i\in I}$ 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.
(,...,)	$(a 1, a 2, ..., a n)$	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.

Miscellaneous signs and symbols

Sign		Example	Meaning and verbal equivalent	Remarks
HTML	TeX	Example	Meaning and verbal equivalent	Remarks
≝	${\stackrel {\mathrm {def} }{=}}$	a ≝ b	a is by definition equal to b [2]	:= is also used
=	$=$	a = b	a equals b	≡ may be used to emphasize that a particular equality is an identity.
≠	$\neq$	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.
≙	${\stackrel {\wedge }{=}}$	a ≙ b	a corresponds to b	On a 1:10⁶ map: 1 cm ≙ 10 km.
≈	$\approx$	a ≈ b	a is approximately equal to b	The symbol ≃ is reserved for "is asymptotically equal to".
∼ ∝	${\begin{matrix}\sim \\\propto \end{matrix}}$	a ∼ b a ∝ b	a is proportional to b
<	$<$	a < b	a is less than b
>	$>$	a > b	a is greater than b
≤	$\leq$	a ≤ b	a is less than or equal to b	The symbol ≦ is also used.
≥	$\geq$	a ≥ b	a is greater than or equal to b	The symbol ≧ is also used.
≪	$\ll$	a ≪ b	a is much less than b
≫	$\gg$	a ≫ b	a is much greater than b
∞	$\infty$		infinity
() [] {} ⟨⟩	${\begin{matrix}()\\{[]}\\\{\}\\\langle \rangle \end{matrix}}$	${\begin{matrix}{(a+b)c}\\{[a+b]c}\\{\{a+b\}c}\\{\langle a+b\rangle c}\end{matrix}}$	$ac+bc$ , 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
⊥	$\perp$	$\mathrm {AB\perp CD}$	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
$f:D\rightarrow C$	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	base of natural logarithms	e = 2.718 28...
e^x	exponential function to the base e of x
log_ax	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 = log_ex
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 = 3.141 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 + i y, where x = Re z and y = Im z
Im z	imaginary part of z	z = x + i y, 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^i φ, 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	$[xyz]=[xyz];$ $[dxdydz];$	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. e_x, e_y, e_z form an orthonormal right-handed system. For the base vectors, i, j, k are also used.
ρ, φ, z	$[x,y,z]=[\rho \cos(\phi ),\rho \sin(\phi ),z]$	cylindrical	e_ρ(φ), e_φ(φ), e_z form an orthonormal right-handed system. lf z= 0, then ρ and φ are the polar coordinates.
r, θ, φ	$[x,y,z]=r[\sin(\theta )\cos(\phi ),\sin(\theta )\sin(\phi ),\cos(\theta )]$	spherical	e_r(θ,φ), e_θ(θ,φ),e_φ(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
$J l (x)$	cylindrical Bessel functions (of the first kind)	...

References and notes

"ISO 80000-2:2009". International Organization for Standardization. Retrieved 1 July 2010.
Thompson, Ambler; Taylor, Barry M (March 2008). Guide for the Use of the International System of Units (SI) — NIST Special Publication 811, 2008 Edition — Second Printing (PDF). Gaithersburg, MD, USA: NIST.
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)

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] "ISO 80000-2:2009". International Organization for Standardization. Retrieved 1 July 2010.

[ISO_31-11-2] Thompson, Ambler; Taylor, Barry M (March 2008). Guide for the Use of the International System of Units (SI) — NIST Special Publication 811, 2008 Edition — Second Printing (PDF). Gaithersburg, MD, USA: NIST.

[3] 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)

ISO standards by standard number
List of ISO standards / ISO romanizations / IEC standards
1–9999	1 2 3 4 5 6 7 9 16 17 31 -0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 68-1 128 216 217 226 228 233 259 261 262 269 302 306 361 428 500 518 519 639 -1 -2 -3 -5 -6 646 657 668 690 704 732 764 838 843 860 898 965 999 1000 1004 1007 1073-1 1073-2 1155 1413 1538 1629 1745 1989 2014 2015 2022 2033 2047 2108 2145 2146 2240 2281 2533 2709 2711 2720 2788 2848 2852 3029 3103 3166 -1 -2 -3 3297 3307 3601 3602 3864 3901 3950 3977 4031 4157 4165 4217 4909 5218 5426 5427 5428 5725 5775 5776 5800 5807 5964 6166 6344 6346 6385 6425 6429 6438 6523 6709 6943 7001 7002 7010 7027 7064 7098 7185 7200 7498 -1 7637 7736 7810 7811 7812 7813 7816 7942 8000 8093 8178 8217 8373 8501-1 8571 8583 8601 8613 8632 8651 8652 8691 8805/8806 8807 8820-5 8859 -1 -2 -3 -4 -5 -6 -7 -8 -8-I -9 -10 -11 -12 -13 -14 -15 -16 8879 9000/9001 9036 9075 9126 9141 9227 9241 9293 9314 9362 9407 9496 9506 9529 9564 9592/9593 9594 9660 9797-1 9897 9899 9945 9984 9985 9995
10000–19999	10005 10006 10007 10116 10118-3 10160 10161 10165 10179 10206 10218 10303 -11 -21 -22 -28 -238 10383 10487 10585 10589 10646 10664 10746 10861 10957 10962 10967 11073 11170 11179 11404 11544 11783 11784 11785 11801 11889 11898 11940 (-2) 11941 11941 (TR) 11992 12006 12182 12207 12234-2 12620 13211 -1 -2 13216 13250 13399 13406-2 13450 13485 13490 13567 13568 13584 13616 13816 14000 14031 14224 14289 14396 14443 14496 -2 -3 -6 -10 -11 -12 -14 -17 -20 14644 14649 14651 14698 14750 14764 14882 14971 15022 15189 15288 15291 15292 15398 15408 15444 -3 15445 15438 15504 15511 15686 15693 15706 -2 15707 15897 15919 15924 15926 15926 WIP 15930 16023 16262 16355-1 16612-2 16750 16949 (TS) 17024 17025 17100 17203 17369 17442 17799 18000 18004 18014 18245 18629 18916 19005 19011 19092 -1 -2 19114 19115 19125 19136 19407 19439 19500 19501 19502 19503 19505 19506 19507 19508 19509 19510 19600 19752 19757 19770 19775-1 19794-5 19831
20000+	20000 20022 20121 20400 21000 21047 21500 21827 22000 22300 22395 23270 23271 23360 24517 24613 24617 24707 25178 25964 26000 26262 26300 26324 27000 series 27000 27001 27002 27005 27006 27729 28000 29110 29148 29199-2 29500 30170 31000 32000 37001 38500 40500 42010 45001 50001 55000 80000 -1
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