A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entierely constitued with symbols of various types, many symbols are needed for expressing all mathematics.
The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the HinduArabic numeral system. Historically, uppercase letters were used for representing points in geometry, and lowercase letters were used for variables and constants. Letters are used for representing many other sort of mathematical objects. As the number of these sorts has dramatically increased in modern mathematics, the Greek alphabet and some Hebrew letters are also used. In mathematical formulas, the standard typeface is italic type for Latin letters and lowercase Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainly boldface script typeface (the lowercase script face is rarely used because of the possible confusion with the standard face), German fraktur and blackboard bold (the other letters are rarely used, or their use is controversial).
In this article we presentthe main symbols that are used in mathematics and their common use.
For the use of letters as symbols for variables, see variable (mathematics). For their use as symbols for constants, see List of mathematical constants.
Several logical symbols are widely used in all mathematics, and are listed here. For symbols that are used only in mathematical logic, or are rarely used, see List of logic symbols.
In this section, the symbols that are listed are used as some sort of punctuation marks in mathematics reasoning, or as abbreviations of English phrases. They are generally not used inside a formula. Some were used in classical logic for indicating the logical dependence between sentences written in plain English. Except for the first one, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. However, they are still used on a black board for indicating relationships between formulas.
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
factorial

means the product . 
 
\to 
function arrow
from ... to

f: X > Y means the function f maps the set X into the set Y.  Let f: Z > N ? {0} be defined by f(x) := x^{2}.  
?

\mapsto 
function arrow
maps to

f: a ? b means the function f maps the element a to the element b.  Let f: x ? x + 1 (the successor function). 
\leftarrow 
.. if ..

a b means that for the propositions a and b, if b implies a, then a is the converse implication of b.a to the element b. This reads as "a if b", or "not b without a". It is not to be confused with the assignment operator in computer science.  
?

\langle 
the bra ...;
the dual of ... 
?? means the dual of the vector ??, a linear functional which maps a ket ?? onto the inner product ????.  
?

\rangle 
the ket ...;
the vector ... 
?? means the vector with label ?, which is in a Hilbert space.  A qubit's state can be represented as ?0?+ ?1?, where ? and ? are complex numbers s.t. ?^{2} + ?^{2} = 1. 
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
{\ \choose\ } 
n choose k

means (in the case of n = positive integer) the number of combinations of k elements drawn from a set of n elements. (This may also be written as C(n, k), C(n; k), _{n}C_{k}, ^{n}C_{k}, or .) 
 
\left(\!\!{\ \choose\ }\!\!\right) 
u multichoose k



\left\{ \begin{array}{lr} \ldots \\ \ldots \end{array}\right. 
is defined as ... if ..., or as ... if ...;
match ... with everywhere

means the function f(x) is defined as a if the condition p(x) holds, or as b if the condition q(x) holds. (The body of a piecewisedefined function can have any finite number (not only just two) expressioncondition pairs.) This symbol is also used in type theory for pattern matching the constructor of the value of an algebraic type. For example does pattern matching on the function's arguments and means that g(x) is defined as a, and g(y) is defined as b. (A pattern matching can have any finite number (not only just two) patternexpression pairs.) 

...

 \ldots  \!\, 
absolute value;
modulus absolute value of; modulus of

x means the distance along the real line (or across the complex plane) between x and zero.  3 = 3 5 = 5 = 5  i  = 1  3 + 4i  = 5 
Euclidean norm or Euclidean length or magnitude
Euclidean norm of

x means the (Euclidean) length of vector x.  For x = (3,4)  
determinant of

A means the determinant of the matrix A  
cardinality of;
size of; order of 
X means the cardinality of the set X. (# may be used instead as described below.) 
{3, 5, 7, 9} = 4.  
?...?

\ \ldots \ \!\, 
norm of;
length of 
? x ? means the norm of the element x of a normed vector space.^{[2]}  ? x + y ? x ? + ? y ? 
nearest integer to

?x? means the nearest integer to x. (This may also be written [x], ?x?, nint(x) or Round(x).) 
?1? = 1, ?1.6? = 2, ?2.4? = 2, ?3.49? = 3  
{\{\ ,\!\ \}} \!\, 
set brackets
the set of ...

{a,b,c} means the set consisting of a, b, and c.^{[3]}  N = { 1, 2, 3, ... }  
{ : }
{  } { ; } 
\{\ :\ \} \!\, \{\ \ \} \!\, \{\ ;\ \} \!\, 
the set of ... such that

{x : P(x)} means the set of all x for which P(x) is true.^{[3]} {x  P(x)} is the same as {x : P(x)}.  {n ? N : n^{2} < 20} = { 1, 2, 3, 4 } 
?...?

\lfloor \ldots \rfloor \!\, 
floor;
greatest integer; entier 
?x? means the floor of x, i.e. the largest integer less than or equal to x. (This may also be written [x], floor(x) or int(x).) 
?4? = 4, ?2.1? = 2, ?2.9? = 2, ?2.6? = 3 
?...?

\lceil \ldots \rceil \!\, 
ceiling

?x? means the ceiling of x, i.e. the smallest integer greater than or equal to x. (This may also be written ceil(x) or ceiling(x).) 
?4? = 4, ?2.1? = 3, ?2.9? = 3, ?2.6? = 2 
?...?

\lfloor \ldots \rceil \!\, 
nearest integer to

?x? means the nearest integer to x. (This may also be written [x], x, nint(x) or Round(x).) 
?2? = 2, ?2.6? = 3, ?3.4? = 3, ?4.49? = 4, ?4.5? = 5 
[ : ]

[\ :\ ] \!\, 
the degree of

[K : F] means the degree of the extension K : F.  [Q(?2) : Q] = 2 [C : R] = 2 [R : Q] = ? 
[\ ] \!\, [\ ,\ ] \!\, 
the equivalence class of

[a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation. [a]_{R} means the same, but with R as the equivalence relation. 
Let a ~ b be true iff a ? b (mod 5).
Then [2] = {..., 8, 3, 2, 7, ...}.  
floor;
greatest integer; entier 
[x] means the floor of x, i.e. the largest integer less than or equal to x. (This may also be written ?x?, floor(x) or int(x). Not to be confused with the nearest integer function, as described below.) 
[3] = 3, [3.5] = 3, [3.99] = 3, [3.7] = 4  
nearest integer to

[x] means the nearest integer to x. (This may also be written ?x?, x, nint(x) or Round(x). Not to be confused with the floor function, as described above.) 
[2] = 2, [2.6] = 3, [3.4] = 3, [4.49] = 4  
1 if true, 0 otherwise

[S] maps a true statement S to 1 and a false statement S to 0.  [0=5]=0, [7>0]=1, [2 ? {2,3,4}]=1, [5 ? {2,3,4}]=0  
image of ... under ...
everywhere

f[X] means { f(x) : x ? X }, the image of the function f under the set X ? dom(f). (This may also be written as f(X) if there is no risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.) 

closed interval

.  0 and 1/2 are in the interval [0,1].  
the commutator of

[g, h] = g^{1}h^{1}gh (or ghg^{1}h^{1}), if g, h ? G (a group). [a, b] = ab  ba, if a, b ? R (a ring or commutative algebra). 
x^{y} = x[x, y] (group theory). [AB, C] = A[B, C] + [A, C]B (ring theory).  
the triple scalar product of

[a, b, c] = a × b · c, the scalar product of a × b with c.  [a, b, c] = [b, c, a] = [c, a, b].  
(\ ) \!\, (\ ,\ ) \!\, 
function application
of

f(x) means the value of the function f at the element x.  If f(x) := x^{2}  5, then f(6) = 6^{2}  5 = 36  5=31.  
image of ... under ...
everywhere

f(X) means { f(x) : x ? X }, the image of the function f under the set X ? dom(f). (This may also be written as f[X] if there is a risk of confusing the image of f under X with the function application f of X. Another notation is Im f, the image of f under its domain.) 

precedence grouping
parentheses
everywhere

Perform the operations inside the parentheses first.  (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.  
everywhere

An ordered list (or sequence, or horizontal vector, or row vector) of values.
(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. Set theorists and computer scientists often use angle brackets ? ? instead of parentheses.) 
(a, b) is an ordered pair (or 2tuple).
(a, b, c) is an ordered triple (or 3tuple). ( ) is the empty tuple (or 0tuple).  
highest common factor;
greatest common divisor; hcf; gcd number theory

(a, b) means the highest common factor of a and b. (This may also be written hcf(a, b) or gcd(a, b).) 
(3, 7) = 1 (they are coprime); (15, 25) = 5.  
( , )
] , [ 
(\ ,\ ) \!\,(\ ,\ ) \!\, ]\ ,\ [ \!\,] 
open interval

.
(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation ]a,b[ can be used instead.) 
4 is not in the interval (4, 18).
(0, +?) equals the set of positive real numbers. 
( , ]
] , ] 
(\ ,\ ] \!\, \ ,\ ] \!\,] 
halfopen interval;
leftopen interval 
.  (1, 7] and (?, 1] 
[ , )
[ , [ 
[\ ,\ ) \!\, [\ ,\ [ \!\, 
halfopen interval;
rightopen interval 
.  [4, 18) and [1, +?) 
?,? 
\langle\ \rangle \!\, \langle\ ,\ \rangle \!\, 
inner product of

?u,v? means the inner product of u and v, where u and v are members of an inner product space. Note that the notation ?u, v? may be ambiguous: it could mean the inner product or the linear span. There are many variants of the notation, such as ?u  v? and (u  v), which are described below. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. As ? and ? can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts. 
The standard inner product between two vectors x = (2, 3) and y = (1, 5) is: ?x, y? = 2 × 1 + 3 × 5 = 13 
average
average of

let S be a subset of N for example, represents the average of all the elements in S.  for a time series :g(t) (t = 1, 2,...)
we can define the structure functions S_{q}():  
the expectation value of

For a single discrete variable of a function , the expectation value of is defined as , and for a single continuous variable the expectation value of is defined as ; where is the PDF of the variable .^{[4]}  
(linear) span of;
linear hull of 
?S? means the span of S ? V. That is, it is the intersection of all subspaces of V which contain S. ?u_{1}, u_{2}, ...? is shorthand for ?{u_{1}, u_{2}, ...}?.

.  
subgroup generated by a set
the subgroup generated by

means the smallest subgroup of G (where S ? G, a group) containing every element of S. is shorthand for . 
In S_{3}, and .  
tuple; ntuple;
ordered pair/triple/etc; row vector; sequence everywhere

An ordered list (or sequence, or horizontal vector, or row vector) of values.
(The notation (a,b) is often used as well.) 
is an ordered pair (or 2tuple).
is an ordered triple (or 3tuple). is the empty tuple (or 0tuple).  
??
() 
\langle\ \ \rangle \!\, (\ \ ) \!\, 
inner product of

?u  v? means the inner product of u and v, where u and v are members of an inner product space.^{[5]} (u  v) means the same. Another variant of the notation is ?u, v? which is described above. For spatial vectors, the dot product notation, x · y is common. For matrices, the colon notation A : B may be used. As ? and ? can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts. 
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
\ast or * 
convolution;
convolved with 
f*g means the convolution of f and g. (Different than f*g, which means the product of g with the complex conjugate of f, as described below.) (Can also be written in text as f ∗ g.) 
.  
Hodge star;
Hodge dual 
*v means the Hodge dual of a vector v. If v is a kvector within an ndimensional oriented quadratic space, then *v is an (nk)vector.  If are the standard basis vectors of ,  
^\ast or ^* 
conjugate

z* means the complex conjugate of z. ( can also be used for the conjugate of z, as described below.) 
.  
the group of units of

R^{*} consists of the set of units of the ring R, along with the operation of multiplication. This may also be written R^{×}as described above, or U(R). 

the (set of) hyperreals

^{*}R means the set of hyperreal numbers. Other sets can be used in place of R.  ^{*}N is the hypernatural numbers.  
Kleene star

Corresponds to the usage of * in regular expressions. If ? is a set of strings, then ?* is the set of all strings that can be created by concatenating members of ?. The same string can be used multiple times, and the empty string is also a member of ?*.  If ? = ('a', 'b', 'c') then ?* includes '', 'a', 'ab', 'aba', 'abac', etc. The full set cannot be enumerated here since it is countably infinite, but each individual string must have finite length.  
given

P(AB) means the probability of the event A occurring given that B occurs.  if X is a uniformly random day of the year P(X is 25  X is in May) = 1/31  
restriction of ... to ...;
restricted to 
f_{A} means the function f is restricted to the set A, that is, it is the function with domain A ? dom(f) that agrees with f.  The function f : R > R defined by f(x) = x^{2} is not injective, but f_{R+} is injective.  
such that
such that;
so that everywhere

 means "such that", see ":" (described below).  S = {(x,y)  0 < y < f(x)} The set of (x,y) such that y is greater than 0 and less than f(x).  

? 
\mid \nmid 
divides

a  b means a divides b. a ? b means a does not divide b. (The symbol  can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar  character is often used instead.) 
Since 15 = 3 × 5, it is true that 3  15 and 5  15. 


\mid\mid 
exact divisibility
exactly divides

p^{a}  n means p^{a} exactly divides n (i.e. p^{a} divides n but p^{a+1} does not).  2^{3}  360. 
?
? ? 
\ Requires the viewer to support Unicode: \unicode{x2225}, \unicode{x2226}, and \unicode{x22D5}. \mathrel{\rlap{\,\parallel}} requires \setmathfont{MathJax}.^{[6]} 
is parallel to

x ? y means x is parallel to y. x ? y means x is not parallel to y. x ? y means x is equal and parallel to y. (The symbol ? can be difficult to type, and its negation is rare, so two regular but slightly longer vertical bar  characters are often used instead.) 
If l ? m and m ? n then l ? n. 
is incomparable to

x ? y means x is incomparable to y.  {1,2} ? {2,3} under set containment.  
\sharp 
cardinality of;
size of; order of 
#X means the cardinality of the set X. (... may be used instead as described above.) 
#{4, 6, 8} = 3  
connected sum of;
knot sum of; knot composition of 
A#B is the connected sum of the manifolds A and B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition.  A#S^{m} is homeomorphic to A, for any manifold A, and the sphere S^{m}.  
primorial

n# is product of all prime numbers less than or equal to n.  12# = 2 × 3 × 5 × 7 × 11 = 2310  
such that
such that;
so that everywhere

: means "such that", and is used in proofs and the setbuilder notation (described below).  ? n ? N: n is even.  
extends;
over 
K : F means the field K extends the field F. This may also be written as K >= F. 
R : Q  
inner product of matrices
inner product of

A : B means the Frobenius inner product of the matrices A and B. The general inner product is denoted by ?u, v?, ?u  v? or (u  v), as described below. For spatial vectors, the dot product notation, x·y is common. See also braket notation. 

index of subgroup

The index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G  
divided by
over everywhere

A : B means the division of A with B (dividing A by B)  10 : 2 = 5  
?

\vdots \!\, 
vertical ellipsis
everywhere

Denotes that certain constants and terms are missing out (e.g. for clarity) and that only the important terms are being listed.  
?

\wr \!\, 
wreath product of ... by ...

A ? H means the wreath product of the group A by the group H. This may also be written A_{ wr }H. 
is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices. 
?
? 
\oplus \!\, \veebar \!\, 
xor

The statement A ? B is true when either A or B, but not both, are true. A ? B means the same.  (¬A) ? A is always true, A ? A is always false. 
direct sum of

The direct sum is a special way of combining several objects into one general object. (The bun symbol ?, or the coproduct symbol ?, is used; ? is only for logic.) 
Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ? W (U = V + W) ? (V ? W = {0})  
?

\Box \!\ 
D'Alembertian;
wave operator nonEuclidean Laplacian

It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions. 
Includes upsidedown letters.
Also called diacritics.
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
\bar{a}, \overline{a} 
overbar;
... bar 
(often read as "x bar") is the mean (average value of ).  .  
finite sequence, tuple

means the finite sequence/tuple .  .  
algebraic closure of

is the algebraic closure of the field F.  The field of algebraic numbers is sometimes denoted as because it is the algebraic closure of the rational numbers .  
conjugate

means the complex conjugate of z. (z^{*}can also be used for the conjugate of z, as described above.) 
.  
(topological) closure of

is the topological closure of the set S. This may also be denoted as cl(S) or Cl(S). 
In the space of the real numbers, (the rational numbers are dense in the real numbers).  
\overset{\rightharpoonup}{a} 
harpoon


â

\hat a 
hat

(pronounced "a hat") is the normalized version of vector , having length 1.  
estimator for

is the estimator or the estimate for the parameter .  The estimator produces a sample estimate for the mean .  
' 
... prime;
derivative of 
f ?(x) means the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. (The singlequote character ' is sometimes used instead, especially in ASCII text.) 
If f(x) := x^{2}, then f ?(x) = 2x.  
\dot{\,} 
... dot;
time derivative of 
means the derivative of x with respect to time. That is .  If x(t) := t^{2}, then . 
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
\mathbb{B} \mathbf{B} 
B;
the (set of) boolean values; the (set of) truth values; 
B means either {0, 1}, {false, true}, {F, T}, or .  (¬False) ? B  
\mathbb{C} \mathbf{C} 
C;
the (set of) complex numbers 
C means {a + b i : a,b ? R}.  i ? C
 
\partial 
partial;
d 
?f/?x_{i} means the partial derivative of f with respect to x_{i}, where f is a function on (x_{1}, ..., x_{n}).  If f(x,y) := x^{2}y, then ?f/?x = 2xy,  
boundary of

?M means the boundary of M  ?{x : x x : x = 2}  
degree of

?f means the degree of the polynomial f. (This may also be written deg f.) 
?(x^{2}  1) = 2  
\mathbb E \mathrm{E} 
expected value

the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained  
\mathbb{F} 
Galois field, or finite field
Field (mathematics) theory

, for any prime p and integer n, is the unique finite field with order , often written , and sometimes also known as , , or , although this last notation is ambiguous.  is , the finite field in whose quadratic extension the popular elliptic curve Curve25519 is computed.  
\mathbb{H} \mathbf{H} 
quaternions or Hamiltonian quaternions
H;
the (set of) quaternions 
H means {a + b i + c j + d k : a,b,c,d ? R}.  
\mathbb{I} \mathbf{I} 
the indicator of

The indicator function of a subset A of a set X is a function defined as :
Note that the indicator function is also sometimes denoted 1. 

\mathbb{N} \mathbf{N} 
the (set of) natural numbers

N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}. The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analysts, set theorists and computer scientists prefer the former. To avoid confusion, always check an author's definition of N. Set theorists often use the notation ? (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation  N = {a : a ? Z} or N = {a > 0: a ? Z}  
?
? 
\circ ^{[7]}^{[8]} \odot 
entrywise product, elementwise product, circled dot

For two matrices (or vectors) of the same dimensions the Hadamard product is a matrix of the same dimensions with elements given by .  
?

\circ 
composed with

f ? g is the function such that (f ? g)(x) = f(g(x)).^{[9]}  if f(x) := 2x, and g(x) := x + 3, then (f ? g)(x) = 2(x + 3). 
O 
bigoh of

The Big O notation describes the limiting behavior of a function, when the argument tends towards a particular value or infinity.  If f(x) = 6x^{4}  2x^{3} + 5 and g(x) = x^{4}, then
 
\mathbb{P} \mathbf{P} 
P;
the set of prime numbers 
P is often used to denote the set of prime numbers.  
P;
the projective space; the projective line; the projective plane 
P means a space with a point at infinity.  ,  
the space of all possible polynomials

P means a_{n}x^{n} + a_{n1}x^{n1}...a_{1}x+a_{0} P_{n} means the space of all polynomials of degree less than or equal to n 

the probability of

P(X) means the probability of the event X occurring. This may also be written as P(X), Pr(X), P[X] or Pr[X]. 
If a fair coin is flipped, P(Heads) = P(Tails) = 0.5.  
the Power set of

Given a set S, the power set of S is the set of all subsets of the set S. The power set of S0 is
denoted by P(S). 
The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence,
P({0, 1, 2}) = {?, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2} }.  
\mathbb{Q} \mathbf{Q} 
Q;
the (set of) rational numbers; the rationals 
Q means {p/q : p ? Z, q ? N}.  3.14000... ? Q ? ? Q  
\mathbb{Q}_p \mathbf{Q}_p 
the (set of) padic numbers;
the padics 
Q means {p/q : p ? Z, q ? N}. 
 
\mathbb{R} \mathbf{R} 
R;
the (set of) real numbers; the reals 
R means the set of real numbers.  ? ? R ?(1) ? R  
{}^\dagger 
conjugate transpose;
adjoint; Hermitian adjoint/conjugate/transpose/dagger 
A^{+} means the transpose of the complex conjugate of A.^{[10]} This may also be written A^{*T}, A^{T*}, A^{*}, A^{T}or A^{T}. 
If A = (a_{ij}) then A^{+} = (a_{ji}).  
{}^{\mathsf{T}} 
transpose

A^{T} means A, but with its rows swapped for columns. This may also be written A?, A^{t}or A^{tr}. 
If A = (a_{ij}) then A^{T} = (a_{ji}).  
\top 
the top element

? means the largest element of a lattice.  ?x : x ? ? = x  
the top type; top

? means the top or universal type; every type in the type system of interest is a subtype of top.  ? types T, T <  
top, verum

The statement ? is unconditionally true.  A => ? is always true.  
\bot 
bottom, falsum, falsity

The statement ? is unconditionally false.  ? => A is always true.  
is perpendicular to

x ? y means x is perpendicular to y; or more generally x is orthogonal to y.  If l ? m and m ? n in the plane, then l  n.  
orthogonal/ perpendicular complement of;
perp 
W^{?} means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W.  Within , .  
is coprime to

x ? y means x has no factor greater than 1 in common with y.  34 ? 55  
is independent of

A ? B means A is an event whose probability is independent of event B. The double perpendicular symbol () is also commonly used for the purpose of denoting this, for instance: (In LaTeX, the command is: "A \perp\!\!\!\perp B".)  If A ? B, then P(AB) = P(A).  
the bottom element

? means the smallest element of a lattice.  ?x : x ? ? = x  
the bottom type;
bot 
? means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system.  ? types T, ? <: T  
is comparable to

x ? y means that x is comparable to y.  {e, ?} ? {1, 2, e, 3, ?} under set containment.
 
\mathbb{U} \mathbf{U} 
U;
the universal set; the set of all numbers; all numbers considered 
U means "the set of all elements being considered." It may represent all numbers both real and complex, or any subset of thesehence the term "universal". 
U = {R,C} includes all numbers. If instead, U = {Z,C}, then ? ? U.  
?

\otimes 
tensor product of

means the tensor product of V and U.^{[11]} means the tensor product of modules V and U over the ring R.  {1, 2, 3, 4} ? {1, 1, 2} = {{1, 1, 2}, {2, 2, 4}, {3, 3, 6}, {4, 4, 8}} 
?
? 
\ltimes \rtimes 
the semidirect product of

N ?_{?}H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to ?. Also, if G = N ?_{?}H, then G is said to split over N. (? may also be written the other way round, as ?, or as ×.) 

the semijoin of

R ? S is the semijoin of the relations R and S, the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names.  R S = _{a1,..,an}(R S)  
?

\bowtie 
the natural join of

R ? S is the natural join of the relations R and S, the set of all combinations of tuples in R and S that are equal on their common attribute names. 

\mathbb{Z} \mathbf{Z} 
the (set of) integers

Z means {..., 3, 2, 1, 0, 1, 2, 3, ...}.
Z^{+} or Z^{>} means {1, 2, 3, ...} . 
Z = {p, p : p ? N ? {0}}  
Z_{n}
Z_{p} Z_{n} Z_{p} 
\mathbb{Z}_n \mathbb{Z}_p \mathbf{Z}_n 
the (set of) integers modulo n

Z_{n} means {[0], [1], [2], ...[n1]} with addition and multiplication modulo n. Note that any letter may be used instead of n, such as p. To avoid confusion with padic numbers, use Z/pZ or Z/(p) instead. 
Z_{3} = {[0], [1], [2]} 
the (set of) padic integers

Note that any letter may be used instead of p, such as n or l. 
Symbol in HTML 
Symbol in TeX 
Name  Explanation  Examples 

Read as  
Category  
\Gamma 
Gamma function


\delta 
Dirac delta of

?(x)  
Kronecker delta of

?_{ij}  
Functional derivative of


?
? ? 
\vartriangle \ominus \oplus 
symmetric difference

A ? B (or A ? B) means the set of elements in exactly one of A or B. (Not to be confused with delta, ?, described below.) 
{1,5,6,8} ? {2,5,8} = {1,2,6} {3,4,5,6} ? {1,2,5,6} = {1,2,3,4} 
\Delta 
delta;
change in 
?x means a (noninfinitesimal) change in x. (If the change becomes infinitesimal, ? and even d are used instead. Not to be confused with the symmetric difference, written ?, above.) 
is the gradient of a straight line.  
Laplace operator

The Laplace operator is a second order differential operator in ndimensional Euclidean space  If ? is a twicedifferentiable realvalued function, then the Laplacian of ? is defined by  
\nabla 
?f (x_{1}, ..., x_{n}) is the vector of partial derivatives (?f / ?x_{1}, ..., ?f / ?x_{n}).  If f (x,y,z) := 3xy + z², then ?f = (3y, 3x, 2z)  
del dot;
divergence of 
If , then .  
curl of

If , then .  
\pi 
pi;
3.1415926...; ?355÷113 
Used in various formulas involving circles; ? is equivalent to the amount of area a circle would take up in a square of equal width with an area of 4 square units, roughly 3.14159. It is also the ratio of the circumference to the diameter of a circle.  A = ?R^{2} = 314.16 > R = 10  
primecounting function of

counts the number of prime numbers less than or equal to .  
Projection of

restricts to the attribute set.  
the nth Homotopy group of

consists of homotopy equivalence classes of base point preserving maps from an ndimensional sphere (with base point) into the pointed space X.  
\prod 
product over ... from ... to ... of

means .  
the Cartesian product of;
the direct product of 
means the set of all (n+1)tuples


\sigma 
population standard deviation

A measure of spread or variation of a set of values in a sample population set. 
In mathematics written in Persian or Arabic, some symbols may be reversed to make righttoleft writing and reading easier.^{[14]}
Some Unicode charts of mathematical operators and symbols:
Some Unicode crossreferences: