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ISO 31-11

## Summary

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 It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-80000-2:2019.[1]

Its definitions include the following:[2]

## 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 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 belong 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 {xiiI}, 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${\displaystyle \mathbb {R} }$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
AB difference between A and B; A minus B The set of elements which belong to A but not to B.
AB = { xxAxB }
AB should not be used.
the empty set
${\displaystyle \mathbb {N} }$ the set of natural numbers; the set of positive integers and zero ${\displaystyle \mathbb {N} }$ = {0, 1, 2, 3, ...}
Exclusion of zero is denoted by an asterisk:
${\displaystyle \mathbb {N} }$* = {1, 2, 3, ...}
${\displaystyle \mathbb {N} }$k = {0, 1, 2, 3, ..., k − 1}
${\displaystyle \mathbb {Z} }$ the set of integers ${\displaystyle \mathbb {Z} }$ = {..., −3, −2, −1, 0, 1, 2, 3, ...}

${\displaystyle \mathbb {Z} }$* = ${\displaystyle \mathbb {Z} }$ ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...}

${\displaystyle \mathbb {Q} }$ the set of rational numbers ${\displaystyle \mathbb {Q} }$* = ${\displaystyle \mathbb {Q} }$ ∖ {0}
${\displaystyle \mathbb {R} }$ the set of real numbers ${\displaystyle \mathbb {R} }$* = ${\displaystyle \mathbb {R} }$ ∖ {0}
${\displaystyle \mathbb {C} }$ the set of complex numbers ${\displaystyle \mathbb {C} }$* = ${\displaystyle \mathbb {C} }$ ∖ {0}
[,] [a,b] closed interval in ${\displaystyle \mathbb {R} }$ from a (included) to b (included) [a,b] = {x${\displaystyle \mathbb {R} }$axb}
],]
(,]
]a,b]
(a,b]
left half-open interval in ${\displaystyle \mathbb {R} }$ from a (excluded) to b (included) ]a,b] = {x${\displaystyle \mathbb {R} }$a < xb}
[,[
[,)
[a,b[
[a,b)
right half-open interval in ${\displaystyle \mathbb {R} }$ from a (included) to b (excluded) [a,b[ = {x${\displaystyle \mathbb {R} }$ax < b}
],[
(,)
]a,b[
(a,b)
open interval in ${\displaystyle \mathbb {R} }$ from a (excluded) to b (excluded) ]a,b[ = {x${\displaystyle \mathbb {R} }$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 }
${\displaystyle \bigcup _{i=1}^{n}A_{i}}$ union of a collection of sets ${\displaystyle \bigcup _{i=1}^{n}A_{i}=A_{1}\cup A_{2}\cup \ldots \cup A_{n}}$, the set of elements belonging to at least one of the sets A1, ..., An. ${\displaystyle \bigcup {}_{i=1}^{n}}$ and ${\displaystyle \bigcup _{i\in I}}$, ${\displaystyle \bigcup {}_{i\in I}}$ 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 }
${\displaystyle \bigcap _{i=1}^{n}A_{i}}$ intersection of a collection of sets ${\displaystyle \bigcap _{i=1}^{n}A_{i}=A_{1}\cap A_{2}\cap \ldots \cap A_{n}}$, the set of elements belonging to all sets A1, ..., An. ${\displaystyle \bigcap {}_{i=1}^{n}}$ and ${\displaystyle \bigcap _{i\in I}}$, ${\displaystyle \bigcap {}_{i\in I}}$ 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 = 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.
(,...,) (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
HTML TeX
${\displaystyle {\stackrel {\mathrm {def} }{=}}}$ ab a is by definition equal to b [2] := is also used
= ${\displaystyle =}$ a = b a equals b ≡ may be used to emphasize that a particular equality is an identity.
${\displaystyle \neq }$ ab a is not equal to b ${\displaystyle a\not \equiv b}$ may be used to emphasize that a is not identically equal to b.
${\displaystyle {\stackrel {\wedge }{=}}}$ ab a corresponds to b On a 1:106 map: 1 cm ≙ 10 km.
${\displaystyle \approx }$ ab a is approximately equal to b The symbol ≃ is reserved for "is asymptotically equal to".

${\displaystyle {\begin{matrix}\sim \\\propto \end{matrix}}}$ ab
ab
a is proportional to b
< ${\displaystyle <}$ a < b a is less than b
> ${\displaystyle >}$ a > b a is greater than b
${\displaystyle \leq }$ ab a is less than or equal to b The symbol ≦ is also used.
${\displaystyle \geq }$ ab a is greater than or equal to b The symbol ≧ is also used.
${\displaystyle \ll }$ ab a is much less than b
${\displaystyle \gg }$ ab a is much greater than b
${\displaystyle \infty }$ infinity
()
[]
{}
⟨⟩
${\displaystyle {\begin{matrix}()\\{[]}\\\{\}\\\langle \rangle \end{matrix}}}$ ${\displaystyle {\begin{matrix}{(a+b)c}\\{[a+b]c}\\{\{a+b\}c}\\{\langle a+b\rangle c}\end{matrix}}}$ ${\displaystyle ac+bc}$, parentheses
${\displaystyle ac+bc}$, square brackets
${\displaystyle ac+bc}$, braces
${\displaystyle ac+bc}$, angle brackets
In ordinary algebra, the sequence of ${\displaystyle (),[],\{\},\langle \rangle }$ in order of nesting is not standardized. Special uses are made of ${\displaystyle (),[],\{\},\langle \rangle }$ in particular fields.
${\displaystyle \|}$ AB ∥ CD the line AB is parallel to the line CD
${\displaystyle \perp }$ ${\displaystyle \mathrm {AB\perp CD} }$ the line AB is perpendicular to the line CD[3]

## Operations

Sign Example Meaning and verbal equivalent Remarks
+ a + b a plus b
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
${\displaystyle f:D\rightarrow C}$ function f has domain D and codomain C Used to explicitly define the domain and codomain of a function.
${\displaystyle f\left(S\right)}$ ${\displaystyle \left\{f\left(x\right)\mid x\in S\right\}}$ 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${\displaystyle x}$ exponential function to the base e of ${\displaystyle x}$
log${\displaystyle a}$${\displaystyle x}$ logarithm to the base ${\displaystyle a}$ of ${\displaystyle x}$
lb ${\displaystyle x}$ binary logarithm (to the base 2) of ${\displaystyle x}$ lb ${\displaystyle x}$ = log2${\displaystyle x}$
ln ${\displaystyle x}$ natural logarithm (to the base e) of ${\displaystyle x}$ ln ${\displaystyle x}$ = loge${\displaystyle x}$
lg ${\displaystyle x}$ common logarithm (to the base 10) of ${\displaystyle x}$ lg ${\displaystyle x}$ = log10${\displaystyle x}$

## 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 + 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 ${\displaystyle [xyz]=[xyz];}$ ${\displaystyle [dxdydz];}$ 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. ex, ey, ez form an orthonormal right-handed system. For the base vectors, i, j, k are also used.
ρ, φ, z ${\displaystyle [x,y,z]=[\rho \cos(\phi ),\rho \sin(\phi ),z]}$ 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 ${\displaystyle [x,y,z]=r[\sin(\theta )\cos(\phi ),\sin(\theta )\sin(\phi ),\cos(\theta )]}$ spherical er(θ,φ), eθ(θ,φ),eφ(φ) form an orthonormal right-handed system.

## Vectors and tensors

Example Meaning and verbal equivalent Remarks
a
${\displaystyle {\vec {a}}}$
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
Jl(x) cylindrical Bessel functions (of the first kind) ...

## References and notes

1. ^ "ISO 80000-2:2019". International Organization for Standardization. Retrieved 4 Oct 2021.
2. ^ a b 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)