Jump to content

User:Alksentrs/Table of mathematical symbols (grouped by kind)

From Wikipedia, the free encyclopedia

Based on Table of mathematical symbols. (Version: 18:12, 13 Oct 2008)

Relations

[edit]
Symbol Name Explanation Examples
Read as
Category
=
equality x = y means x and y represent the same thing or value. 1 + 1 = 2
is equal to; equals
everywhere


<>

!=
inequation x ≠ y means that x and y do not represent the same thing or value.

(The symbols != and <> are primarily from computer science. They are avoided in mathematical texts.)
1 ≠ 2
is not equal to; does not equal
everywhere
<

>



strict inequality x < y means x is less than y.

x > y means x is greater than y.

x ≪ y means x is much less than y.

x ≫ y means x is much greater than y.
3 < 4
5 > 4

0.003 ≪ 1000000

is less than, is greater than, is much less than, is much greater than
order theory

<=


>=
inequality x ≤ y means x is less than or equal to y.

x ≥ y means x is greater than or equal to y.

(The symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.)
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is less than or equal to, is greater than or equal to
order theory
cover x <• y means that x is covered by y. {1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment.
is covered by
order theory
proportionality yx means that y = kx for some constant k. if y = 2x, then yx
is proportional to; varies as
everywhere
~
probability distribution X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
has distribution
statistics
Row equivalence A~B means that B can be generated by using a series of elementary row operations on A
is row equivalent to
Matrix theory
same order of magnitude m ~ n means the quantities m and n have the same order of magnitude, or general size.

(Note that ~ is used for an approximation that is poor, otherwise use ≈ .)
2 ~ 5

8 × 9 ~ 100

but π2 ≈ 10
roughly similar; poorly approximates
Approximation theory
asymptotically equivalent f ~ g means . x ~ x+1

is asymptotically equivalent to
Asymptotic analysis
Equivalence relation a ~ b means (and equivalently ). 1 ~ 5 mod 4

are in the same equivalence class
everywhere
approximately equal x ≈ y means x is approximately equal to y. π ≈ 3.14159
is approximately equal to
everywhere
isomorphism G ≈ H means that group G is isomorphic to group H. Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group.
is isomorphic to
group theory
normal subgroup N ◅ G means that N is a normal subgroup of group G. Z(G) ◅ G
is a normal subgroup of
group theory
ideal I ◅ R means that I is an ideal of ring R. (2) ◅ Z
is an ideal of
ring theory
:=



:⇔
definition x := y or x ≡ y means x is defined to be another name for y

(Some writers useto mean congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.
cosh x := (1/2)(exp(x)+exp(-x))
is defined as
everywhere
delta equal to means equal by definition. When is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡. .
equal by definition
everywhere
congruence △ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
is congruent to
geometry
congruence relation a ≡ b (mod n) means a − b is divisible by n 5 ≡ 11 (mod 3)
... is congruent to ... modulo ...
modular arithmetic


set membership a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S. (1/2)−1 ∈ ℕ

2−1 ∉ ℕ
is an element of; is not an element of
everywhere, set theory


subset (subset) A ⊆ B means every element of A is also element of B.

(proper subset) A ⊂ B means A ⊆ B but A ≠ B.

(Some writers use the symbol as if it were the same as ⊆.)
(A ∩ B) ⊆ A

ℕ ⊂ ℚ

ℚ ⊂ ℝ
is a subset of
set theory


superset A ⊇ B means every element of B is also element of A.

A ⊃ B means A ⊇ B but A ≠ B.

(Some writers use the symbol as if it were the same as .)
(A ∪ B) ⊇ B

ℝ ⊃ ℚ
is a superset of
set theory
<:
subtype T1 <: T2 means that T1 is a subtype of T2. If S <: T and T <: U then S <: U (transitivity).
is a subtype of
type theory
||
parallel x || y means x is parallel to y. If l || m and m ⊥ n then l ⊥ n. In physics this is also used to express
is parallel to
geometry, physics
incomparability x || y means x is incomparable to y. {1,2} || {2,3} under set containment.
is incomparable to
order theory
exact divisibility pa || n means pa exactly divides n (i.e. pa divides n but pa+1 does not). 23 || 360.
exactly divides
number theory
perpendicular 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.
is perpendicular to
geometry
coprime x ⊥ y means x has no factor in common with y. 34  ⊥  55.
is coprime to
number theory
comparability xy means that x is comparable to y. {eπ} ⊥ {1, 2, e, 3, π} under set containment.
is comparable to
Order theory

Nullary operators (constants)

[edit]
Symbol Name Explanation Examples
Read as
Category


{ }
empty set ∅ means the set with no elements. { } means the same. {n ∈ ℕ : 1 < n2 < 4} = ∅
the empty set
set theory


N
natural numbers N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention. ℕ = {|a| : a ∈ ℤ, a ≠ 0}
N
numbers


Z
integers ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ+ means {1, 2, 3, ...} = ℕ. ℤ = {p, -p : p ∈ ℕ} ∪ {0}
Z
numbers


Q
rational numbers ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. 3.14000... ∈ ℚ

π ∉ ℚ
Q
numbers


R
real numbers ℝ means the set of real numbers. π ∈ ℝ

√(−1) ∉ ℝ
R
numbers


C
complex numbers ℂ means {a + b i : a,b ∈ ℝ}. i = √(−1) ∈ ℂ
C
numbers
arbitrary constant C can be any number, most likely unknown; usually occurs when calculating antiderivatives. if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C, where F'(x) = f(x)
C
integral calculus
𝕂

K
real or complex numbers K means the statement holds substituting K for R and also for C.

because


and

.
K
linear algebra
infinity ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.
infinity
numbers
top element x = ⊤ means x is the largest element. x : x ∨ ⊤ = ⊤
the top element
lattice theory
top type The top or universal type; every type in the type system of interest is a subtype of top. ∀ types T, T <: ⊤
the top type; top
type theory
bottom element x = ⊥ means x is the smallest element. x : x ∧ ⊥ = ⊥
the bottom element
lattice theory
bottom type 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
the bottom type; bot
type theory

Unary operators

[edit]
Symbol Name Explanation Examples
Read as
Category
+
positive sign +a means the same as a. +∞ is the positive infinity. +1
positive; plus
arithmetic
negative sign −3 means the negative of the number 3. −(−5) = 5
negative; minus; the opposite of
arithmetic
square root means the positive number whose square is .
the principal square root of; square root
real numbers
complex square root if is represented in polar coordinates with , then .
the complex square root of …

square root
complex numbers
|…|
absolute value or modulus |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
absolute value (modulus) of
numbers
Euclidean distance |x – y| means the Euclidean distance between x and y. For x = (1,1), and y = (4,5),
|x – y| = √([1–4]2 + [1–5]2) = 5
Euclidean distance between; Euclidean norm of
Geometry
Determinant |A| means the determinant of the matrix A
determinant of
Matrix theory
Cardinality |X| means the cardinality of the set X. |{3, 5, 7, 9}| = 4.
cardinality of
set theory
!
factorial n! is the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
T

tr
transpose Swap rows for columns If then .
transpose
matrix operations
||…||
norm || x || is the norm of the element x of a normed vector space. || x  + y || ≤  || x ||  +  || y ||
norm of

length of
linear algebra
orthogonal complement If W is a subspace of the inner product space V, then W is the set of all vectors in V orthogonal to every vector in W. Within , .
orthogonal/perpendicular complement of; perp
linear algebra


mean (often read as "x bar") is the mean (average value of ). .
overbar, … bar
statistics


complex conjugate is the complex conjugate of z.
conjugate
complex numbers

Binary operators

[edit]
Symbol Name Explanation Examples
Read as
Category
+
addition 4 + 6 means the sum of 4 and 6. 2 + 7 = 9
plus
arithmetic
disjoint union A1 + A2 means the disjoint union of sets A1 and A2. A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒
A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
the disjoint union of ... and ...
set theory
subtraction 9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5
minus
arithmetic
set-theoretic complement A − B means the set that contains all the elements of A that are not in B.

∖ can also be used for set-theoretic complement as described below.
{1,2,4} − {1,3,4}  =  {2}
minus; without
set theory
×
multiplication 3 × 4 means the multiplication of 3 by 4. 7 × 8 = 56
times
arithmetic
Cartesian product X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
the Cartesian product of ... and ...; the direct product of ... and ...
set theory
cross product u × v means the cross product of vectors u and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
cross
vector algebra
·
multiplication 3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56
times
arithmetic
dot product u · v means the dot product of vectors u and v (1,2,5) · (3,4,−1) = 6
dot
vector algebra
÷

division 6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = .5

12 ⁄ 4 = 3
divided by
arithmetic
quotient group G / H means the quotient of group G modulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
mod
group theory
quotient set A/~ means the set of all ~ equivalence classes in A. If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then
ℝ/~ = {{x + n : n ∈ ℤ} : x ∈ (0,1]}
mod
set theory
±
plus-minus 6 ± 3 means both 6 + 3 and 6 - 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
plus or minus
arithmetic
plus-minus 10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
plus or minus
measurement
minus-plus 6 ± (3 ∓ 5) means both 6 + (3 - 5) and 6 - (3 + 5). cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
minus or plus
arithmetic
set-theoretic union A ∪ B means the set of those elements which are either in A, or in B, or in both. A ⊆ B  ⇔  (A ∪ B) = B (inclusive)
the union of … or …

union
set theory
set-theoretic intersection A ∩ B means the set that contains all those elements that A and B have in common. {x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
intersected with; intersect
set theory
symmetric difference A ∆ B means the set of elements in exactly one of A or B. {1,5,6,8} ∆ {2,5,8} = {1,2,6}
symmetric difference
set theory
set-theoretic complement A ∖ B means the set that contains all those elements of A that are not in B.

− can also be used for set-theoretic complement as described above.
{1,2,3,4} ∖ {3,4,5,6} = {1,2}
minus; without
set theory
o
function composition fog is the function, such that (fog)(x) = f(g(x)). if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).
composed with
set theory
〈,〉

( | )

< , >

·

:
inner product x,y〉 means the inner product of x and y as defined in an inner product space.

For spatial vectors, the dot product notation, x·y is common.
For matrices, the colon notation may be used.
Note that the notation 〈x, y〉 may be ambiguous: it could mean the inner product or the linear span.

The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is:
〈x, y〉 = 2 × −1 + 3 × 5 = 13

inner product of
linear algebra
tensor product, tensor product of modules means the tensor product of V and U. means the tensor product of modules V and U over the ring R. {1, 2, 3, 4} ⊗ {1, 1, 2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
tensor product of
linear algebra
*
convolution f * g means the convolution of f and g.
convolution, convolved with
functional analysis

n-ary operators

[edit]
Symbol Name Explanation Examples
Read as
Category
{ , }
set brackets {a,b,c} means the set consisting of a, b, and c. ℕ = { 1, 2, 3, …}
the set of …
set theory
( , )
tuple An ordered list (or sequence) of values.

Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval.

(a, b) is an ordered pair (or 2-tuple).

(a, b, c) is an ordered triple (or 3-tuple).

( ) is the empty tuple (or 0-tuple).

tuple; n-tuple; ordered pair/triple/etc.
everywhere
〈 , 〉

< , >

Sp
linear span If u,v,wV then 〈u, v, w〉 means the span of u, v and w, as does Sp(u, v, w). That is, it is the intersection of all subspaces of V which contain u, v and w.

Note that the notation 〈u, v〉 may be ambiguous: it could mean the inner product or the span.

(linear) span of; linear hull of
linear algebra

Logic

[edit]
Symbol Name Explanation Examples
Read as
Category
therefore Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
therefore
everywhere
because Sometimes used in proofs before reasoning. 3331 is prime ∵ it has no positive factors other than itself and one.
because
everywhere




material implication AB 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 meaning for functions given below.

⊃ may mean the same as ⇒, or it may have the meaning for superset given below.
x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2).
implies; if … then
propositional logic, Heyting algebra


material equivalence A ⇔ B means A is true if B is true and A is false if B is false. x + 5 = y +2  ⇔  x + 3 = y
if and only if; iff
propositional logic
¬

˜
logical negation The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.

(The symbol ~ has many other uses, so ¬ or the slash notation is preferred.)
¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y)
not
propositional logic
logical conjunction or meet in a lattice The statement AB is true if A and B are both true; else it is false.

For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).

(Old notation) uv means the cross product of vectors u and v.
n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.
and; min
propositional logic, lattice theory
logical disjunction or join in a lattice The statement AB is true if A or B (or both) are true; if both are false, the statement is false.

For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).
n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
or; max
propositional logic, lattice theory



exclusive or The statement AB is true when either A or B, but not both, are true. AB means the same. A) ⊕ A is always true, AA is always false.
xor
propositional logic, Boolean algebra
direct sum The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).

Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = VW ⇔ (U = V + W) ∧ (VW = {0})
direct sum of
Abstract algebra
universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n.
for all; for any; for each
predicate logic
existential quantification ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even.
there exists
predicate logic
∃!
uniqueness quantification ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ ℕ: n + 5 = 2n.
there exists exactly one
predicate logic
entailment A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. A ⊧ A ∨ ¬A
entails
model theory
inference x ⊢ y means y is derivable from x. A → B ⊢ ¬B → ¬A
infers or is derived from
propositional logic, predicate logic

The rest

[edit]
Symbol Name Explanation Examples
Read as
Category
|
divides A single vertical bar is used to denote divisibility.
a|b means a divides b.
Since 15 = 3×5, it is true that 3|15 and 5|15.
divides
Number theory
Conditional probability A single vertical bar is used to describe the probability of an event given another event happening.
P(A|B) means a given b.
If P(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4
Given
Probability
{ : }

{ | }
set builder notation {x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4}
the set of … such that
set theory
( )
function application f(x) means the value of the function f at the element x. If f(x) := x2, then f(3) = 32 = 9.
of
set theory
precedence grouping Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
parentheses
everywhere
f:XY
function arrow fX → Y means the function f maps the set X into the set Y. Let f: ℤ → ℕ be defined by f(x) := x2.
from … to
set theory,type theory
[ , ]
closed interval . [0,1]
closed interval
order theory
( , )

] , [
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.

(4,18)
open interval
order theory
( , ]

] , ]
left-open interval . (−1, 7] and (−∞, −1]
half-open interval; left-open interval
order theory
[ , )

[ , [
right-open interval . [4, 18) and [1, +∞)
half-open interval; right-open interval
order theory
summation

means a1 + a2 + … + an.

= 12 + 22 + 32 + 42 

= 1 + 4 + 9 + 16 = 30
sum over … from … to … of
arithmetic
product

means a1a2···an.

= (1+2)(2+2)(3+2)(4+2)

= 3 × 4 × 5 × 6 = 360
product over … from … to … of
arithmetic
Cartesian product

means the set of all (n+1)-tuples

(y0, …, yn).

the Cartesian product of; the direct product of
set theory
coproduct
coproduct over … from … to … of
category theory


derivative 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
… prime

derivative of
calculus
indefinite integral or antiderivative ∫ f(x) dx means a function whose derivative is f. x2 dx = x3/3 + C
indefinite integral of

the antiderivative of
calculus
definite integral ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. ab x2  dx = b3/3 - a3/3;
integral from … to … of … with respect to
calculus
contour integral or closed line integral Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∰.

The contour integral can also frequently be found with a subscript capital letter C, ∮C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮S, is used to denote that the integration is over a closed surface.

If C is a Jordan curve about 0, then .
contour integral of
calculus
gradient f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn). If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
del, nabla, gradient of
vector calculus
divergence If , then .
del dot, divergence of
vector calculus
curl
If , then .
curl of
vector calculus
partial differential With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant. If f(x,y) := x2y, then ∂f/∂x = 2xy
partial, d
calculus
boundary M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
boundary of
topology
δ
Dirac delta function δ(x)
Dirac delta of
hyperfunction
Kronecker delta δij
Kronecker delta of
hyperfunction