# ISO 31-11

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 It was superseded in 2009 by ISO 80000-2.[1]

Its definitions include the following:[2]

## Mathematical logic

Sign Example Name Meaning and verbal equivalent Remarks
pqconjunction signp and q
pqdisjunction signp or q (or both)
¬¬ pnegation signnegation of p; not p; non p
pqimplication signif p then q; p implies qCan also be written as qp. Sometimes → is used.
xA p(x)
(∀xA) p(x)
universal quantifierfor every x belonging to A, the proposition p(x) is trueThe "∈A" can be dropped where A is clear from context.
xA p(x)
(∃xA) p(x)
existential quantifierthere exists an x belonging to A for which the proposition p(x) is trueThe "∈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
xAx belongs to A; x is an element of the set A
xAx does not belong to A; x is not an element of the set AThe negation stroke can also be vertical.
Axthe set A contains x (as an element)same meaning as xA
Axthe set A does not contain x (as an element)same meaning as xA
{ }{x1, x2, ..., xn}set with elements x1, x2, ..., xnalso {xiiI}, where I denotes a set of indices
{ ∣ }{xAp(x)}set of those elements of A for which the proposition p(x) is trueExample: {x ∈ ℝ ∣ x > 5}
The ∈A can be dropped where this set is clear from the context.
cardcard(A)number of elements in A; cardinal of A
AB difference between A and B; A minus BThe set of elements which belong to A but not to B.
AB = { xxAxB }
AB should not be used.
the empty set
the set of natural numbers; the set of positive integers and zeroℕ = {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 integersℤ = {..., −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}
BAB is included in A; B is a subset of AEvery element of B belongs to A. ⊂ is also used.
BAB 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".
CAC is not included in A; C is not a subset of A⊄ is also used.
ABA 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".
ACA does not include C (as subset)⊅ is also used. AC means the same as CA.
ABunion of A and BThe set of elements which belong to A or to B or to both A and B.
AB = { xxAxB }
union of a collection of sets, the set of elements belonging to at least one of the sets A1, …, An. and , are also used, where I denotes a set of indices.
ABintersection of A and BThe set of elements which belong to both A and B.
AB = { xxAxB }
intersection of a collection of sets, the set of elements belonging to all sets A1, …, An. and , are also used, where I denotes a set of indices.
ABcomplement of subset B of AThe 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 = AB.
(,)(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-tuplea1, a2, …, an⟩ is also used.
×A × Bcartesian product of A and BThe 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.
ΔΔAset of pairs (a, a) ∈ A × A where aA; diagonal of the set A × AΔA = { (a, a) ∣ aA }
idA is also used.

## Miscellaneous signs and symbols

Sign Example Meaning and verbal equivalent Remarks

aba 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.
aba is not equal to b may be used to emphasize that a is not identically equal to b.
aba corresponds to bOn a 1:106 map: 1 cm ≙ 10 km.
aba is approximately equal to bThe symbol ≃ is reserved for "is asymptotically equal to".

ab
ab
a is proportional to b
<a < ba is less than b
>a > ba is greater than b
aba is less than or equal to bThe symbol ≦ is also used.
aba is greater than or equal to bThe symbol ≧ is also used.
aba is much less than b
aba is much greater than b
infinity
()
[]
{}
(a+b)c
[a+b]c
{a+b}c
a+bc
ac+bc, parentheses
ac+bc, square brackets
ac+bc, braces
ac+bc, angle brackets
In ordinary algebra, the sequence of (), [], {}, in order of nesting is not standardized. Special uses are made of (), [], {}, in particular fields.[3]
AB ∥ CDthe line AB is parallel to the line CD
ABCDthe line AB is perpendicular to the line CD[4]

## Operations

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

## Functions

Example Meaning and verbal equivalent Remarks
function f has domain D and codomain C Used to explicitly define the domain and codomain of a function.
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
ebase of natural logarithmse = 2.718 28...
exexponential function to the base e of x
logaxlogarithm to the base a of x
lb xbinary logarithm (to the base 2) of xlb x = log2x
ln xnatural logarithm (to the base e) of xln x = logex
lg xcommon logarithm (to the base 10) of xlg x = log10x
.........

## Circular and hyperbolic functions

Example Meaning and verbal equivalent Remarks
πratio of the circumference of a circle to its diameterπ = 3.141 59...
.........

## Complex numbers

Example Meaning and verbal equivalent Remarks
i   jimaginary unit; i² = −1In electrotechnology, j is generally used.
Re zreal part of z z = x + iy, where x = Re z and y = Im z
Im zimaginary part of z
zabsolute value of z; modulus of zmod z is also used
arg zargument of z; phase of zz = 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 zsometimes a bar above z is used instead of z*
sgn zsignum zsgn z = z / ∣z∣ = exp(i arg z) for z ≠ 0, sgn 0 = 0

## Matrices

Example Meaning and verbal equivalent Remarks
Amatrix 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, ez 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)] sphericaler(θ,φ), eθ(θ,φ),eφ(φ) form an orthonormal right-handed system.

## Vectors and tensors

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