# Úsáideoir:Cliste/Teimpléid/nodaireacht na Mata

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=
${\displaystyle =\!\,}$
cothroime
tá 'a' cothrom le
gach réimse
Ciallaíonn x = y go bhfuil x agus y cothrom, nó go léiríonn siad an rud/luach céanna. 1 + 1 = 2
${\displaystyle \neq \!\,}$
éagothromóid
gach réimse
Ciallaíonn x ≠ y nach léiríonn x agus y an luach/rud céanna.

(Úsáidtear !=, /= nó <> go minic sna Teanga ríomhaireachta.)
2 + 2 ≠ 5
<

>
${\displaystyle <\!\,}$

${\displaystyle >\!\,}$
éagothroime dian
níos lú ná, níos mó ná
Teoiric Ord
Ciallaíonn x < y go bhfuil x níos lú ná y.

Ciallaíonn x > y go bhfuil x níos mó ná y.
3 < 4
5 > 4

${\displaystyle \leq \!\,}$

${\displaystyle \geq \!\,}$
éagothroime
níos lú ná nó cothrom le, níos mó ná nó cothrom le
Teoiric Ord
Ciallaíonnxy go bhfuil x níos lú ná nó cothrom le y.

Ciallaíonnx'≥ ygo bhfuil x níos mó ná nó cothrom le y.

(Úsáidtear <= agus >= go minic sna Teanga ríomhaireachta.)
3 ≤ 4 agus 5 ≤ 5
5 ≥ 4 agus 5 ≥ 5
${\displaystyle \propto \!\,}$
comhréireach
tá sé comhréireach le;
everywhere
Ciallaíonn yx go bhfuil y = kx fíor do tairiseach k. má tá y = 2x, is féidir yx a scríobh
+
${\displaystyle +\!\,}$
suimiú
cuir le; suimigh
uimhríocht
Ciallaíonn 4 + 6 an suim do 4 agus 6. 2 + 7 = 9
aontas scartha
an aontas scartha de ... agus ...
teoiric na dtacar
Ciallaíonn A1 + A2 an aontas scartha de na tacar A1 agus A2. 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)}
${\displaystyle -\!\,}$
dealú
bain _ ó _; thóg _ ó _
uimhríocht
Ciallaíonn 9 − 4 bain 4 ó 9. 8 − 3 = 5
uimhir dhiúltach
dhiúltach; inbhéartach suimiúcháin
uimhríocht
Ciallaíonn −3 an uimhir dhiúltach don uimhir 3. −(−5) = 5

×
${\displaystyle \times \!\,}$
iolrúchán
arna iolrú; iolraigh;
uimhríocht
Ciallaíonn 3 × 4 an iolrúchán 3 fá 4. 7 × 8 = 56
Iolrachán Cairtéiseach
an iolrachán Cairtéiseach de... gus ...
Teoiric na dtacair
Ciallaíonn X×Y an tacar iomlán na beirte leis an céad ball ó X agus an dara ball ó Y. {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
trasiolrach
ailgéabar veicteoir
Ciallaíonn u × v an trasiolrach na veicteoir u agus v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
·
${\displaystyle \cdot \!\,}$
iolrúchán
arna iolrú; iolraigh;
uimhríocht
Ciallaíonn 3 · 4 an iolrachán de 3 fá 4. 7 · 8 = 56
ponc iolrach
an ponc iolrach de
ailgéabar veicteoir
Ciallaíonn u · v an ponciolrach do na veicteoir u agus v (1,2,5) · (3,4,−1) = 6
÷

${\displaystyle \div \!\,}$

${\displaystyle /\!\,}$
líon
roinnt, thar
uimhríocht
Ciallaíonn 6 ÷ 3 nó 6 ⁄ 3 go bhfuil 6 roinnte ag a 3. 2 ÷ 4 = .5

12 ⁄ 4 = 3
±
${\displaystyle \pm \!\,}$
comhartha suimithe nó dealaithe
comhartha suimithe nó dealaithe
uimhríocht
Ciallaíonn 6 ± 3 na dhá rud 6 + 3 agus 6 − 3. Tá dhá réiteach ag an cothromóid ${\displaystyle x=5\pm {\sqrt {4}};x=7agusx=3.}$
comhartha suimithe nó dealaithe
comhartha suimithe nó dealaithe
tomhas
Ciallaíonn 10 ± 2 nó a chomhluach 10 ± 20% an raon idir 10 − 2 agus 10 + 2. Má tá a = 100 ± 1 mm, ciallaíonn sé go bhfuil a ≥ 99 mm agus a ≤ 101 mm.
${\displaystyle \mp \!\,}$
comhartha suimithe nó dealaithe
comhartha suimithe nó dealaithe
uimhríocht
Ciallaíonn 6 ± (3 5) na dhá rud 6 + (3 − 5) agus 6 − (3 + 5). cos(x ± y) = cos(x) cos(y) sin(x) sin(y).
${\displaystyle \surd \!\,}$

${\displaystyle {\sqrt {\ }}\!\,}$
fréamh chearnach
an fréamh chearnach
Ciallaíonn ${\displaystyle {\sqrt {x}}}$ an uimhir deimhneach leis an cearnach ${\displaystyle x}$. ${\displaystyle {\sqrt {4}}=2}$
the (complex) square root of
if ${\displaystyle z=r\,\exp(i\phi )}$ is represented in polar coordinates with ${\displaystyle -\pi <\phi \leq \pi }$, then ${\displaystyle {\sqrt {z}}={\sqrt {r}}\exp(i\phi /2)}$. ${\displaystyle {\sqrt {-1}}=i}$
|…|
${\displaystyle |\ldots |\!\,}$
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 distance between; Euclidean norm of
|x – y| means the Euclidean distance between x and y. For x = (1,1), and y = (4,5),
|x – y| = v([1–4]2 + [1–5]2) = 5
determinant of
|A| means the determinant of the matrix A ${\displaystyle {\begin{vmatrix}1&2\\2&4\\\end{vmatrix}}=0}$
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.
||…||
${\displaystyle \|\ldots \|\!\,}$
norm of; length of
|| x || means the norm of the element x of a normed vector space.[1] || 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

${\displaystyle \mid \!\,}$

${\displaystyle \nmid \!\,}$
divides
a|b means a divides b.
ab 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.)
Since 15 = 3×5, it is true that 3|15 and 5|15.
given
P(A|B) 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 May 25 | X is in May) = 1/31
restriction of … to …; restricted to
f|A means the function f restricted to the set A, that is, it is the function with domain A n dom(f) that agrees with f. The function f : R ? R defined by f(x) = x2 is not injective, but f|R+ is injective.
||
${\displaystyle \|\!\,}$
is parallel to
x || y means x is parallel to y. 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.
exactly divides
pa || n means pa exactly divides n (i.e. pa divides n but pa+1 does not). 23 || 360.
#
${\displaystyle \#\!\,}$
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#Sm is homeomorphic to A, for any manifold A, and the sphere Sm.
${\displaystyle \aleph \!\,}$
aleph
a represents an infinite cardinality (specifically, the a-th one, where a is an ordinal). |N| = ℵ0, which is called aleph-null.
${\displaystyle \beth \!\,}$
beth
a represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ). ${\displaystyle \beth _{1}=|P(\mathbb {N} )|=2^{\aleph _{0}}.}$
C
${\displaystyle {\mathfrak {c}}\!\,}$
cardinality of the continuum; cardinality of the real numbers; c;
The cardinality of ${\displaystyle \mathbb {R} }$ is denoted by ${\displaystyle |\mathbb {R} |}$ or by the symbol ${\displaystyle {\mathfrak {c}}}$ (a lowercase Fraktur letter C). ${\displaystyle {\mathfrak {c}}={\beth }_{1}}$
:
${\displaystyle :\!\,}$
such that
such that; so that
everywhere
: means “such that”, and is used in proofs and the set-builder 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 ?uv?, ?u | v? or (u | v), as described below. For spatial vectors, the dot product notation, x·y is common. See also Bra-ket notation.
${\displaystyle A:B=\sum _{i,j}A_{ij}B_{ij}\!\,}$
!
${\displaystyle !\!\,}$
factorial
n! means the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24
not
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.)
!(!A) ? A
x ? y  ?  !(x = y)
~
${\displaystyle \sim \!\,}$
has distribution
X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
is row equivalent to
A~B means that B can be generated by using a series of elementary row operations on A ${\displaystyle {\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\sim {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}}$
roughly similar; poorly approximates
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 p2 ˜ 10
is asymptotically equivalent to
f ~ g means ${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1}$. x ~ x+1

are in the same equivalence class
everywhere
a ~ b means ${\displaystyle b\in [a]}$ (and equivalently ${\displaystyle a\in [b]}$). 1 ~ 5 mod 4

˜
${\displaystyle \approx \!\,}$
approximately equal
is approximately equal to
everywhere
x ˜ y means x is approximately equal to y. p ˜ 3.14159
is isomorphic to
G ˜ H means that group G is isomorphic (structurally identical) to group H.

( can also be used for isomorphic, as described below.)
Q / {1, −1} ˜ V,
where Q is the quaternion group and V is the Klein four-group.
?
${\displaystyle \wr \!\,}$
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.
${\displaystyle S_{n}\wr Z_{2}}$ is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.

${\displaystyle \triangleleft \!\,}$

${\displaystyle \triangleright \!\,}$
is a normal subgroup of
N  G means that N is a normal subgroup of group G. Z(G G
is an ideal of
I  R means that I is an ideal of ring R. (2)  Z
the antijoin of
R  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. R ${\displaystyle \triangleright }$ S = R - R ${\displaystyle \ltimes }$ S

${\displaystyle \ltimes \!\,}$

${\displaystyle \rtimes \!\,}$
the semidirect product of
Nf H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to f. Also, if G = N f H, then G is said to split over N.

( may also be written the other way round, as , or as ×.)
${\displaystyle D_{2n}\cong C_{n}\rtimes C_{2}}$
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 ${\displaystyle \ltimes }$ S = ${\displaystyle \Pi }$a1,..,an(R ${\displaystyle \bowtie }$ S)
${\displaystyle \bowtie \!\,}$
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.
?
${\displaystyle \therefore \!\,}$
therefore; so; hence
everywhere
Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ? Socrates is mortal.
?
${\displaystyle \because \!\,}$
because; since
everywhere
Sometimes used in proofs before reasoning. 3331 is prime ? it has no positive integer factors other than itself and one.
¦

?

?

?

?
${\displaystyle \blacksquare \!\,}$

${\displaystyle \Box \!\,}$

${\displaystyle \blacktriangleright \!\,}$
QED; tombstone; Halmos symbol
everywhere
Used to mark the end of a proof.

(May also be written Q.E.D.)
?

?

?
${\displaystyle \Rightarrow \!\,}$

${\displaystyle \rightarrow \!\,}$

${\displaystyle \supset \!\,}$
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 meaning for functions given below.)

(? may mean the same as ?,[2] 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).
?

?
${\displaystyle \Leftrightarrow \!\,}$

${\displaystyle \leftrightarrow \!\,}$
if and only if; iff
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
¬

˜
${\displaystyle \neg \!\,}$

${\displaystyle \sim \!\,}$
not
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.)
¬(¬A) ? A
x ? y  ?  ¬(x =  y)
?
${\displaystyle \land \!\,}$
and; min; meet
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)).
n < 4  ?  n >2  ?  n = 3 when n is a natural number.
wedge product; exterior product
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.)
${\displaystyle u\wedge v=u\times v,{\mbox{ if }}u,v\in \mathbb {R} ^{3}}$
… (raised) to the power of …
everywhere
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.)
2^3 = 23 = 8
?
${\displaystyle \lor \!\,}$
or; max; join
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)).
n = 4  ?  n = 2  ? n ? 3 when n is a natural number.
?

${\displaystyle \oplus \!\,}$

${\displaystyle \veebar \!\,}$
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 modules into one general module.

(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 n W = {0})
${\displaystyle \forall \!\,}$
for all; for any; for each
∀ x: P(x) means P(x) is true for all x. ∀ n ? : n2 = n.
${\displaystyle \exists \!\,}$
there exists; there is; there are
∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ? : n is even.
∃!
${\displaystyle \exists !\!\,}$
there exists exactly one
∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ? : n + 5 = 2n.
:=

=

:?

?

?

?
${\displaystyle :=\!\,}$

${\displaystyle \equiv \!\,}$

${\displaystyle :\Leftrightarrow \!\,}$

${\displaystyle \triangleq \!\,}$

${\displaystyle {\overset {\underset {\mathrm {def} }{}}{=}}\!\,}$

${\displaystyle \doteq \!\,}$
is defined as; equal by definition
everywhere
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.
${\displaystyle \cosh x:={\frac {e^{x}+e^{-x}}{2}}}$
${\displaystyle \cong \!\,}$
is congruent to
?ABC ?DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
is isomorphic to
G  H means that group G is isomorphic (structurally identical) to group H.

can also be used for isomorphic, as described above.)
${\displaystyle \mathbb {R} ^{2}\cong \mathbb {C} }$.
=
${\displaystyle \equiv \!\,}$
... is congruent to ... modulo ...
a = b (mod n) means ab is divisible by n 5 = 2 (mod 3)
{ , }
${\displaystyle {\{\ ,\!\ \}}\!\,}$
set brackets
the set of …
{a,b,c} means the set consisting of a, b, and c.[3]  = { 1, 2, 3, …}
{ : }

{ | }
${\displaystyle \{\ :\ \}\!\,}$

${\displaystyle \{\ |\ \}\!\,}$
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 ?  : n2 < 20} = { 1, 2, 3, 4}

{ }
${\displaystyle \emptyset \!\,}$

${\displaystyle \varnothing \!\,}$

${\displaystyle \{\}\!\,}$
the empty set
means the set with no elements.[3] { } means the same. {n ?  : 1 < n2 < 4} =
?

${\displaystyle \in \!\,}$

${\displaystyle \notin \!\,}$
set membership
is an element of; is not an element of
everywhere, set theory
a ? S means a is an element of the set S;[3] a  S means a is not an element of S.[3] (1/2)−1 ?

2−1
?

?
${\displaystyle \subseteq \!\,}$

${\displaystyle \subset \!\,}$
is a subset of
(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 ?.)
(A n B) ? A

?

?
?

?
${\displaystyle \supseteq \!\,}$

${\displaystyle \supset \!\,}$
is a superset of
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 ?.)
(A ? B) ? B

?
?
${\displaystyle \cup \!\,}$
the union of … or …; union
A ? B means the set of those elements which are either in A, or in B, or in both.[4] A ? B  ?  (A ? B) = B
n
${\displaystyle \cap \!\,}$
intersected with; intersect
A n B means the set that contains all those elements that A and B have in common.[4] {x ?  : x2 = 1} n  = {1}
?
${\displaystyle \vartriangle \!\,}$
symmetric difference
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}
${\displaystyle \setminus \!\,}$
minus; without
A  B means the set that contains all those elements of A that are not in B.[4]

(− can also be used for set-theoretic complement as described above.)
{1,2,3,4}  {3,4,5,6} = {1,2}
?
${\displaystyle \to \!\,}$
function arrow
from … to
fX ? Y means the function f maps the set X into the set Y. Let f ? ?{0} be defined by f(x) := x2.
?
${\displaystyle \mapsto \!\,}$
function arrow
maps to
fa ? b means the function f maps the element a to the element b. Let fx ? x+1 (the successor function).
°
${\displaystyle \circ \!\,}$
composed with
f°g is the function, such that (f°g)(x) = f(g(x)).[5] if f(x) := 2x, and g(x) := x + 3, then (f°g)(x) = 2(x + 3).

N
${\displaystyle \mathbb {N} \!\,}$

${\displaystyle \mathbf {N} \!\,}$
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 ? to denote the set of natural numbers (including zero), along with the standard ordering relation =.
= {|a| : a ? }

Z
${\displaystyle \mathbb {Z} \!\,}$

${\displaystyle \mathbf {Z} \!\,}$
Z; the (set of) integers
means {..., −3, −2, −1, 0, 1, 2, 3, ...}.

+ or > means {1, 2, 3, ...} . = means {0, 1, 2, 3, ...} .

= {p, −p : p ?  ? {0}​}
n

p

Zn

Zp
${\displaystyle \mathbb {Z} _{n}\!\,}$

${\displaystyle \mathbb {Z} _{p}\!\,}$

${\displaystyle \mathbf {Z} _{n}\!\,}$

${\displaystyle \mathbf {Z} _{p}\!\,}$
Zn; the (set of) integers modulo n
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 /p or /(p) instead.
3 = {[0], [1], [2]}

Note that any letter may be used instead of p, such as n or l.

P
${\displaystyle \mathbb {P} \!\,}$

${\displaystyle \mathbf {P} \!\,}$
P; the projective space, the projective line, the projective plane
means a space with a point at infinity. ${\displaystyle \mathbb {P} ^{1}}$,${\displaystyle \mathbb {P} ^{2}}$
the probability of
(X) means the probability of the event X occurring.

This may also be written as P(X) or Pr(X).
If a fair coin is flipped, (Heads) = (Tails) = 0.5.

Q
${\displaystyle \mathbb {Q} \!\,}$

${\displaystyle \mathbf {Q} \!\,}$
Q; the (set of) rational numbers; the rationals
means {p/q : p ? , q ? }. 3.14000... ?

R
${\displaystyle \mathbb {R} \!\,}$

${\displaystyle \mathbf {R} \!\,}$
R; the (set of) real numbers; the reals
R means the set of real numbers. p ?

v(−1)

C
${\displaystyle \mathbb {C} \!\,}$

${\displaystyle \mathbf {C} \!\,}$
C; the (set of) complex numbers
means {a + b i : a,b ? }. i = v(−1) ?
O
${\displaystyle O}$
big-oh 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) = 6x4 - 2x3 + 5 and g(x) = x4 , then ${\displaystyle f(x)=O(g(x)){\mbox{ as }}x\to \infty \,}$
8
${\displaystyle \infty \!\,}$
infinity
8 is an element of the extended number line that is greater than all real numbers; it often occurs in limits. ${\displaystyle \lim _{x\to 0}{\frac {1}{|x|}}=\infty }$
?…?
${\displaystyle \lfloor \ldots \rfloor \!\,}$
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
?…?
${\displaystyle \lceil \ldots \rceil \!\,}$
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
?…?
${\displaystyle \lfloor \ldots \rceil \!\,}$
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
[ : ]
${\displaystyle [\ :\ ]\!\,}$
the degree of
[K : F] means the degree of the extension K : F. [Q(v2) : Q] = 2

[C : R] = 2

[R : Q] = 8
[ ]

[ , ]

[ , , ]
${\displaystyle [\ ]\!\,}$

${\displaystyle [\ ,\ ]\!\,}$

${\displaystyle [\ ,\ ,\ ]\!\,}$
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}]=0, [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.)
${\displaystyle \sin[\mathbb {R} ]=[-1,1]}$
closed interval
${\displaystyle [a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}$. [0,1]
the commutator of
[gh] = g−1h−1gh (or ghg−1h−1), if g, h ? G (a group).

[ab] = ab − ba, if a, b ? R (a ring or commutative algebra).
xy = x[xy] (group theory).

[ABC] = A[BC] + [AC]B (ring theory).
the triple scalar product of
[abc] = a × b · c, the scalar product of a × b with c. [abc] = [bca] = [cab].
( )

( , )
${\displaystyle (\ )\!\,}$

${\displaystyle (\ ,\ )\!\,}$
function application
of
f(x) means the value of the function f at the element x. If f(x) := x2, then f(3) = 32 = 9.
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.)
${\displaystyle \sin(\mathbb {R} )=[-1,1]}$
precedence grouping
parentheses
everywhere
Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
tuple; n-tuple; ordered pair/triple/etc; row vector; sequence
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 2-tuple).

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

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

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.
( , )

] , [
${\displaystyle (\ ,\ )\!\,}$

${\displaystyle ]\ ,\ [\!\,}$
open interval
${\displaystyle (a,b)=\{x\in \mathbb {R} :a.

(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)
( , ]

] , ]
${\displaystyle (\ ,\ ]\!\,}$

${\displaystyle ]\ ,\ ]\!\,}$
half-open interval; left-open interval
${\displaystyle (a,b]=\{x\in \mathbb {R} :a. (−1, 7] and (−8, −1]
[ , )

[ , [
${\displaystyle [\ ,\ )\!\,}$

${\displaystyle [\ ,\ [\!\,}$
half-open interval; right-open interval
${\displaystyle [a,b)=\{x\in \mathbb {R} :a\leq x. [4, 18) and [1, +8)
??

?,?
${\displaystyle \langle \ \rangle \!\,}$

${\displaystyle \langle \ ,\ \rangle \!\,}$
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
(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.
?u1u2, …?is shorthand for ?{u1u2, …}?.

Note that the notation ?uv? may be ambiguous: it could mean the inner product or the linear span.

The span of S may also be written as Sp(S).

${\displaystyle \left\langle \left({\begin{smallmatrix}1\\0\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\1\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\0\\1\end{smallmatrix}}\right)\right\rangle =\mathbb {R} ^{3}}$.
subgroup generated by a set
the subgroup generated by
${\displaystyle \langle S\rangle }$ means the smallest subgroup of G (where S ? G, a group) containing every element of S.
${\displaystyle \langle g_{1},g_{2},...\rangle }$ is shorthand for ${\displaystyle \langle {g_{1},g_{2},...}\rangle }$.
In S3, ${\displaystyle \langle (1\;2)\rangle =\{id,\;(1\;2)\}}$ and ${\displaystyle \langle (1\;2\;3)\rangle =\{id,\;(1\;2\;3),(1\;2\;3))\}}$.
tuple; n-tuple; 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.)

${\displaystyle \langle a,b\rangle }$ is an ordered pair (or 2-tuple).

${\displaystyle \langle a,b,c\rangle }$ is an ordered triple (or 3-tuple).

${\displaystyle \langle \rangle }$ is the empty tuple (or 0-tuple).

?|?

(|)
${\displaystyle \langle \ |\ \rangle \!\,}$

${\displaystyle (\ |\ )\!\,}$
inner product of
?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 ?uv? 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.
|?
${\displaystyle |\ \rangle \!\,}$
the ket …; the vector …
|f? means the vector with label f, which is in a Hilbert space. A qubit's state can be represented as a|0?+ ß|1?, where a and ß are complex numbers s.t. |a|2 + |ß|2 = 1.
?|
${\displaystyle \langle \ |\!\,}$
the bra …; the dual of …
?f| means the dual of the vector |f?, a linear functional which maps a ket |?? onto the inner product ?f|??.
?
${\displaystyle \sum \!\,}$
sum over … from … to … of
${\displaystyle \sum _{k=1}^{n}{a_{k}}}$ means a1 + a2 + … + an. ${\displaystyle \sum _{k=1}^{4}{k^{2}}}$ = 12 + 22 + 32 + 42
= 1 + 4 + 9 + 16 = 30
?
${\displaystyle \prod \!\,}$
product over … from … to … of
${\displaystyle \prod _{k=1}^{n}a_{k}}$ means a1a2···an. ${\displaystyle \prod _{k=1}^{4}(k+2)}$ = (1+2)(2+2)(3+2)(4+2)
= 3 × 4 × 5 × 6 = 360
the Cartesian product of; the direct product of
${\displaystyle \prod _{i=0}^{n}{Y_{i}}}$ means the set of all (n+1)-tuples
(y0, …, yn).
${\displaystyle \prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}}$
${\displaystyle \coprod \!\,}$
coproduct over … from … to … of
A general construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism.
'

${\displaystyle '\!\,}$

${\displaystyle {\dot {\,}}\!\,}$
… 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 dot notation indicates a time derivative. That is ${\displaystyle {\dot {x}}(t)={\frac {\partial }{\partial t}}x(t)}$.

If f(x) := x2, then f '(x) = 2x
?
${\displaystyle \int \!\,}$
indefinite integral of

the antiderivative of
f(x) dx means a function whose derivative is f. ?x2 dx = x3/3 + C
integral from … to … of … with respect to
?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;
line/path/curve integral of … along …
?C f ds means the integral of f along the curve C, ${\displaystyle \textstyle \int _{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,dt}$, where r is a parametrization of C.

(If the curve is closed, the symbol ? may be used instead, as described below.)
?
${\displaystyle \oint \!\,}$
contour integral of
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 ${\displaystyle \oint _{C}{1 \over z}\,dz=2\pi i}$.
?
${\displaystyle \nabla \!\,}$ ?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 dot, divergence of
${\displaystyle \nabla \cdot {\vec {v}}={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}}$ If ${\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }$, then ${\displaystyle \nabla \cdot {\vec {v}}=3y+2yz}$.
curl of
${\displaystyle \nabla \times {\vec {v}}=\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right)\mathbf {i} }$
${\displaystyle +\left({\partial v_{x} \over \partial z}-{\partial v_{z} \over \partial x}\right)\mathbf {j} +\left({\partial v_{y} \over \partial x}-{\partial v_{x} \over \partial y}\right)\mathbf {k} }$
If ${\displaystyle {\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k} }$, then ${\displaystyle \nabla \times {\vec {v}}=-y^{2}\mathbf {i} -3x\mathbf {k} }$.
?
${\displaystyle \partial \!\,}$
partial, d
?f/?xi means the partial derivative of f with respect to xi, where f is a function on (x1, …, xn). If f(x,y) := x2y, then ?f/?x = 2xy
boundary of
?M means the boundary of M ?{x : ||x|| = 2} = {x : ||x|| = 2}
degree of
?f means the degree of the polynomial f.

(This may also be written deg f.)
?(x2 − 1) = 2
?
${\displaystyle \Delta \!\,}$
delta; change in
?x means a (non-infinitesimal) change in x.

(If the change becomes infinitesimal, d and even d are used instead. Not to be confused with the symmetric difference, written ?, above.)
${\displaystyle {\tfrac {\Delta x}{\Delta y}}}$ is the gradient of a straight line
d
${\displaystyle \delta \!\,}$
Dirac delta of
${\displaystyle \delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}}$ d(x)
Kronecker delta of
${\displaystyle \delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}}$ dij
p
${\displaystyle \pi \!\,}$
Projection of
${\displaystyle \pi _{a_{1},...,a_{n}}(R)}$ restricts ${\displaystyle R}$ to the ${\displaystyle \{a_{1},...,a_{n}\}}$ attribute set. ${\displaystyle \pi _{Age,Weight}(Person)}$
s
${\displaystyle \sigma \!\,}$
Selection of
The selection ${\displaystyle \sigma _{a\theta b}(R)}$ selects all those tuples in ${\displaystyle R}$ for which ${\displaystyle \theta }$ holds between the ${\displaystyle a}$ and the ${\displaystyle b}$ attribute. The selection ${\displaystyle \sigma _{a\theta v}(R)}$ selects all those tuples in ${\displaystyle R}$ for which ${\displaystyle \theta }$ holds between the ${\displaystyle a}$ attribute and the value ${\displaystyle v}$. ${\displaystyle \sigma _{Age\geq 34}(Person)}$
${\displaystyle \sigma _{Age=Weight}(Person)}$
<:

${\displaystyle <:\!\,}$

${\displaystyle {<}{\cdot }\!\,}$
is covered by
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 a subtype of
T1 <: T2 means that T1 is a subtype of T2. If S <: T and T <: U then S <: U (transitivity).

${\displaystyle {}^{\dagger }\!\,}$
A means the transpose of the complex conjugate of A.[7]

If A = (aij) then A = (Teimpléad:Overline).
T
${\displaystyle {}^{\mathsf {T}}\!\,}$
transpose
AT means A, but with its rows swapped for columns.

This may also be written At or Atr.
If A = (aij) then AT = (aji).

?
${\displaystyle \bot \!\,}$
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 ${\displaystyle \mathbb {R} ^{3}}$, ${\displaystyle (\mathbb {R} ^{2})^{\perp }\cong \mathbb {R} }$.
is coprime to
x ? y means x has no factor in common with y. 34  ?  55.
the bottom element
? means the smallest element of a lattice. ?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. {ep} ? {1, 2, e, 3, p} under set containment.

*
${\displaystyle *\!\,}$
convolution, convolved with
f * g means the convolution of f and g. ${\displaystyle (f*g)(t)=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau }$.
conjugate
z* means the complex conjugate of z.

(${\displaystyle {\bar {z}}}$ can also be used for the conjugate of z, as described below.)
${\displaystyle (3+4i)^{\ast }=3-4i}$.
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).
{\displaystyle {\begin{aligned}(\mathbb {Z} /5\mathbb {Z} )^{\ast }&=\{[1],[2],[3],[4]\}\\&\cong C_{4}\\\end{aligned}}}
hyperreal numbers
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.
${\displaystyle {\bar {x}}\!\,}$

overbar, … bar
${\displaystyle {\bar {x}}}$ (often read as “x bar”) is the mean (average value of ${\displaystyle x_{i}}$). ${\displaystyle x=\{1,2,3,4,5\};{\bar {x}}=3}$.
conjugate
${\displaystyle {\overline {z}}}$ means the complex conjugate of z.

(z* can also be used for the conjugate of z, as described above.)
${\displaystyle {\overline {3+4i}}=3-4i}$.
algebraic closure of
${\displaystyle {\overline {F}}}$ is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as ${\displaystyle {\overline {\mathbb {Q} }}}$ because it is the algebraic closure of the rational numbers ${\displaystyle {\mathbb {Q} }}$.
(topological) closure of
${\displaystyle {\overline {S}}}$ 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, ${\displaystyle {\overline {\mathbb {Q} }}=\mathbb {R} }$ (the rational numbers are dense in the real numbers).

## Variations

Occasionally, variant forms of mathematical symbols are used. In mathematics written in Arabic, some symbols may be reversed to make right-to-left reading easier. [8]