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Table of mathematical symbols
From Wikipedia, the free encyclopedia
For the HTML codes of mathematical symbols see mathematical HTML.
Note: This article contains special characters.
The following table lists many specialized symbols commonly used in mathematics.
Basic mathematical symbols
Name
Symbol
Read as
Explanation
Examples
Category
equality
=
is equal to;
equals
x = y means x and y
represent the same
thing or value.
1+1=2
everywhere
≠
<>
!=
<
inequation
x ≠ y means that x and y
do not represent the
same thing or value.
is not equal to;
1≠2
does not equal (The symbols != and <>
are primarily from
computer science. They
are avoided in
everywhere mathematical texts.)
strict inequality
x < y means x is less
than y.
>
≪
≫
is less than, is
greater than, is
much less than,
is much greater
than
x ≫ y means x is much
greater than y.
order theory
inequality
≤
<=
x > y means x is greater
3<4
than y.
5 > 4.
x ≪ y means x is much
0.003 ≪ 1000000
less than y.
is less than or
x ≤ y means x is less
than or equal to y.
x ≥ y means x is
greater than or equal to
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≥
>=
equal to, is
greater than or
equal to
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y.
(The symbols <= and
3 ≤ 4 and 5 ≤ 5
>= are primarily from
5 ≥ 4 and 5 ≥ 5
computer science. They
order theory are avoided in
mathematical texts.)
proportionality
∝
is proportional y ∝ x means that y = kx
if y = 2x, then y ∝ x
to; varies as
for some constant k.
everywhere
addition
plus
4 + 6 means the sum of
2+7=9
4 and 6.
arithmetic
+
disjoint union
the disjoint
union of ...
and ...
A1 + A2 means the
A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒
disjoint union of sets A1 A + A = {(1,1), (2,1), (3,1), (4,1), (2,2),
1
2
and A2.
(4,2), (5,2), (7,2)}
set theory
subtraction
minus
9 − 4 means the
8−3=5
subtraction of 4 from 9.
arithmetic
negative sign
−
negative ; minus
−3 means the negative
of the number 3.
−(−5) = 5
arithmetic
set-theoretic
complement
A − B means the set
that contains all the
minus; without elements of A that are
set theory not in B.
{1,2,4} − {1,3,4} = {2}
multiplication
3 × 4 means the
times
multiplication of 3 by
arithmetic 4.
7 × 8 = 56
Cartesian
product
×
X×Y means the set of
the Cartesian all ordered pairs with
product of ... the first element of each
{1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
and ...; the direct pair selected from X
product of ... and the second element
and ...
selected from Y.
set theory
cross product
u × v means the cross
product of vectors u
vector algebra and v
cross
(1,2,5) × (3,4,−1) =
(−22, 16, − 2)
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multiplication
3 · 4 means the
multiplication of 3 by
arithmetic 4.
times
·
dot product
u · v means the dot
product of vectors u
vector algebra and v
dot
÷
6 ÷ 3 or 6 ⁄ 3 means the
division of 6 by 3.
2 ÷ 4 = .5
12 ⁄ 4 = 3
arithmetic
plus-minus
plus or minus
±
(1,2,5) · (3,4,−1) = 6
division
divided by
⁄
7 · 8 = 56
arithmetic
6 ± 3 means both 6 + 3 The equation x = 5 ± √4, has two
and 6 - 3.
solutions, x = 7 and x = 3.
plus-minus
10 ± 2 or eqivalently 10 If a = 100 ± 1 mm, then a is ≥ 99 mm
± 20% means the range
and ≤ 101 mm.
measurement from 10 − 2 to 10 + 2.
plus or minus
∓
minus-plus
6 ± (3 5) means both
6 + (3 - 5) and 6 - (3 + cos(x ± y) = cos(x) cos(y)
arithmetic 5).
minus or plus
sin(x) sin(y).
square root
the principal
square root of;
square root
√x means the positive
number whose square is √4 = 2
x.
real numbers
√
|…|
complex square
root
if z = r exp(iφ) is
the complex represented in polar
square root of … coordinates with -π < φ
√(-1) = i
≤ π, then √z = √r exp
square root
(i φ/2).
complex
numbers
absolute value or
|x| means the distance
modulus
along the real line (or
absolute value across the complex
(modulus) of plane) between x and
zero.
numbers
|3| = 3
|–5| = |5|
|i|=1
| 3 + 4i | = 5
Euclidean
distance
Euclidean
distance
|x – y| means the
For x = (1,1), and y = (4,5),
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between;
Euclidean distance
Euclidean norm between x and y.
of
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|x – y| = √([1–4]2 + [1–5]2) = 5
Geometry
Determinant
|A| means the
determinant of determinant of the
Matrix theory matrix A
A single vertical bar is
used to denote
divides
divisibility.
Number Theory a|b means a divides b.
divides
|
Since 15 = 3×5, it is true that 3|15 and
5|15.
factorial
!
factorial
n ! is the product 1 ×
2× ... × n.
4! = 1 × 2 × 3 × 4 = 24
combinatorics
transpose
T
transpose
matrix
operations
Swap rows for columns Aij = (AT)ji
probability
distribution
~
⇒
→
⊃
⇔
X ~ D, means the
random variable X has
has distribution the probability
statistics distribution D.
X ~ N(0,1), the standard normal
distribution
Row equivalence A~B means that B can
is row equivalent be generated by using a
to
series of elementary
Matrix theory row operations on A
material
implication
implies; if …
then
A ⇒ B means if A is
true then B is also true;
if A is false then
nothing is said about B.
→ may mean the same
as ⇒, or it may have the x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x =
meaning for functions 2 is in general false (since x could be −2).
given below.
propositional ⊃ may mean the same
logic, Heyting
as ⇒, or it may have the
algebra
meaning for superset
given below.
material
equivalence
A ⇔ B means A is true
if B is true and A is
if and only if; iff
false if B is false.
x + 5 = y +2 ⇔ x + 3 = y
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↔
propositional
logic
logical negation
¬
˜
∧
not
The statement ¬A is
true if and only if A is
false.
A slash placed through
another operator is the ¬(¬A) ⇔ A
same as "¬" placed in
x ≠ y ⇔ ¬(x = y)
front.
propositional (The symbol ~ has
logic many other uses, so ¬
or the slash notation is
preferred.)
The statement A ∧ B is
logical
conjunction or
true if A and B are both
meet in a lattice true; else it is false.
and; min
For functions A(x) and
propositional B(x), A(x) ∧ B(x) is
logic, lattice used to mean min(A(x),
theory B(x)).
logical
disjunction or
join in a lattice
∨
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or; max
The statement A ∨ B is
true if A or B (or both)
are true; if both are
false, the statement is
false.
n < 4 ∧ n >2 ⇔ n = 3 when n is a
natural number.
n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a
natural number.
For functions A(x) and
propositional B(x), A(x) ∨ B(x) is
logic, lattice
theory used to mean max(A
(x), B(x)).
exclusive or
The statement A ⊕ B is
true when either A or
(¬A) ⊕ A is always true, A ⊕ A is
propositional B, but not both, are
always false.
logic, Boolean true. A B means the
algebra same.
xor
⊕
The direct sum is a
special way of
combining several one
modules into one
direct sum of
general module (the
symbol ⊕ is used, is
only
for logic).
Abstract algebra
direct sum
universal
quantification
Most commonly, for vector spaces U, V,
and W, the following consequence is
used:
U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W =
∅)
∀n∈
: n 2 ≥ n.
for all; for any; ∀ x: P(x) means P(x) is
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∀
∃
for each
true for all x.
predicate logic
existential
quantification
there exists
predicate logic
∃!
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uniqueness
quantification
there exists
exactly one
∃ x: P(x) means there is
at least one x such that ∃ n ∈
P(x) is true.
∃! x: P(x) means there
is exactly one x such
that P(x) is true.
∃! n ∈
: n is even.
: n + 5 = 2n.
predicate logic
:=
≡
:⇔
definition
is defined as
x := y or x ≡ y means x
is defined to be another
name for y
(Some writers use ≡ to
mean congruence).
cosh x := (1/2)(exp x + exp (−x))
A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
P :⇔ Q means P is
everywhere defined to be logically
equivalent to Q.
△ABC △DEF means
triangle ABC is
is congruent to congruent to (has the
same measurements as)
geometry triangle DEF.
congruence
congruence
relation
≡
... is congruent a ≡ b (mod n) means a
to ... modulo ... − b is divisible by n
modular
arithmetic
5 ≡ 11 (mod 3)
set brackets
{,}
{:}
{|}
∅
{a,b,c} means the set
the set of … consisting of a, b, and
set theory c.
set builder
notation
the set of …
such that
= { 1, 2, 3, …}
{x : P(x)} means the set
of all x for which P(x)
{n ∈
is true. {x | P(x)} is the
same as {x : P(x)}.
: n2 < 20} = { 1, 2, 3, 4}
∅ means the set with
no elements. { } means {n ∈
the same.
: 1 < n2 < 4} = ∅
set theory
empty set
the empty set
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{}
∈
∉
⊆
set theory
set membership
is an element of; a ∈ S means a is an
(1/2)−1 ∈
is not an element element of the set S; a ∉
of
S means a is not an
2−1 ∉
everywhere, set element of S.
theory
(subset) A ⊆ B means
every element of A is
also element of B.
subset
is a subset of
⊂
⊇
superset
(A ∩ B) ⊆ A
⊂
⊂
A ⊇ B means every
element of B is also
element of A.
A ⊃ B means A ⊇ B but (A ∪ B) ⊇ B
A ≠ B.
⊃
(Some writers use the
set theory symbol ⊃ as if it were
the same as ⊇.)
set-theoretic
union
∪
(proper subset) A ⊂ B
means A ⊆ B but A ≠ B.
(Some writers use the
set theory symbol ⊂ as if it were
the same as ⊆.)
is a superset of
⊃
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the union of …
and
(exclusive) A ∪ B
means the set that
contains all the
elements from A, or all
the elements from B,
but not both.
"A or B, but not both."
A ⊆ B ⇔ (A ∪ B) = B (inclusive)
(inclusive) A ∪ B means
the set that contains all
union
the elements from A, or
all the elements from B,
or all the elements from
set theory both A and B.
"A or B or both".
set-theoretic
intersection
A ∩ B means the set
that contains all those
intersected with; elements that A and B
{x ∈
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∩
intersect
have in common.
set theory
symmetric
difference
∆
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symmetric
difference
A∆B means the set of
elements in exactly one {1,5,6,8} ∆ {2,5,8} = {1,2,6}
of A or B.
set theory
set-theoretic
complement
A B means the set
that contains all those
minus; without elements of A that are
set theory not in B.
function
application
of
()
{1,2,3,4}
{3,4,5,6} = {1,2}
f(x) means the value of
the function f at the
If f(x) := x2, then f(3) = 32 = 9.
element x.
set theory
precedence
grouping
parentheses
Perform the operations
inside the parentheses (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
first.
everywhere
function arrow
f:X→Y
o
f: X → Y means the
function f maps the set
set theory,type X into the set Y.
theory
from … to
Let f:
→
be defined by f(x) := x2.
function
composition
fog is the function, such if f(x) := 2x, and g(x) := x + 3, then (fog)
composed with that (fog)(x) = f(g(x)).
(x) = 2(x + 3).
set theory
natural numbers
N
N
numbers
integers
Z
Z
numbers
N means { 1, 2, 3, ...},
but see the article on
natural numbers for a
different convention.
= {|a| : a ∈
means {..., −3, −2,
−1, 0, 1, 2, 3, ...} and
+ means {1, 2, 3,
...} = .
= {p, -p : p ∈
rational numbers
Q
Q
means {p/q : p ∈
q ∈ }.
,
, a ≠ 0}
} ∪ {0}
3.14000... ∈
π
numbers
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real numbers
R
R
means the set of real
numbers.
numbers
complex
numbers
C
means {a + b i :
a,b ∈ }.
π∈
√(−1)
i = √(−1) ∈
numbers
C
C can be any number,
most likely unknown;
usually occurs when
C
calculating
integral calculus antiderivatives.
arbitrary
constant
real or complex
numbers
K
K
if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² +
C, where F'(x) = f(x)
because
K means the statement
holds substituting K for
R and also for C.
and
linear algebra
.
∞ is an element of the
extended number line
infinity
that is greater than all
real numbers; it often
numbers occurs in limits.
infinity
∞
limx 0 1/|x| = ∞
→
norm
||…||
norm of
length of
|| x || is the norm of the
element x of a normed
vector space.
|| x + y || ≤ || x || + || y ||
linear algebra
summation
∑
sum over …
from … to … of
arithmetic
means a1 + a2 +
… + an.
= 1 2 + 2 2 + 3 2 + 42
= 1 + 4 + 9 + 16 = 30
product
∏
product over …
from … to … of
means
a a ···a .
arithmetic 1 2 n
= (1+2)(2+2)(3+2)(4+2)
= 3 × 4 × 5 × 6 = 360
Cartesian
product
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the Cartesian
product of; the
direct product of
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means the set of
all (n+1)-tuples
set theory
(y0, …, yn).
coproduct
coproduct over
… from … to …
of
category theory
derivative
′
… prime
derivative of
•
f ′(x) is the derivative of
the function f at the
point x, i.e., the slope of
the tangent to f at x.
The dot notation
indicates a time
derivative. That is
If f(x) := x2, then f ′(x) = 2x
.
calculus
indefinite
integral or
antiderivative
indefinite
integral of
∫
∫ f(x) dx means a
function whose
derivative is f.
∫x2 dx = x3/3 + C
the antiderivative
of
calculus
definite integral
∫ab f(x) dx means the
integral from … signed area between the
to … of … with x-axis and the graph of ∫0b x2 dx = b3/3;
respect to
the function f between
calculus x = a and x = b.
∇f (x1, …, xn) is the
vector of partial
If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x,
del, nabla,
f
x
derivatives
(
/
,
…,
∂ ∂ 1
gradient of
2z)
calculus ∂f / ∂xn).
gradient
∇
partial
differential
∂
With f (x1, …, xn),
∂f/∂x is the derivative
i
2
of f with respect to xi, If f(x,y) := x y, then ∂f/∂x = 2xy
partial, d
with all other variables
calculus kept constant.
boundary
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∂M means the boundary
∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
topology of M
boundary of
perpendicular
⊥
x ⊥ y means x is
is perpendicular perpendicular to y; or
to
more generally x is
geometry orthogonal to y.
If l ⊥ m and m ⊥ n then l || n.
bottom element
the bottom
element
x = ⊥ means x is the
smallest element.
∀x : x ∧ ⊥ = ⊥
lattice theory
parallel
||
is parallel to
x || y means x is parallel
If l || m and m ⊥ n then l ⊥ n.
to y.
geometry
A B means the
sentence A entails the
A
entails
sentence B, that is
every model in which A
model theory is true, B is also true.
entailment
A ∨ ¬A
inference
infers or is
derived from
x y means y is
derived
from x.
propositional
logic, predicate
logic
A→B
¬B → ¬A
normal subgroup
is a normal
subgroup of
N G means that N is
a normal subgroup of
group G.
Z(G)
G
group theory
quotient group
G/H means the quotient
{0, a, 2a, b, b+a, b+2a} / {0, b} = {{0,
of group G modulo its
b}, {a, b+a}, {2a, b+2a}}
group theory subgroup H.
mod
/
quotient set
A/~ means the set of all If we define ~ by x~y ⇔ x-y∈Z, then
mod
~ equivalence classes in
R/~ = {{x+n : n∈Z} : x ∈ (0,1]}
A.
set theory
isomorphism
G ≈ H means that
is isomorphic to group G is isomorphic
group theory to group H
≈
Q / {1, −1} ≈ V,
where Q is the quaternion group and V is
the Klein four-group.
approximately
equal
x ≈ y means x is
is approximately approximately equal to π ≈ 3.14159
equal to
y
everywhere
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same order of
magnitude
~
m ~ n, means the
roughly similar quantities m and n have 2 ~ 5
the general size.
poorly
8 × 9 ~ 100
approximates (Note that ~ is used for
Approximation an approximation that but π2 ≈ 10
theory is poor, otherwise use
≈ .)
〈,〉
inner product
〈x,y〉 means the
inner product of x and y The standard inner product between two
as defined in an inner vectors x = (2, 3) and y = (−1, 5) is:
product space.
〈x, y〉 = 2×−1 + 3×5 = 13
(|)
<,>
inner product of
·
For spatial vectors, the A:B =
∑ AijBij
dot product notation,
x·y is common.
i,j
For matricies, the colon
notation may be used.
vector algebra
:
tensor product
tensor product of
V U means the tensor {1, 2, 3, 4} {1,1,2} =
product of V and U.
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
linear algebra
*
convolution
convolution
mean
overbar
statistics
f * g means the
convolution of f and g.
is the mean (average
value of xi).
means equal by
definition. When is
used, equality is not
true generally, but
equal by
rather equality is true
definition
under certain
assumptions that are
everywhere taken in context.
.
delta equal to
.
See also
Mathematical alphanumeric symbols
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Table of logic symbols
Physical constants
Variables commonly used in physics
ISO 31-11
External links
Jeff Miller: Earliest Uses of Various Mathematical Symbols
(http://members.aol.com/jeff570/mathsym.html)
TCAEP - Institute of Physics (http://www.tcaep.co.uk/science/symbols/maths.htm)
GIF and PNG Images for Math Symbols (http://us.metamath.org/symbols/symbols.html)
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