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Iso 31-11

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Title: Iso 31-11  
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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 It was superseded in 2009 by ISO 80000-2.[1]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 1
  • Sets 2
  • Miscellaneous signs and symbols 3
  • Operations 4
  • Functions 5
  • Exponential and logarithmic functions 6
  • Circular and hyperbolic functions 7
  • Complex numbers 8
  • Matrices 9
  • Coordinate systems 10
  • Vectors and tensors 11
  • Special functions 12
  • See also 13
  • References and notes 14

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
{ } {x1, x2, ..., xn} set with elements x1, x2, ..., xn also {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 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, ...}
ℕk = {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 }
⋃ \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}, \bigcup{}_{i \in I} 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 }
⋂ \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}, ⋂i∈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 = 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.
(,…,) (a1, a2, …, 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 a ∈ A and b ∈ B.
A × B = { (a, b) ∣ a ∈ A ∧ b ∈ B }
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 a ∈ A; diagonal of the set A × A ΔA = { (a, a) ∣ a ∈ A }
idA is also used.

Miscellaneous signs and symbols

Sign Example Meaning and verbal equivalent Remarks
≝
\ \stackrel{\mathrm{def}}{=}\ 		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 a \not\equiv b may be used to emphasize that a is not identically equal to b.
≙ a ≙ b a corresponds to b On a 1:106 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
()
[]
{}
\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.[3]
∥ AB ∥ CD the line AB is parallel to the line CD
\perp AB\perpCD 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
f:D \rightarrow C function f has domain D and codomain C Used to explicitly define the domain and codomain of a function.
f\left(S\right) \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...
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 = 3.141 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 [x y z] = [x y z]; [dx dy dz]; 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 [x, y, z] = [ρ cos(φ), ρ sin(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 [x, y, z] = r [sin(θ)cos(φ), sin(θ)sin(φ), cos(θ)] 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
\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 80000-2:2009".  
  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:  
  3. ^ 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: 〈 〉.
  4. ^ 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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