This article contains special characters. Without proper rendering support, you may see question marks, boxes, or other symbols. 
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 
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. 
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 , 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. 
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]} 
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 
...  ...  ...  ... 
⋮ 
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.  
⋮ 
Example  Meaning and verbal equivalent  Remarks 

e  base of natural logarithms  e = 2.71828... 
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 
...  ...  ... 
⋮ 
Example  Meaning and verbal equivalent  Remarks 

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

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

J_{l}(x)  cylindrical Bessel functions (of the first kind)  ... 
...  ...  ... 
⋮ 
