ISO 31-11

From Wikipedia, the free encyclopedia

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]

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 ≝ b	a is by definition equal to b^[1]	:= 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.^[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] 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. http://physics.nist.gov/cuu/pdf/sp811.pdf Guide for the Use of the International System of Units (SI).
^ 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: &x3008h; &x3009h;.
^ 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

[ISO_31-11-0] 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. http://physics.nist.gov/cuu/pdf/sp811.pdf Guide for the Use of the International System of Units (SI).

[1] 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: &x3008h; &x3009h;.

[2] 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)

[1]

[2]

[3]

ISO 31-11

From Wikipedia, the free encyclopedia

Contents

[edit] Mathematical logic

[edit] Sets

[edit] Miscellaneous signs and symbols

[edit] Operations

[edit] Functions

[edit] Exponential and logarithmic functions

[edit] Circular and hyperbolic functions

[edit] Complex numbers

[edit] Matrices

[edit] Coordinate systems

[edit] Vectors and tensors

[edit] Special functions

[edit] References and notes

[edit] See also

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