ISO 31-11

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ISO 31-11 is the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology.

Its definitions include:[1]

Contents

[edit] 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 only exactly one x exists for which p(x) is true.

[edit] 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 belongs 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 {xi : iI}, 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 ∈ 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
the empty set
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:
* = {1, 2, 3, ...}
k = {0, 1, 2, 3, ..., k − 1}
the set of integers ℤ = {..., −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}
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 }
\bigcup_{i=1}^n A_i union of a collection of sets \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. \bigcup{}_{i=1}^n and \bigcup_{i\in I}, ⋃iI 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 }
\bigcap_{i=1}^n A_i intersection of a collection of sets \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. \bigcap{}_{i=1}^n and \bigcap_{i\in I}, ⋂iI are also used, where I denotes a set of indices.
\ A \ B difference between A and B; A minus B The set of elements which belong to A but not to B.
A \ B = { xxAxB }
AB should not be used.
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 = 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.
(,…,) (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.

[edit] Miscellaneous signs and symbols

Sign Example Meaning and verbal equivalent Remarks

\ \stackrel{\mathrm{def}}{=}\
ab a is by definition equal to b[1] := is also used
= a = b a equals b ≡ may be used to emphasize that a particular equality is an identity.
ab a is not equal to b a \not\equiv b may be used to emphasize that a is not identically equal to b.
ab a corresponds to b On a 1:106 map: 1 cm ≙ 10 km.
ab a is approximately equal to b The symbol ≃ is reserved for "is asymptotically equal to".

ab
ab
a is proportional to b
< a < b a is less than b
> a > b a is greater than b
ab a is less than or equal to b The symbol ≦ is also used.
ab a is greater than or equal to b The symbol ≧ is also used.
ab a is much less than b
ab a is much greater than b
infinity
()
[]
{}
\langle \rangle
(a+b)c
[a+b]c
{a+b}c
\langlea+b\ranglec
ac+bc, parentheses
ac+bc, square brackets
ac+bc, braces
ac+bc, angle brackets
In ordinary algebra, the sequence of (), [], {}, \langle \rangle in order of nesting is not standardized. Special uses are made of (), [], {}, \langle \rangle in particular fields.[2]
AB ∥ CD the line AB is parallel to the line CD
\perp AB{}^\perpCD the line AB is perpendicular to the line CD[3]

[edit] 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
... ... ... ...

[edit] Functions

Example Meaning and verbal equivalent Remarks
f function f ...
... ... ...

[edit] Exponential and logarithmic functions

Example Meaning and verbal equivalent Remarks
ax exponential function to the base a of x ...
e base of natural logarithms e = 2.718 281 8...
... ... ...

[edit] Circular and hyperbolic functions

Example Meaning and verbal equivalent Remarks
π ratio of the circumference of a circle to its diameter π = 3.141 592 6...
... ... ...

[edit] 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

[edit] Matrices

Example Meaning and verbal equivalent Remarks
A matrix A ...
... ... ...

[edit] Coordinate systems

Coordinates Position vector and its differential Name of coordinate system Remarks
x, y, z ... cartesian coordinates ...
ρ, φ, z ... cylindrical coordinates ...
r, θ, φ ... spherical coordinates ...

[edit] Vectors and tensors

Example Meaning and verbal equivalent Remarks
a
\vec a
vector a Instead of boldface, vectors can also be indicated by an arrow above the letter symbol. Any vector a can be multiplied by a number k, i.e. ka.
... ... ...

[edit] Special functions

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

[edit] References and notes

  1. ^ a b Barry M. Taylor. Guide for the Use of the International System of Units (SI) ";Standardized mathematical signs; §10.1.2 p. 33: an abbreviated list of symbols from ISO 31-11". NIST. http://physics.nist.gov/cuu/pdf/sp811.pdf Guide for the Use of the International System of Units (SI). 
  2. ^ 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: &x3008h; &x3009h;.
  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)

[edit] See also

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