! style="text-align:right;" |Category{{row of table of mathematical symbols
|symbol ==
|tex =
=
|rowspan =1
|name =
equality |readas =is equal to; equals
|category =everywhere
|explain =
x =
y means
x and
y represent the same thing or value.
|examples =1 + 1 = 2
| symbol =≠
| tex =
≠
| rowspan =1
| name =
inequation | readas =is not equal to; does not equal
| category =everywhere
| explain =
x ≠
y means that
x and
y do not represent the same thing or value.
(The forms !=, /= or <> are generally used in programming languages where ease of typing and use of
ASCII text is preferred.)
| examples =2 + 2 ≠ 5
| symbol =<
>
| tex =
<
>
| rowspan =2
| name =
strict inequality | readas =is less than, is greater than
| category =
order theory | explain =
x <
y means
x is less than
y.
x >
y means
x is greater than
y.
| examples =3 < 4
5 > 4
| name =
proper subgroup | readas =is a proper subgroup of
| category =
group theory | explain =
H <
G means
H is a proper subgroup of
G.
| examples =5
Z <
Z A
3 3
| symbol =≪
≫
| tex =
ll
gg
| rowspan =2
| name =(very)
strict inequality | readas =is much less than, is much greater than
| category =
order theory | explain =
x ≪
y means
x is much less than
y.
x ≫
y means
x is much greater than
y.
| examples =0.003 ≪ 1000000
| name =asymptotic comparison
| readas =of smaller (greater) order than
| category =
analytic number theory | explain =
f ≪
g means the growth of
f is asymptotically bounded by
g.
(
This is I. M. Vinogradov's notation. Another notation is the Big O notation, which looks like f = O(
g).)
| examples =
x ≪ e
x
| symbol =≤
≥
| tex =
≤
≥
| rowspan =3
| name =
inequality | readas =is less than or equal to, is greater than or equal to
| category =
order theory | explain =
x ≤
y means
x is less than or equal to
y.
x ≥
y means
x is greater than or equal to
y.
(The forms <= and >= are generally used in programming languages where ease of typing and use of
ASCII text is preferred.)
| examples =3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
| name =
subgroup | readas =is a subgroup of
| category =
group theory | explain =
H ≤
G means
H is a subgroup of
G.
| examples =
Z ≤
Z A
3 ≤S
3
| name =
reduction | readas =is reducible to
| category =
computational complexity theory | explain =
A ≤
B means the
problem A can be reduced to the problem
B. Subscripts can be added to the ≤ to indicate what kind of reduction.
| examples =If
eξsts f ∈ F . ∀ x ∈ N . x ∈ A ⇔ f(x) ∈ B
then
}}{{row of table of mathematical symbols
| symbol =≺
| tex =
(rec
| rowspan =1
| name =
Karp reduction | readas =is Karp reducible to; is polynomial-time many-one reducible to
| category =
computational complexity theory | explain =
L1 ≺
L2 means that the problem
L1 is Karp reducible to
L2.
(1) | examples =If
L1 ≺
L2 and
L2 ∈
P, then
L1 ∈
P.
| symbol =∝
| tex =
(ro(→
| rowspan =1
| name =
proportionality | readas =is proportional to; varies as
| category =everywhere
| explain =
y ∝
x means that
y =
kx for some constant
k.
| examples =if
y = 2
x, then
y ∝
x
| symbol =+
| tex =
+
| rowspan =2
| name =
addition | readas =
plus; add
| category =
arithmetic | explain =4 + 6 means the sum of 4 and 6.
| examples =2 + 7 = 9
| name =
disjoint union | readas =the disjoint union of ... and ...
| category =
set theory | explain =
A1 +
A2 means the disjoint union of sets
A1 and
A2.
| examples =
A1 = {3, 4, 5, 6} ∧
A2 = {7, 8, 9, 10} ⇒
A1 +
A2 = {(3,1), (4,1), (5,1), (6,1), (7,2), (8,2), (9,2), (10,2)}
| symbol =−
| tex =
-
| rowspan =3
| name =
subtraction | readas =
minus; take; subtract
| category =
arithmetic | explain =9 − 4 means the subtraction of 4 from 9.
| examples =8 − 3 = 5
| name =
negative sign | readas =negative; minus; the opposite of
| category =
arithmetic | explain =−3 means the negative of the number 3.
| examples =−(−5) = 5
| name =
set-theoretic complement | readas =minus; without
| category =
set theory | explain =
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.)
| examples ={1,2,4} − {1,3,4} = {2}
| symbol =×
| tex =
⋅
| rowspan =4
| name =
multiplication | readas =times; multiplied by
| category =
arithmetic | explain =3 × 4 means the multiplication of 3 by 4.
| examples =7 × 8 = 56
| name =
Cartesian product | readas =the Cartesian product of ... and ...; the direct product of ... and ...
| category =
set theory | explain =
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.
| examples ={1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
| name =
cross product | readas =cross
| category =
vector algebra | explain =
u ×
v means the cross product of
vectors
u and
v | examples =(1,2,5) × (3,4,−1) =
(−22, 16, − 2)
| name =
group of units | readas =the group of units of
| category =
ring theory | explain =
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 below, or U(
R).
| examples =
beg∈align (Z / 5Z)⋅ & = [1] [2] [3] [4] & ≅ C4 endalign
| symbol =·
| tex =
cderiv(⋅)
| rowspan =2
| name =
multiplication | readas =times; multiplied by
| category =
arithmetic | explain =3 · 4 means the multiplication of 3 by 4.
| examples =7 · 8 = 56
| symbol =÷
⁄
| tex =
div
/
| rowspan =3
| name =
division (
Obelus)
| readas =divided by; over
| category =
arithmetic | explain =6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.
| examples =2 ÷ 4 = .5
12 ⁄ 4 = 3
| name =
quotient group | readas =mod
| category =
group theory | explain =
G /
H means the quotient of group
G modulo its subgroup
H.
| examples ={0,
a, 2
a,
b,
b+
a,
b+2
a} / {0,
b} =
{{0,
b}, {
a,
b+
a}, {2
a,
b+2
a}}
| name =quotient set
| readas =mod
| category =
set theory | explain =
A/~ means the set of all ~
equivalence classes in
A.
| examples =If we define ~ by x ~ y ⇔ x − y ∈ {{Unicode|ℤ}}, then
{{Unicode|ℝ}}/~ =
{x +
n :
n ∈ {{Unicode|ℤ}} : x ∈ (0,1]}
| symbol =±
| tex =
&(lusmn;
| rowspan =2
| name =
plus-minus | readas =plus or minus
| category =
arithmetic | explain =6 ± 3 means both 6 + 3 and 6 − 3.
| examples =The equation
x = 5 ± √4, has two solutions,
x = 7 and
x = 3.
| name =
plus-minus | readas =plus or minus
| category =
measurement | explain =10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.
| examples =If
a = 100 ± 1
mm, then
a ≥ 99 mm and
a ≤ 101 mm.
| symbol ={{Unicode|∓{edih}
| tex =
&(lusmn;
| rowspan =1
| name =
minus-plus | readas =minus or plus
| category =
arithmetic | explain =6 ± (3 {{Unicode|∓}} 5) means both 6 + (3 − 5) and 6 − (3 + 5).
| examples =cos(
x ±
y) = cos(
x) cos(
y) {{Unicode|∓}} sin(
x) sin(
y).
| symbol =√
| tex =
surd
√
| rowspan =2
| name =
square root | readas =the (principal) square root of
| category =
real numbers | explain =
√x
means the positive number whose square is
x
.
| examples =
√4=2
| symbol =
|…| | tex =
|| lderiv(⋅)s ||
| rowspan =4
| name =
absolute value or
modulus | readas =absolute value of; modulus of
| category =
numbers
| explain =
|x| means the distance along the
real line (or across the
complex plane) between
x and
zero.
| examples =
|3| = 3
|–5| =
|5| = 5
| i | = 1
| 3 + 4
i | = 5
| name =
Euclidean distance | readas =Euclidean distance between; Euclidean norm of
| category =
geometry | explain =
|x –
y| means the Euclidean distance between
x and
y.
| examples =For
x = (1,1), and
y = (4,5),
|x –
y| = √([1–4]
2 + [1–5]
2) = 5
| name =
determinant | readas =determinant of
| category =
matrix theory | explain =
|A| means the determinant of the matrix
A | examples =
| name =
cardinality | readas =cardinality of; size of; order of
| category =
set theory | explain =
|X| means the cardinality of the set
X.
(#
may be used instead as described below.)
| examples =
|{3, 5, 7, 9}| = 4.
| symbol =
||…|| | tex =
|| lderiv(⋅)s ||
| rowspan =2
| name =
norm | readas =norm of; length of
| category =
linear algebra | explain =
|| x || means the
norm of the element
x of a
normed vector space.
(2) | examples =
|| x +
y || ≤
|| x || +
|| y ||
| name =
nearest integer function | readas =nearest integer to
| category =
numbers
| explain =
||x|| means the nearest integer to
x.
(
This may also be written [
x], ⌊
x⌉, nint(
x)
or Round(
x).)
| examples =
||1|| = 1, ||1.6|| = 2, ||−2.4|| = −2, ||3.49|| = 3
| symbol =∣
∤
| tex =
mid
nmid
| rowspan =3
| name =
divisor,
divides | readas =divides
| category =
number theory | explain =
a|b means
a divides
b.
a∤
b means
a does not divide
b.
(
This symbol can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar | character can be used.)
| examples =Since 15 = 3×5, it is true that 3
|15 and 5
|15.
| name =
conditional probability | readas =given
| category =
probability | explain =
P(
A|B) means the probability of the event
a occurring given that
b occurs.
| examples =if X is a uniformly random day of the year
P(X is May 25
| X is in May) = 1/31
| name =
restriction | readas =restriction of … to …; restricted to
| category =
set theory | explain =
f|A means the function
f restricted to the set
A, that is, it is the function with
domain A ∩ dom(
f) that agrees with
f.
| examples =The function
f :
R →
R defined by
f(
x) =
x2 is not injective, but
f|R+ is injective.
| symbol =
|| | tex =
||
| rowspan =3
| name =
parallel | readas =is parallel to
| category =
geometry | explain =
x || y means
x is parallel to
y.
| examples =If
l || m and
m ⊥
n then
l ⊥
n.
| name =
incomparability | readas =is incomparable to
| category =
order theory | explain =
x || y means
x is incomparable to
y.
| examples ={1,2}
|| {2,3} under set containment.
| name =exact
divisibility | readas =exactly divides
| category =
number theory | explain =
pa || n means
pa exactly divides
n (i.e.
pa divides
n but
pa+1 does not).
| examples =2
3 || 360.
| symbol =# | tex =
#
| rowspan =2
| name =
cardinality | readas =cardinality of; size of; order of
| category =
set theory | explain =#
X means the cardinality of the set
X.
(
|…| may be used instead as described above.)
| examples =#{4, 6, 8} = 3
| name =
connected sum | readas =connected sum of; knot sum of; knot composition of
| category =
topology,
knot theory | explain =
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.
| examples =
A#
Sm is
homeomorphic to
A, for any manifold
A, and the sphere
Sm.
| symbol =ℵ
| tex =
a≤(h
| rowspan =1
| name =
aleph number | readas =aleph
| category =
set theory | explain =ℵ
α represents an infinite cardinality (specifically, the
α-th one, where
α is an ordinal).
| examples =
|ℕ| = ℵ
0, which is called aleph-null.
| symbol =ℶ
| tex =
beth
| rowspan =1
| name =
beth number | readas =beth
| category =
set theory | explain =ℶ
α represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ).
| examples =
beth1 = ||P(N)|| = 2a≤(h0.
| symbol =𝔠
| tex =
Fraktur c
| rowspan =1
| name =
cardinality of the continuum | readas =cardinality of the continuum; cardinality of the real numbers; c;
| category =
set theory | explain =The cardinality of
R
is denoted by
|| R||
or by the symbol
Fraktur c
(a lowercase
Fraktur letter C).
| examples =
Fraktur c = beth1
| symbol =:
| tex =
:
| rowspan =3
| name =such that
| readas =such that; so that
| category =everywhere
| explain =: means “such that”, and is used in proofs and the
set-builder notation (
described below).
| examples =∃
n ∈ ℕ:
n is even.
| name =
field extension | readas =extends; over
| category =
field theory | explain =
K :
F means the field
K extends the field
F.
This may also be written as K ≥
F.
| examples =ℝ : ℚ
| name =
inner product of matrices
| readas =inner product of
| category =
linear algebra | explain =
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
Bra-ket notation.
| examples =
A:B = Σij AijBij
| symbol =!
| tex =
| rowspan =2
| name =
factorial | readas =factorial
| category =
combinatorics | explain =
n! means the product 1 × 2 × ... ×
n.
| examples =4! = 1 × 2 × 3 × 4 = 24
| name =
logical negation | readas =not
| category =
propositional logic | explain =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 !
is primarily from computer science. It is avoided in mathematical texts, where the notation ¬
A is preferred.)
| examples =!(!
A) ⇔
A x ≠
y ⇔ !(
x =
y)
1&2
0&0
end{bmatrix}{edih}{{row of table of mathematical symbols
}}{{row of table of mathematical symbols
}}{{row of table of mathematical symbols
| name =
equivalence relation | readas =are in the same equivalence class
| category =everywhere
| explain =
a ~
b means
b ∈ [a]
(and equivalently
a ∈ [b]
).
| examples =1 ~ 5 mod 4
| symbol =≈
| tex =
&asy&(lusmn;;
| rowspan =2
| name =approximately equal
| readas =is approximately equal to
| category =everywhere
| explain =
x ≈
y means
x is approximately equal to
y.
| examples =π ≈ 3.14159
| name =
isomorphism | readas =is isomorphic to
| category =
group theory | explain =
G ≈
H means that group
G is isomorphic (structurally identical) to group
H.
({{Unicode|≅}}
can also be used for isomorphic, as described below.)
| examples =
Q / {1, −1} ≈
V,
where
Q is the
quaternion group and
V is the
Klein four-group.
| symbol =≀ | tex =
wr
| rowspan =1
| name =
wreath product | readas =wreath product of … by …
| category =
group theory | explain =
A ≀
H means the wreath product of the group
A by the group
H.
This may also be written A wr H.
| examples =
Sn wr Z2
is isomorphic to the
automorphism group of the
complete bipartite graph on (
n,
n) vertices.
| symbol ={{Unicode|◅{edih}
{{Unicode|▻}}
| tex =
triang≤(&nbs(;
triang≤&nbs(;)
| rowspan =3
| name =
normal subgroup | readas =is a normal subgroup of
| category =
group theory | explain =
N {{Unicode|◅}}
G means that
N is a normal subgroup of group
G.
| examples =
Z(
G) {{Unicode|◅}}
G
| name =
ideal | readas =is an ideal of
| category =
ring theory | explain =
I {{Unicode|◅{edih}
R means that
I is an ideal of ring
R.
| examples =(2) {{Unicode|◅}}
Z
| name =
antijoin | readas =the antijoin of
| category =
relational algebra | explain =
R {{Unicode|▻{edih}
S means the antijoin of the relations
R and
S, the tuples in
R for which there is not a tuple in
S that is equal on their common attribute names.
| examples =
R triang≤&nbs(;)
S =
R -
R l⋅
S
| symbol ={{Unicode|⋉{edih}
{{Unicode|⋊}}
| tex =
l⋅
r⋅
| rowspan =2
| name =
semidirect product | readas =the semidirect product of
| category =
group theory | explain =
N ⋊
φ H is the semidirect product of
N (a normal subgroup) and
H (a subgroup), with respect to φ. Also, if
G =
N {{Unicode|⋊}}
φ H, then
G is said to split over
N.
({{Unicode|⋊}}
may also be written the other way round, as {{Unicode|⋉}},
or as ×.)
| examples =
D2n ≅ Cn r⋅ C2
| name =
semijoin | readas =the semijoin of
| category =
relational algebra | explain =
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.
| examples =
R l⋅
S =
Π
a1,..,an(
R bowtie
S)
| symbol ={{Unicode|⋈{edih}
| tex =
bowtie
| rowspan =1
| name =
natural join | readas =the natural join of
| category =
relational algebra | explain =
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.
| examples =
| symbol =∴
| tex =
&#there4;
| rowspan =1
| name =
therefore | readas =therefore; so; hence
| category =everywhere
| explain =Sometimes used in proofs before
logical consequences.
| examples =All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
| symbol =∵
| tex =
because
| rowspan =1
| name =(wiktionary:because|because)
| readas =because; since
| category =everywhere
| explain =Sometimes used in proofs before reasoning.
| examples =3331 is
prime ∵ it has no positive integer factors other than itself and one.
| symbol =■
□
∎
▮
‣
| tex =
blacksquare
Box
blacktriang≤&nbs(;)
| rowspan =1
| name =
end of proof | readas =
QED;
tombstone; Halmos symbol
| category =everywhere
| explain =Used to mark the end of a proof.
(
May also be written Q.E.D.)
| examples =
| symbol =⇒
→
⊃
| tex =
⇒
→
su(set
| rowspan =1
| name =
material implication | readas =implies; if … then
| category =
propositional logic,
Heyting algebra | explain =
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 meaning for functions given below.)
(⊃
may mean the same as ⇒
,)
| examples =
x = 2 ⇒
x2 = 4 is true, but
x2 = 4 ⇒
x = 2 is in general false (since
x could be −2).
| symbol =⇔
↔
| tex =
⇔
↔
| rowspan =1
| name =
material equivalence | readas =if and only if;
iff | category =
propositional logic | explain =
A ⇔
B means
A is true if
B is true and
A is false if
B is false.
| examples =
x + 5 =
y +2 ⇔
x + 3 =
y
| symbol =¬
˜
| tex =
≠g
∼
| rowspan =1
| name =
logical negation | readas =not
| category =
propositional logic | explain =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. Computer scientists will often use !
but this is avoided in mathematical texts.)
| examples =¬(¬
A) ⇔
A x ≠
y ⇔ ¬(
x =
y)
| symbol =∧
| tex =
and
| rowspan =3
| name =
logical conjunction or
meet in a
lattice | readas =and; min; meet
| category =
propositional logic,
lattice theory | explain =The statement
A ∧
B 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)).
| examples =
n < 4 ∧
n >2 ⇔
n = 3 when
n is a
natural number.
| name =
wedge product | readas =wedge product; exterior product
| category =
linear algebra | explain =
u ∧
v means the wedge product of
vectors
u and
v. This generalizes the cross product to higher dimensions.
(
For vectors in R3, ×
can also be used.)
| examples =
u ∧ v = u ⋅ v if u v ∈ R3
| name =
exponentiation | readas =… (raised) to the power of …
| category =everywhere
| explain =
a ^
b means
a raised to the power of
b(
a ^
b is more commonly written ab.
The symbol ^
is generally used in programming languages where ease of typing and use of plain ASCII text is preferred.)
| examples =2^3 = 2
3 = 8
| symbol =∨
| tex =
or
| rowspan =1
| name =
logical disjunction or
join in a
lattice | readas =or; max; join
| category =
propositional logic,
lattice theory | explain =The statement
A ∨
B 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)).
| examples =
n ≥ 4 ∨
n ≤ 2 ⇔
n ≠ 3 when
n is a
natural number.
| symbol =⊕
{{Unicode|⊻{edih}
| tex =
&o(lus;
∨
| rowspan =2
| name =
exclusive or | readas =xor
| category =
propositional logic,
Boolean algebra | explain =The statement
A ⊕
B is true when either A or B, but not both, are true.
A {{Unicode|⊻}}
B means the same.
| examples =(¬
A) ⊕
A is always true,
A ⊕
A is always false.
| name =
direct sum | readas =direct sum of
| category =
abstract algebra | explain =The direct sum is a special way of combining several objects into one general object.
(
The bun symbol ⊕,
or the coproduct symbol {{Unicode|∐}}, is used;
{{Unicode|⊻}} is only for logic.'')
| examples =Most commonly, for vector spaces
U,
V, and
W, the following consequence is used:
U =
V ⊕
W ⇔ (
U =
V +
W) ∧ (
V ∩
W = {0})
| symbol =∀
| tex =
∀
| rowspan =1
| name =
universal quantification | readas =for all; for any; for each
| category =
predicate logic | explain =∀
x:
P(
x) means
P(
x) is true for all
x.
| examples =∀
n ∈ {{Unicode|ℕ}}:
n2 ≥
n.
| symbol =∃
| tex =
eξsts
| rowspan =1
| name =
existential quantification | readas =there exists; there is; there are
| category =
predicate logic | explain =∃
x:
P(
x) means there is at least one
x such that
P(
x) is true.
| examples =∃
n ∈ {{Unicode|ℕ}}:
n is even.
| symbol =∃!
| tex =
eξsts
| rowspan =1
| name =
uniqueness quantification | readas =there exists exactly one
| category =
predicate logic | explain =∃!
x:
P(
x) means there is exactly one
x such that
P(
x) is true.
| examples =∃!
n ∈ {{Unicode|ℕ}}:
n + 5 = 2
n.
| symbol =:=
≡
:⇔
≜
≝
≐
| tex =
:=
≡
:⇔
triang≤q
/setundersetmathrmdef//=
deriv(⋅)eq
| rowspan =1
| name =
definition | readas =is defined as; equal by definition
| category =everywhere
| explain =
x :=
y or
x ≡
y means
x is defined to be another name for
y, under certain assumptions taken in context.
(
Some writers use ≡
to mean congruence).
P :⇔
Q means
P is defined to be
logically equivalent to
Q.
| examples =
cosh x := ex + e-x/2
| symbol ={{Unicode|≅{edih}
| tex =
≅
| rowspan =2
| name =
congruence | readas =is congruent to
| category =
geometry | explain =△ABC {{Unicode|≅}} △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
| examples =
| name =
isomorphic | readas =is isomorphic to
| category =
abstract algebra | explain =
G {{Unicode|≅{edih}
H means that group
G is isomorphic (structurally identical) to group
H.
(≈
can also be used for isomorphic, as described above.)
| examples =
R2 ≅ C
.
| symbol =≡
| tex =
≡
| rowspan =1
| name =
congruence relation | readas =... is congruent to ... modulo ...
| category =
modular arithmetic | explain =
a ≡
b (mod
n) means
a −
b is divisible by
n | examples =5 ≡ 2 (mod 3)
| symbol ={ , }
| tex =
| rowspan =1
| name =
set brackets
| readas =the set of …
| category =
set theory | explain ={
a,
b,
c} means the set consisting of
a,
b, and
c.
(3) | examples ={{Unicode|ℕ}} = { 1, 2, 3, …}
| symbol ={ : }
{
| }
| tex =
:
||
| rowspan =1
| name =
set builder notation | readas =the set of … such that
| category =
set theory | explain ={
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)}.
| examples ={
n ∈ {{Unicode|ℕ}} :
n2 < 20} = { 1, 2, 3, 4}
| symbol ={{unicode|∅{edih}
{ }
| tex =
e&(lusmn;ty
varnoth∈g
| rowspan =1
| name =
empty set | readas =the empty set
| category =
set theory | explain ={{unicode|∅}} means the set with no elements.
{ } means the same.
| examples ={
n ∈ {{Unicode|ℕ}} : 1 <
n2 < 4} = {{unicode|∅}}
| symbol =∈
{{Unicode|∉{edih}
| tex =
∈
not∈
| rowspan =1
| name =set membership
| readas =is an element of; is not an element of
| category =everywhere,
set theory | explain =
a ∈
S means
a is an element of the set
S;
{{Unicode|∉}}
S means
a is not an element of
S.
| examples =(1/2)
−1 ∈ {{Unicode|ℕ}}
2
−1 {{Unicode|∉}} {{Unicode|ℕ}}
| symbol =⊆
⊂
| tex =
&su(;eq
&su(;
| rowspan =1
| name =
subset | readas =is a subset of
| category =
set theory | explain =(subset)
A ⊆
B means every element of
A is also an element of
B.
(4) (proper subset)
A ⊂
B means
A ⊆
B but
A ≠
B.
(
Some writers use the symbol ⊂
as if it were the same as ⊆.)
| examples =(
A ∩
B) ⊆
A{{Unicode|ℕ}} ⊂ {{Unicode|ℚ}}
{{Unicode|ℚ}} ⊂ {{Unicode|ℝ}}
| symbol =⊇
⊃
| tex =
su(seteq
su(set
| rowspan =1
| name =
superset | readas =is a superset of
| category =
set theory | explain =
A ⊇
B means every element of
B is also an element of
A.
A ⊃
B means
A ⊇
B but
A ≠
B.
(
Some writers use the symbol ⊃
as if it were the same as ⊇
.)
| examples =(
A ∪
B) ⊇
B{{Unicode|ℝ}} ⊃ {{Unicode|ℚ}}
| symbol =∪
| tex =
&cu(;
| rowspan =1
| name =
set-theoretic union | readas =the union of … or …; union
| category =
set theory | explain =
A ∪
B means the set of those elements which are either in
A, or in
B, or in both.
| examples =
A ⊆
B ⇔ (
A ∪
B) =
B
| symbol =∩
| tex =
&ca(;
| rowspan =1
| name =
set-theoretic intersection | readas =intersected with; intersect
| category =
set theory | explain =
A ∩
B means the set that contains all those elements that
A and
B have in common.
| examples ={
x ∈ {{Unicode|ℝ}} :
x2 = 1} ∩ {{Unicode|ℕ}} = {1}
| symbol =∆
| tex =
vartriang≤
| rowspan =1
| name =
symmetric difference | readas =symmetric difference
| category =
set theory | explain =A ∆ B means the set of elements in exactly one of
A or
B.
(
Not to be confused with delta, Δ,
described below.)
| examples ={1,5,6,8} ∆ {2,5,8} = {1,2,6}
| symbol ={{Unicode|∖{edih}
| tex =
setmiνs
| rowspan =1
| name =
set-theoretic complement | readas =minus; without
| category =
set theory | explain =
A {{Unicode|∖}}
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.)
| examples ={1,2,3,4} {{Unicode|∖}} {3,4,5,6} = {1,2}
| symbol =→
| tex =
→
| rowspan =1
| name =
function arrow
| readas =from … to
| category =
set theory,
type theory | explain =
f:
X →
Y means the function
f maps the set
X into the set
Y.
| examples =Let
f: {{Unicode|ℤ}} → {{Unicode|ℕ}}∪{0} be defined by
f(
x) :=
x2.
| symbol =↦
| tex =
ma(s→
| rowspan =1
| name =
function arrow
| readas =maps to
| category =
set theory | explain =
f:
a ↦
b means the function
f maps the element
a to the element
b.
| examples =Let
f:
x ↦
x+1 (the successor function).
| symbol =∘ | tex =
ο
| rowspan =1
| name =
function composition | readas =composed with
| category =
set theory | explain =
f∘
g is the function, such that (
f∘
g)(
x) =
f(
g(
x)).
(5) | examples =if
f(
x) := 2
x, and
g(
x) :=
x + 3, then (
f∘
g)(
x) = 2(
x + 3).
| symbol ={{Unicode|ℕ{edih}
N | tex =
N
N
| rowspan =1
| name =
natural numbers
| readas =N; the (set of) natural numbers
| category =
numbers
| explain =
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 ω
to denote the set of natural numbers (including zero), along with the standard ordering relation ≤.
| examples ={{Unicode|ℕ}} =
{|a| :
a ∈ {{Unicode|ℤ}}}
| symbol ={{Unicode|ℤ{edih}
Z | tex =
Z
Z
| rowspan =1
| name =
integers
| readas =Z; the (set of) integers
| category =
numbers
| explain ={{Unicode|ℤ}} means {..., −3, −2, −1, 0, 1, 2, 3, ...}.
| examples ={{Unicode|ℤ}} = {p, −p : p ∈ {{Unicode|ℕ}} ∪ {0}}
}}{hide}row of table of mathematical symbols
| symbol ={{Unicode|ℤ{edih}
n{{Unicode|ℤ}}
pZnZp | tex =
Zn
Z(
Zn
Z(
| rowspan =2
| name =
integers mod n | readas =Z
n; the (set of) integers modulo
n | category =
numbers
| explain ={{Unicode|ℤ}}
n means {[0], [1], [2], ...[
n−1]} with addition and multiplication modulo
n.
Note that any letter may be used instead of n,
such as p.
To avoid confusion with p-adic numbers, use {{Unicode|ℤ}}/
p{{Unicode|ℤ}}
or {{Unicode|ℤ}}/(
p)
instead. | examples ={{Unicode|ℤ}}
3 = {[0], [1], [2]}
| name =
p-adic integers | readas =the (set of)
p-adic integers
| category =
numbers
| explain =
Note that any letter may be used instead of p,
such as n or l.
| examples =
| symbol ={{Unicode|ℙ{edih}
P | tex =
P
P
| rowspan =2
| name =
projective space | readas =P; the projective space, the projective line, the projective plane
| category =
topology | explain ={{Unicode|ℙ}} means a space with a point at infinity.
| examples =
P1
,
P2
| name =
probability | readas =the probability of
| category =
probability theory | explain ={{Unicode|ℙ{edih}(
X) means the probability of the event
X occurring.
This may also be written as P(
X)
or Pr(
X).
| examples =If a fair coin is flipped, {{Unicode|ℙ}}(Heads) = {{Unicode|ℙ}}(Tails) = 0.5.
| symbol ={{Unicode|ℚ{edih}
Q | tex =
Q
Q
| rowspan =1
| name =
rational numbers
| readas =Q; the (set of) rational numbers; the rationals
| category =
numbers
| explain ={{Unicode|ℚ}} means {
p/
q :
p ∈ {{Unicode|ℤ}},
q ∈ {{Unicode|ℕ}}}.
| examples =3.14000... ∈ {{Unicode|ℚ}}
π {{Unicode|∉}} {{Unicode|ℚ}}
| symbol ={{Unicode|ℝ{edih}
R | tex =
R
R
| rowspan =1
| name =
real numbers
| readas =R; the (set of) real numbers; the reals
| category =
numbers
| explain ={{Unicode|ℝ}} means the set of real numbers.
| examples =π ∈ {{Unicode|ℝ}}
√(−1) {{Unicode|∉}} {{Unicode|ℝ}}
| symbol ={{Unicode|ℂ{edih}
C | tex =
C
C
| rowspan =1
| name =
complex numbers
| readas =C; the (set of) complex numbers
| category =
numbers
| explain ={{Unicode|ℂ}} means {
a +
b i :
a,
b ∈ {{Unicode|ℝ}}}.
| examples =
i = √(−1) ∈ {{Unicode|ℂ}}
| symbol ={{Unicode|ℍ{edih}
H | tex =
H
H
| rowspan =1
| name =
quaternions or Hamiltonian quaternions
| readas =H; the (set of) quaternions
| category =
numbers
| explain ={{Unicode|ℍ}} means {
a +
b i +
c j +
d k :
a,
b,
c,
d ∈ {{Unicode|ℝ}}}.
| examples =
| symbol =∞
| tex =
&∈f∈;
| rowspan =1
| name =
infinity | readas =infinity
| category =
numbers
| explain =∞ is an element of the
extended number line that is greater than all real numbers; it often occurs in
limits.
| examples =
limx→ 0 1/||x|| = &∈f∈;
| symbol =⌊…⌋
| tex =
⌊ lderiv(⋅)s ⌋
| rowspan =1
| name =
floor | readas =floor; greatest integer; entier
| category =
numbers
| explain =⌊
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).)
| examples =⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3
| symbol =⌈…⌉
| tex =
⌈ lderiv(⋅)s ⌉
| rowspan =1
| name =
ceiling | readas =ceiling
| category =
numbers
| explain =⌈
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).)
| examples =⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2
| symbol =⌊…⌉
| tex =
⌊ lderiv(⋅)s ⌉
| rowspan =1
| name =
nearest integer function | readas =nearest integer to
| category =
numbers
| explain =⌊
x⌉ means the nearest integer to
x.
(
This may also be written [
x],
||x||, nint(
x)
or Round(
x).)
| examples =⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊-3.4⌉ = -3, ⌊4.49⌉ = 4
| symbol =[ : ]
| tex =
[ : ]
| rowspan =1
| name =
degree of a field extension | readas =the degree of
| category =
field theory | explain =[
K :
F] means the degree of the extension
K :
F.
| examples =[ℚ(√2) : ℚ] = 2
[ℂ : ℝ] = 2
[ℝ : ℚ] = ∞
| symbol =[ ]
[ , ]
[ , , ]
| tex =
[ ]
[ ]
[ ]
| rowspan =8
| name =
equivalence class | readas =the equivalence class of
| category =
abstract algebra | explain =[
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.
| examples =Let
a ~
b be true
iff a ≡
b (
mod 5).
| name =
floor | readas =floor; greatest integer; entier
| category =
numbers
| explain =[
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.)
| examples =[3] = 3, [3.5] = 3, [3.99] = 3, [−3.7] = −4
| name =
nearest integer function | readas =nearest integer to
| category =
numbers
| explain =[
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.)
| examples =[2] = 2, [2.6] = 3, [-3.4] = -3, [4.49] = 4
| name =
Iverson bracket | readas =1 if true, 0 otherwise
| category =
propositional logic | explain =[
S] maps a true statement
S to 1 and a false statement
S to 0.
| examples =[0=5]=0, [7>0]=1, [2 ∈ {2,3,4}]=0, [5 ∈ {2,3,4}]=0
| name =
image | readas =image of … under …
| category =everywhere
| explain =
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.)
| examples =
s∈ [R] = [-1 1]
| name =
closed interval | readas =closed interval
| category =
order theory | explain =
[ab] = x ∈ R : a ≤ x ≤ b
.
| examples =[0,1]
| name =
commutator | readas =the commutator of
| category =
group theory,
ring theory | explain =[
g,
h] =
g−1h−1gh (or
ghg−1h−1), if
g,
h ∈
G (a
group).
/>[
a,
b] =
ab −
ba, if
a,
b ∈
R (a
ring or
commutative algebra).
| examples =xy = x[x, y] (group theory).
[AB, C] = A[B, C] + [A, C]B (ring theory).
}}{{row of table of mathematical symbols
}}{{row of table of mathematical symbols
| symbol =( )
( , )
| tex =
( )
( )
| rowspan =5
| name =
function application
| readas =of
| category =
set theory | explain =
f(
x) means the value of the function
f at the element
x.
| examples =If
f(
x) :=
x2, then
f(3) = 3
2 = 9.
}}{{row of table of mathematical symbols
| name =
image | readas =image of … under …
| category =everywhere
| explain =
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.)
| examples =
s∈ (R) = [-1 1]
}}{{row of table of mathematical symbols
| name =precedence grouping
| readas =parentheses
| category =everywhere
| explain =Perform the operations inside the parentheses first.
| examples =(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
}}{{row of table of mathematical symbols
| name =
tuple | readas =tuple;
n-tuple; ordered pair/triple/etc; row vector; sequence
| category =everywhere
| explain =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.)
| examples =(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).}}{{row of table of mathematical symbols
| name =
highest common factor | readas =highest common factor; greatest common divisor; hcf; gcd
| category =number theory
| explain =(
a,
b) means the highest common factor of
a and
b.
(
This may also be written hcf(
a,
b)
or gcd(
a,
b).)
| examples =(3, 7) = 1 (they are coprime); (15, 25) = 5.
}}{{row of table of mathematical symbols
| symbol =( , )
] , [
| tex =
( )
] [
| rowspan =1
| name =
open interval | readas =open interval
| category =
order theory | explain =
(ab) = x ∈ R : a < x < b
.
(
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.)
| examples =(4,18)
}}{{row of table of mathematical symbols
| symbol =( , ]
] , ]
| tex =
( ]
] ]
| rowspan =1
| name =
left-open interval | readas =half-open interval; left-open interval
| category =
order theory | explain =
(ab] = x ∈ R : a < x ≤ b
.
| examples =(−1, 7] and (−∞, −1]
}}{{row of table of mathematical symbols
| symbol =[ , )
[ , [
| tex =
[ )
[ [
| rowspan =1
| name =
right-open interval | readas =half-open interval; right-open interval
| category =
order theory | explain =
[ab) = x ∈ R : a ≤ x < b
.
| examples =[4, 18) and [1, +∞)
}}{{row of table of mathematical symbols
| symbol =⟨⟩
⟨,⟩
| tex =
lang≤ rang≤
lang≤ rang≤
| rowspan =4
| name =
inner product | readas =inner product of
| category =
linear algebra | explain =⟨
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. | examples =The
standard inner product between two vectors
x = (2, 3) and
y = (−1, 5) is:
⟨x, y⟩ = 2 × −1 + 3 × 5 = 13
}}{{row of table of mathematical symbols
| name =
linear span | readas =(linear) span of;
linear hull of
| category =
linear algebra | explain =⟨
S⟩ means the span of
S ⊆
V. That is, it is the intersection of all subspaces of
V which contain
S.
⟨
u1,
u2, …⟩is shorthand for ⟨{
u1,
u2, …}⟩.
⟨
u,
v⟩
may be ambiguous: it could mean the inner product or the linear span.The span of S may also be written as Sp(
S).
| examples =
(lang (( beg∈smallmatrix 1 0 0 endsmallmatrix &nbs(;)) (( beg∈smallmatrix 0 1 0 endsmallmatrix &nbs(;)) (( beg∈smallmatrix 0 0 1 endsmallmatrix &nbs(;)) &nbs(;)rang = R3
.
}}{{row of table of mathematical symbols
| name =subgroup
generated by a set
| readas =the subgroup generated by
| category =
group theory | explain =
lang≤ S rang≤
means the smallest subgroup of
G (where
S ⊆
G, a group) containing every element of
S.
lang≤ g1 g2 ... rang≤
is shorthand for
lang≤ g1 g2 ... rang≤
.
| examples =In
S3,
lang≤(1 ; 2) rang≤ = id; (1 ; 2)
and
lang≤ (1 ; 2 ; 3) rang≤ = id ; (1 ; 2 ; 3)(1 ; 2 ; 3))
.
}}{{row of table of mathematical symbols
| name =
tuple | readas =tuple;
n-tuple; ordered pair/triple/etc; row vector; sequence
| category =everywhere
| explain =An ordered list (or sequence, or horizontal vector, or row vector) of values.
(
The notation (
a,
b)
is often used as well.)
| examples =
lang≤ a b rang≤
is an ordered pair (or 2-tuple).
lang≤ a b c rang≤
is an ordered triple (or 3-tuple).
lang≤ rang≤
is the
empty tuple (or 0-tuple).}}{{row of table of mathematical symbols
| symbol =⟨
|⟩
(
|)
| tex =
lang≤ || rang≤
( || )
| rowspan =1
| name =
inner product | readas =inner product of
| category =
linear algebra | explain =⟨
u | v⟩ means the inner product of
u and
v, where
u and
v are members of an
inner product space.
(6) (
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. | examples =
}}{{row of table of mathematical symbols
| symbol =
|⟩
| tex =
|| rang≤
| rowspan =1
| name =
ket vector | readas =the ket …; the vector …
| category =
Dirac notation | explain =
|φ⟩ means the vector with label
φ, which is in a
Hilbert space.
| examples =A
qubit's state can be represented as
α|0⟩+
β|1⟩, where
α and
β are complex numbers s.t.
|α|2 +
|β|2 = 1.
}}{{row of table of mathematical symbols
| symbol =⟨
| | tex =
lang≤ ||
| rowspan =1
| name =
bra vector | readas =the bra …; the dual of …
| category =
Dirac notation | explain =⟨
φ| means the dual of the vector
|φ⟩, a
linear functional which maps a ket
|ψ⟩ onto the inner product ⟨
φ|ψ⟩.
| examples =
}}{{row of table of mathematical symbols
| symbol =∑
| tex =
Σ
| rowspan =1
| name =
summation | readas =sum over … from … to … of
| category =
arithmetic | explain =
Σk=1n/ak
means
a1 +
a2 + … +
an.
| examples =
Σk=14/k2
= 1
2 + 2
2 + 3
2 + 4
2
}}{{row of table of mathematical symbols
| symbol =∏
| tex =
&(rod;
| rowspan =2
| name =
product | readas =product over … from … to … of
| category =
arithmetic | explain =
&(rod;k=1nak
means
a1a2···
an.
| examples =
&(rod;k=14(k+2)
= (1+2)(2+2)(3+2)(4+2)
}}{{row of table of mathematical symbols
| examples =
&(rod;n=13/R = R⋅R⋅R = R3
}}{hide}row of table of mathematical symbols
}}{{row of table of mathematical symbols
| symbol =′
• | tex =
'
deriv(⋅)
| rowspan =1
| name =
derivative | readas =… prime
derivative of
| category =
calculus | explain =
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 dot notation indicates a time derivative. That is
deriv(⋅)x(t)=&(art;/&(art; tx(t)
.
| examples =If f(x) := x2, then f ′(x) = 2x
}}{{row of table of mathematical symbols
| symbol =∫
| tex =
∈t
| rowspan =3
| name =
indefinite integral or
antiderivative | readas =indefinite integral of
the antiderivative of
| category =
calculus | explain =∫
f(
x) d
x means a function whose derivative is
f.
| examples =∫
x2 d
x =
x3/3 +
C
}}{{row of table of mathematical symbols
| name =
definite integral | readas =integral from … to … of … with respect to
| category =
calculus | explain =∫
ab f(
x) d
x means the signed
area between the
x-axis and the
graph of the
function f between
x =
a and
x =
b.
| examples =∫
ab x2 d
x =
b3/3 −
a3/3;
}}{{row of table of mathematical symbols
| name =
line integral | readas =line/path/curve integral of … along …
| category =
calculus | explain =∫
C f d
s means the integral of
f along the curve
C,
textsty≤ ∈tab f(r(t)) ||r'(t)|| dt
, where
r is a parametrization of
C.
(
If the curve is closed, the symbol ∮
may be used instead, as described below.)
| examples =
}}{{row of table of mathematical symbols
| symbol =∮
| tex =
o∈t
| rowspan =1
| name =
contour integral or closed
line integral | readas =contour integral of
| category =
calculus | explain =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 {{Unicode|∯}} would be more appropriate. A third related symbol is the closed
volume integral, denoted by the symbol {{Unicode|∰}}.
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.
| examples =If
C is a
Jordan curve about 0, then
o∈tC 1 / zdz = 2&(i; i
.
}}{{row of table of mathematical symbols
| symbol =∇
| tex =
∇
| rowspan =3
| name =
gradient | readas =
del,
nabla,
gradient of
| category =
vector calculus | explain =∇
f (x
1, …, x
n) is the vector of partial derivatives (
∂f /
∂x1, …,
∂f /
∂xn).
| examples =If
f (
x,
y,
z) := 3
xy +
z², then ∇
f = (3
y, 3
x, 2
z)
}}{{row of table of mathematical symbols
| name =
divergence | readas =del dot, divergence of
| category =
vector calculus | explain =
∇ cderiv(⋅) → v = &(art; vx / &(art; x + &(art; vy / &(art; y + &(art; vz / &(art; z
| examples =If
→ v := 3xyi+y2 zj+5k
, then
∇ cderiv(⋅) → v = 3y + 2yz
.
}}{{row of table of mathematical symbols
| name =
curl | readas =curl of
| category =
vector calculus | explain =
∇ ⋅ → v = (( &(art; vz / &(art; y - &(art; vy / &(art; z &nbs(;)) i
+ (( &(art; vx / &(art; z - &(art; vz / &(art; x &nbs(;)) j + (( &(art; vy / &(art; x - &(art; vx / &(art; y &nbs(;)) k
| examples =If
→ v := 3xyi+y2 zj+5k
, then
∇⋅→ v = -y2i - 3xk
.
}}{{row of table of mathematical symbols
| symbol =∂
| tex =
&(art;
| rowspan =3
| name =
partial derivative | readas =partial, d
| category =
calculus | explain =∂
f/∂
xi means the partial derivative of
f with respect to
xi, where
f is a function on (
x1, …,
xn).
| examples =If
f(
x,
y) :=
x2y, then ∂
f/∂
x = 2
xy
}}{hide}row of table of mathematical symbols
| name =
boundary | readas =boundary of
| category =
topology | explain =∂
M means the boundary of
M | examples =∂{
x :
||x|| ≤ 2} = {
x :
||x|| = 2}
{edih}{{row of table of mathematical symbols
| name =
degree of a polynomial | readas =degree of
| category =
algebra | explain =∂
f means the degree of the polynomial
f.
(
This may also be written deg
f.)
| examples =∂(
x2 − 1) = 2
}}{{row of table of mathematical symbols
| symbol =Δ
| tex =
Δ
| rowspan =1
| name =
delta | readas =delta; change in
| category =
calculus | explain =Δ
x means a (non-infinitesimal) change in
x.
(
If the change becomes infinitesimal, δ
and even d
are used instead. Not to be confused with the symmetric difference, written ∆,
above.)
| examples =
tΔ x/Δ y
is the gradient of a straight line
}}{hide}row of table of mathematical symbols
| symbol =δ
| tex =
δ
| rowspan =2
| name =
Dirac delta function | readas =Dirac delta of
| category =
hyperfunction | explain =
δ(x) = beg∈cases &∈f∈; & x = 0 0 & x ≠ 0 endcases
| examples =δ(x)
{edih}{hide}row of table of mathematical symbols
| name =
Kronecker delta | readas =Kronecker delta of
| category =
hyperfunction | explain =
δij = beg∈cases 1 & i = j 0 & i ≠ j endcases
| examples =δ
ij
{edih}{{row of table of mathematical symbols
| symbol =π
| tex =
&(i;
| rowspan =1
| name =
projection | readas =Projection of
| category =
Relational algebra | explain =
&(i;a1 ...an( R )
restricts
R
to the
a1...an
attribute set.
| examples =
&(i;A≥Weight(Person)
}}{{row of table of mathematical symbols
| symbol =σ
| tex =
σ
| rowspan =1
| name =
selection | readas =Selection of
| category =
Relational algebra | explain =The selection
σa thη b( R )
selects all those
tuples in
R
for which
thη
holds between the
a
and the
b
attribute. The selection
σa thη v( R )
selects all those tuples in
R
for which
thη
holds between the
a
attribute and the value
v
.
| examples =
σA≥ ≥ 34( Person )
σA≥ = Weight( Person )
}}{{row of table of mathematical symbols
| symbol =<:
<·
| tex =
<:
</cderiv(⋅)
| rowspan =2
| name =
cover | readas =is covered by
| category =
order theory | explain =
x <•
y means that
x is covered by
y.
| examples ={1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment.
}}{{row of table of mathematical symbols
| name =
subtype | readas =is a subtype of
| category =
type theory | explain =
T1 <:
T2 means that
T1 is a subtype of
T2.
| examples =If
S <:
T and
T <:
U then
S <:
U (
transitivity).
}}{{row of table of mathematical symbols
| symbol =
† | tex =
&dag≥r;
| rowspan =1
| name =
conjugate transpose | readas =conjugate transpose; Hermitian adjoint/conjugate/transpose; adjoint
| category =
matrix operations
| explain =
A† means the transpose of the complex conjugate of
A.
(7)This may also be written A*T,
AT*,
A*, {{overline|
A}}
T or {{overline|
AT}}.
| examples =If
A = (
aij) then
A† = ({{overline|
aji}}).
}}{{row of table of mathematical symbols
| symbol =
T | tex =
mathsfT
| rowspan =1
| name =
transpose | readas =transpose
| category =
matrix operations
| explain =
AT means
A, but with its rows swapped for columns.
This may also be written At or Atr.
| examples =If
A = (
aij) then
AT = (
aji).
}}{{row of table of mathematical symbols
| symbol =⊤
| tex =
→(
| rowspan =2
| name =
top element | readas =the top element
| category =
lattice theory | explain =⊤ means the largest element of a lattice.
| examples =∀
x :
x ∨ ⊤ = ⊤
}}{{row of table of mathematical symbols
| name =
top type | readas =the top type; top
| category =
type theory | explain =⊤ means the top or universal type; every type in the
type system of interest is a subtype of top.
| examples =∀ types
T,
T <: ⊤
}}{{row of table of mathematical symbols
| symbol =⊥
| tex =
bot
| rowspan =6
| name =
perpendicular | readas =is perpendicular to
| category =
geometry | explain =
x ⊥
y means
x is perpendicular to
y; or more generally
x is
orthogonal to
y.
| examples =If
l ⊥
m and
m ⊥
n in the plane then
l || n.
}}{{row of table of mathematical symbols
| name =
orthogonal complement | readas =orthogonal/perpendicular complement of; perp
| category =
linear algebra | explain =
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.
| examples =Within
R3
,
(R2)&(er(; ≅ R
.
}}{{row of table of mathematical symbols
| name =
coprime | readas =is coprime to
| category =
number theory | explain =
x ⊥
y means
x has no factor in common with
y.
| examples =34 ⊥ 55.
}}{{row of table of mathematical symbols
| name =
bottom element | readas =the bottom element
| category =
lattice theory | explain =⊥ means the smallest element of a lattice.
| examples =∀
x :
x ∧ ⊥ = ⊥
}}{{row of table of mathematical symbols
| name =
bottom type | readas =the bottom type; bot
| category =
type theory | explain =⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the
type system.
| examples =∀ types
T, ⊥ <:
T
}}{{row of table of mathematical symbols
| name =
comparability | readas =is comparable to
| category =
order theory | explain =
x ⊥
y means that
x is comparable to
y.
| examples ={
e,
π} ⊥ {1, 2,
e, 3,
π} under set containment.
}}{hide}row of table of mathematical symbols
| symbol ={{Unicode|⊧{edih}
| tex =
vDash
| rowspan =1
| name =
entailment | readas =entails
| category =
model theory | explain =
A {{Unicode|⊧}}
B means the sentence
A entails the sentence
B, that is in every model in which
A is true,
B is also true.
| examples =
A {{Unicode|⊧}}
A ∨ ¬
A
}}{hide}row of table of mathematical symbols
| symbol ={{Unicode|⊢{edih}
| tex =
vdash
| rowspan =1
| name =
inference | readas =infers; is derived from
| category =
propositional logic,
predicate logic | explain =
x {{Unicode|⊢}}
y means
y is derivable from
x.
| examples =
A →
B {{Unicode|⊢}} ¬
B → ¬
A.
}}{hide}row of table of mathematical symbols
| symbol ={{Unicode|⊗{edih}
| tex =
&o⋅;
| rowspan =1
| name =
tensor product,
tensor product of modules | readas =tensor product of
| category =
linear algebra | explain =
V &o⋅; U
means the tensor product of
V and
U.
(8) V &o⋅;R U
means the tensor product of modules
V and
U over the
ring R.
| examples ={1, 2, 3, 4} {{Unicode|⊗}} {1, 1, 2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
}}{{row of table of mathematical symbols
| symbol =*
| tex =
*
| rowspan =4
| name =
convolution | readas =convolution, convolved with
| category =
functional analysis | explain =
f *
g means the convolution of
f and
g.
| examples =
(f * g)(t) = ∈t-&∈f∈;&∈f∈; f(τ) g(t - τ) dτ
.
}}{{row of table of mathematical symbols
| name =
complex conjugate | readas =conjugate
| category =
complex numbers | explain =
z* means the complex conjugate of
z.
(
z
can also be used for the conjugate of z, as described below.)
| examples =
(3+4i)ast = 3-4i
.
}}{{row of table of mathematical symbols
| name =
group of units | readas =the group of units of
| category =
ring theory | explain =
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).
| examples =
beg∈align (Z / 5Z)ast & = [1] [2] [3] [4] & ≅ C4 endalign
}}{{row of table of mathematical symbols
}}{hide}row of table of mathematical symbols
| symbol ={{overline|
x{edih}
| tex =
x
| rowspan =4
| name =
mean | readas =overbar, … bar
| category =
statistics | explain =
x
(often read as “x bar”) is the
mean (average value of
xi
).
| examples =
x = 12345; x = 3
.
}}{{row of table of mathematical symbols
| name =
complex conjugate | readas =conjugate
| category =
complex numbers | explain =
z
means the complex conjugate of
z.
(
z*
can also be used for the conjugate of z, as described above.)
| examples =
3+4i = 3-4i
.
}}{{row of table of mathematical symbols
}}{{row of table of mathematical symbols
| name =
topological closure | readas =(topological) closure of
| category =
topology | explain =
S
is the topological closure of the set
S.
This may also be denoted as cl(
S)
or Cl(
S).
| examples =In the space of the real numbers,
Q = R
(the rational numbers are
dense in the real numbers).
}}
