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]

Mathematical logic

Sign Example Name Meaning and verbal equivalent Remarks
pq conjunction sign p and q
pq disjunction sign p or q (or both)
¬ ¬ p negation sign negation of p; not p; non p
pq implication sign if p then q; p implies q Can also be written as qp. Sometimes → is used.
xA p(x)
(∀xA) 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.
xA p(x)
(∃xA) 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.

Sets

Sign Example Meaning and verbal equivalent Remarks
xA x belongs to A; x is an element of the set A
xA x does not belongs to A; x is not an element of the set A The negation stroke can also be vertical.
Ax the set A contains x (as an element) same meaning as xA
Ax the set A does not contain x (as an element) same meaning as xA
{ } {x1, x2, ..., xn} set with elements x1, x2, ..., xn also {xi : iI}, where I denotes a set of indices
{ ∣ } {xAp(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
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:
* = {1, 2, 3, ...}
k = {0, 1, 2, 3, ..., k − 1}
the set of integers ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...}

* = ℤ \ {0} = {..., −3, −2, −1, 1, 2, 3, ...}

the set of rational numbers * = ℚ \ {0}
the set of real numbers * = ℝ \ {0}
the set of complex numbers * = ℂ \ {0}
[,] [a,b] closed interval in ℝ from a (included) to b (included) [a,b] = {x ∈ ℝ ∣ axb}
],]
(,]
]a,b]
(a,b]
left half-open interval in ℝ from a (excluded) to b (included) ]a,b] = {x ∈ ℝ ∣ a < xb}
[,[
[,)
[a,b[
[a,b)
right half-open interval in ℝ from a (included) to b (excluded) [a,b[ = {x ∈ ℝ ∣ ax < b}
],[
(,)
]a,b[
(a,b)
open interval in ℝ from a (excluded) to b (excluded) ]a,b[ = {x ∈ ℝ ∣ a < x < b}
BA B is included in A; B is a subset of A Every element of B belongs to A. ⊂ is also used.
BA 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".
CA C is not included in A; C is not a subset of A ⊄ is also used.
AB A includes B (as subset) A contains every element of B. ⊃ is also used. BA means the same as AB.
AB. 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".
AC A does not include C (as subset) ⊅ is also used. AC means the same as CA.
AB union of A and B The set of elements which belong to A or to B or to both A and B.
AB = { xxAxB }
$\bigcup_{i=1}^n A_i$ 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 $\bigcup_{i\in I}$, ⋃iI are also used, where I denotes a set of indices.
AB intersection of A and B The set of elements which belong to both A and B.
AB = { xxAxB }
$\bigcap_{i=1}^n A_i$ 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 $\bigcap_{i\in I}$, ⋂iI 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 = { xxAxB }
AB 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.
(,…,) (a1a2, …, 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 aA and bB.
A × B = { (a, b) ∣ aAbB }
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 aA; diagonal of the set A × A ΔA = { (a, a) ∣ aA }
idA is also used.

Miscellaneous signs and symbols

Sign Example Meaning and verbal equivalent Remarks

$\ \stackrel{\mathrm{def}}{=}\$
ab 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.
ab a is not equal to b $a \not\equiv b$ may be used to emphasize that a is not identically equal to b.
ab a corresponds to b On a 1:106 map: 1 cm ≙ 10 km.
ab a is approximately equal to b The symbol ≃ is reserved for "is asymptotically equal to".

ab
ab
a is proportional to b
< a < b a is less than b
> a > b a is greater than b
ab a is less than or equal to b The symbol ≦ is also used.
ab a is greater than or equal to b The symbol ≧ is also used.
ab a is much less than b
ab 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]

Operations

Sign Example Meaning and verbal equivalent Remarks
+ a + b a plus b
ab a minus b
± a ± b a plus or minus b
ab a minus or plus b −(a ± b) = −ab
... ... ... ...

Functions

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

Exponential and logarithmic functions

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

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...
... ... ...

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 ... cartesian coordinates ...
ρ, φ, z ... cylindrical coordinates ...
r, θ, φ ... spherical coordinates ...

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.
... ... ...

Special functions

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

References and notes

1. ^ 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).
2. ^ 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;.
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)