ISO 3111
From Infogalactic: the planetary knowledge core
ISO 3111 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 It was superseded in 2009 by ISO 800002.^{[1]}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 halfopen interval in ℝ from a (excluded) to b (included)  ]a,b] = {x ∈ ℝ ∣ a < x ≤ b} 
[,[ [,) 
[a,b[ [a,b) 
right halfopen 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 , ⋂_{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 ntuple  ⟨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 ∼ 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  
() [] {} 
(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 nmensional space. e_{x}, e_{y}, e_{z} form an orthonormal righthanded 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 righthanded 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 righthanded 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 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 800002:2009". International Organization for Standardization. Retrieved 1 July 2010.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ ^{2.0} ^{2.1} 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.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ 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 ChineseJapaneseKorean (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)