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ISO 3111
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ISO 3111 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 

∧  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 only exactly one x exists for which p(x) is true. 
[edit] 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 belongs 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  
∅  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 , ⋃_{i∈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 } 
⋂  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  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. 
∁  ∁_{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. 
[edit] Miscellaneous signs and symbols
Sign  Example  Meaning and verbal equivalent  Remarks  

≝ 
a ≝ b  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.  
≠  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.^{[2]}  
∥  AB ∥ CD  the line AB is parallel to the line CD  
ABCD  the line AB is perpendicular to the line CD^{[3]} 
[edit] 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 
...  ...  ...  ... 
⋮ 
[edit] Functions
Example  Meaning and verbal equivalent  Remarks 

f  function f  ... 
...  ...  ... 
⋮ 
[edit] Exponential and logarithmic functions
Example  Meaning and verbal equivalent  Remarks 

a^{x}  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 = 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 
[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 
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 

J_{l}(x)  cylindrical Bessel functions (of the first kind)  ... 
...  ...  ... 
⋮ 
[edit] References and notes
 ^ ^{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 3111". NIST. http://physics.nist.gov/cuu/pdf/sp811.pdf Guide for the Use of the International System of Units (SI).
 ^ 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: &x3008h; &x3009h;.
 ^ 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)