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 It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-80000-2:2019.

Its definitions include the following:

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 ∈ Ax belongs to A; x is an element of the set A
∉ x ∉ Ax does not belong to A; x is not an element of the set AThe negation stroke can also be vertical.
∋ A ∋ xthe set A contains x (as an element)same meaning as x ∈ A
∌ A ∌ xthe set A does not contain x (as an element)same meaning as x ∉ A
{ }{x1, x2, ..., xn}set with elements x1, x2, ..., xnalso {xi | 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 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
∖ A ∖ B difference between A and B; A minus BThe set of elements which belong to A but not to B.
A ∖ B = { x | x ∈ A ∧ x ∉ B }
A − B can also be used.
∅the empty set
ℕthe set of natural numbers; the set of positive integers and zeroℕ = {0, 1, 2, 3, ...}
Exclusion of zero is denoted by an asterisk:
ℕ* = {1, 2, 3, ...}
ℕk = {0, 1, 2, 3, ..., k − 1}
ℤthe set of integersℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...}
ℤ* = ℤ ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...}
ℚthe set of rational numbersℚ* = ℚ ∖ {0}
ℝthe set of real numbersℝ* = ℝ ∖ {0}
ℂthe set of complex numbersℂ* = ℂ ∖ {0}
closed interval in ℝ from a (included) to b (included) = {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,ba,b := is also used
= = {\displaystyle =} a = b a equals b ≡ may be used to emphasize that a particular equality is an identity.
≠ ≠ {\displaystyle \neq } a ≠ b a is not equal to b a ≢ b {\displaystyle a\not \equiv b} may be used to emphasize that a is not identically equal to b.
≙ = ∧ {\displaystyle {\stackrel {\wedge }{=}}} a   = ∧   b {\displaystyle a\ {\stackrel {\wedge }{=}}\ b} a corresponds to b On a 1:106 map: 1  cm  = ∧   10  km {\displaystyle 1{\text{ cm }}{\stackrel {\wedge }{=}}\ 10{\text{ km}}} .
≈ ≈ {\displaystyle \approx } a ≈ b a is approximately equal to b The symbol ≃ is reserved for "is asymptotically equal to".
∼
∝		 ∼ ∝ {\displaystyle {\begin{matrix}\sim \\\propto \end{matrix}}} a ∼ b
a ∝ b		 a is proportional to b
< < {\displaystyle <} a < b a is less than b
> > {\displaystyle >} a > b a is greater than b
≤ ≤ {\displaystyle \leq } a ≤ b a is less than or equal to b The symbol ≦ is also used.
≥ ≥ {\displaystyle \geq } a ≥ b a is greater than or equal to b The symbol ≧ is also used.
≪ ≪ {\displaystyle \ll } a ≪ b a is much less than b
≫ ≫ {\displaystyle \gg } a ≫ b a is much greater than b
∞ ∞ {\displaystyle \infty } infinity
()

{}
⟨⟩		 ( ) { } ⟨ ⟩ {\displaystyle {\begin{matrix}()\\{}\\\{\}\\\langle \rangle \end{matrix}}} ( a + b ) c c { a + b } c ⟨ a + b ⟩ c {\displaystyle {\begin{matrix}{(a+b)c}\\{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 ( ) , , { } , ⟨ ⟩ {\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 } AB ⊥ CD the line AB is perpendicular to the line CD

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 → Cfunction f has domain D and codomain C Used to explicitly define the domain and codomain of a function.
f(S){ f(x) | x ∈ S }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...
ex exponential function to the base e of x
logax 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 = logex
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 ≈ 3.14159

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 x1, x2, x3 for the coordinates and e1, e2, e3 for the base vectors are also used. This notation easily generalizes to n-dimensional space. ex, ey, ez form an orthonormal right-handed system. For the base vectors, i, j, k are also used.
ρ, 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 = r 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
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) ...

See also

References and notes

  1. ^ "ISO 80000-2:2019". International Organization for Standardization. 19 May 2020. 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)
ISO standards by standard number
List of ISO standards – ISO romanizations – IEC standards
1–9999
10000–19999
20000–29999
30000+
Common mathematical notation, symbols, and formulas
Lists of Unicode and LaTeX mathematical symbols
Lists of Unicode symbols
General
Alphanumeric
Arrows and Geometric Shapes
Operators
Supplemental Math Operators
Miscellaneous
  • A
  • B
  • Technical
  • ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology)
Typographical conventions and notations
Language
Letters
Notation
Meanings of symbols