# 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 It was superseded in 2009 by ISO 80000-2.^{[1]}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 |
---|---|---|---|---|

∧ | 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 ∈ A | x belongs to A; x is an element of the set A | |

∉ | x ∉ A | x does not belong 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 |

{ } | {x_{1}, x_{2}, ..., x_{n}} | set with elements x_{1}, x_{2}, ..., 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 | |

∖ | 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. |

∅ | 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 | |

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 } |

⋃ | union of a collection of sets | , the set of elements belonging to at least one of the sets A_{1}, …, A_{n}. and , 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 } |

⋂ | intersection of a collection of sets | , the set of elements belonging to all sets A_{1}, …, A_{n}. and , are also used, where I denotes a set of indices. | |

∁ | ∁_{A}B | 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 ∁_{A}B = 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. |

(,…,) | (a_{1}, a_{2}, …, a_{n}) | ordered n-tuple | ⟨a_{1}, a_{2}, …, 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 A^{n}, 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. |

## Miscellaneous signs and symbols

Sign | Example | Meaning and verbal equivalent | Remarks |
---|---|---|---|

≝ | a ≝ b | a 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. |

≠ | a ≠ b | a is not equal to b | may be used to emphasize that a is not identically equal to b. |

≙ | a ≙ b | a corresponds to b | On a 1:10^{6} map: 1 cm ≙ 10 km. |

≈ | a ≈ b | a is approximately equal to b | The symbol ≃ is reserved for "is asymptotically equal to". |

∼ ∝ | a ∼ ba ∝ 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 | ||

() [] {} | (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 ∥ CD | the line AB is parallel to the line CD | |

ABCD | the line AB is perpendicular to the line CD^{[4]} |

## 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 |
---|---|---|

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 |
---|---|---|

e | base of natural logarithms | e = 2.718 28... |

e^{x} | exponential function to the base e of x | |

log_{a}x | logarithm to the base a of x | |

lb x | binary logarithm (to the base 2) of x | lb x = log_{2}x |

ln x | natural logarithm (to the base e) of x | ln x = log_{e}x |

lg x | common logarithm (to the base 10) of x | lg x = log_{10}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 = 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 |

## 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 |
[x y z] = [x y z]; [dx dy dz]; | cartesian | x_{1}, x_{2}, x_{3} for the coordinates and e_{1}, e_{2}, e_{3} for the base vectors are also used. This notation easily generalizes to n-mensional space. e_{x}, e_{y}, e_{z} form an orthonormal right-handed system. For the base vectors, , i, j are also used.k |

ρ, φ, z |
[x, y, z] = [ρ cos(φ), ρ sin(φ), z] | cylindrical | e_{ρ}(φ), e_{φ}(φ), e_{z} 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)] | spherical | e_{r}(θ,φ), e_{θ}(θ,φ),e_{φ}(φ) form an orthonormal right-handed system. |

## Vectors and tensors

Example | Meaning and verbal equivalent | Remarks |
---|---|---|

a | vector a | Instead of italic boldface, vectors can also be indicated by an arrow above the letter symbol. Any vector can be multiplied by a scalar ak, i.e. k.a |

... | ... | ... |

⋮ |

## Special functions

Example | Meaning and verbal equivalent | Remarks |
---|---|---|

J_{l}(x) | cylindrical Bessel functions (of the first kind) | ... |

... | ... | ... |

⋮ |

## See also

## References and notes

- ↑ "ISO 80000-2:2009". International Organization for Standardization. Retrieved 1 July 2010.
- 1 2 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. - ↑ 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: 〈h; 〉h;.
- ↑ 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)

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