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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 It was superseded in 2009 by ISO 80000-2.^{[1]}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]}
Contents
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^{*} = ℤ ∖ {0} = {..., −3, −2, −1, 1, 2, 3, ...} | |
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 } |
⋃ | <math>\bigcup_{i=1}^n A_i</math> | union of a collection of sets | <math>\bigcup_{i=1}^n A_i=A_1\cup A_2\cup\ldots\cup A_n</math>, the set of elements belonging to at least one of the sets A_{1}, …, A_{n}. <math>\bigcup{}_{i=1}^n</math> and <math>\bigcup_{i\in I}</math>, <math>\bigcup{}_{i \in I}</math> 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 } |
⋂ | <math>\bigcap_{i=1}^n A_i</math> | intersection of a collection of sets | <math>\bigcap_{i=1}^n A_i=A_1\cap A_2\cap\ldots\cap A_n</math>, the set of elements belonging to all sets A_{1}, …, A_{n}. <math>\bigcap{}_{i=1}^n</math> and <math>\bigcap_{i\in I}</math>, ⋂_{i∈I} 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 |
---|---|---|---|
≝ <math>\ \stackrel{\mathrm{def}}{=}\ </math> |
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 | <math>a \not\equiv b</math> 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 ∼ b a ∝ 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 | ||
() [] {} <math>\langle \rangle</math> |
(a+b)c [a+b]c {a+b}c <math>\langle</math>a+b<math>\rangle</math>c |
ac+bc, parentheses ac+bc, square brackets ac+bc, braces ac+bc, angle brackets |
In ordinary algebra, the sequence of (), [], {}, <math>\langle \rangle </math> in order of nesting is not standardized. Special uses are made of (), [], {}, <math>\langle \rangle </math> in particular fields.^{[3]} |
∥ | AB ∥ CD | the line AB is parallel to the line CD | |
<math>\perp</math> | AB<math>\perp</math>CD | 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 |
---|---|---|
<math>f:D \rightarrow C</math> | function f has domain D and codomain C | Used to explicitly define the domain and codomain of a function. |
<math>f\left(S\right)</math> | <math>\left\{f\left(x\right)\mid x\in S\right\}</math> | 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, k are also used. |
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 <math>\vec a</math> |
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 |
---|---|---|
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.
- ^ ^{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.
- ^ 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, ⟂h;, does not display correctly, it is similar to ⊥h; (up tack: sometimes meaning orthogonal to) and it also appears similar to ⏊h; (the dentistry: symbol light up and horizontal)