x < y means x is less than y.

x > y means x is greater than y. 3 < 4
5 > 4 proper subgroup

is a proper subgroup of

H < G means H is a proper subgroup of G. 5Z < Z
A3 < S3 ≪

≫

(very) strict inequality

is much less than,
is much greater than

x ≪ y means x is much less than y.

x ≫ y means x is much greater than y. 0.003 ≪ 1000000 asymptotic comparison

is of smaller order than,
is of greater order than

analytic number theory

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).) x ≪ ex ≤

≥

inequality

is less than or equal to,
is greater than or equal to

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.) 3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5 subgroup

is a subgroup of

computational complexity theory

H ≤ G means H is a subgroup of G. Z ≤ Z
A3 ≤ S3 reduction

is reducible to

A ≤ B means the problemA can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction. If

then

≺

Karp reduction

is Karp reducible to;
is polynomial-time many-one reducible to

computational complexity theory

L1 ≺ L2 means that the problem L1 is Karp reducible to L2.[1]If L1 ≺ L2 and L2 ∈ , then L1 ∈ P. ∝

proportionality

is proportional to;
varies as

everywhere

y ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x. Karp reduction [2]

is Karp reducible to;
is polynomial-time many-one reducible to

computational complexity theory

A ∝ B means the problemA can be polynomially reduced to the problem B. If L1 ∝ L2 and L2 ∈ , then L1 ∈ P. +

addition

plus;
add

4 + 6 means the sum of 4 and 6. 2 + 7 = 9 disjoint union

the disjoint union of ... and ...

A1 + A2 means the disjoint union of sets A1 and 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)} −

subtraction

minus;
take;
subtract

9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5 negative sign

negative;
minus;
the opposite of

−3 means the negative of the number 3. −(−5) = 5 set-theoretic complement

minus;
without

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} ±

plus-minus

plus or minus

6 ± 3 means both 6 + 3 and 6 − 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3. plus-minus

plus or minus

measurement

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. ∓

minus-plus

minus or plus

6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5). cos(x± y) = cos(x) cos(y) ∓ sin(x) sin(y). ×

multiplication

times;
multiplied by

3 × 4 means the multiplication of 3 by 4.

(The symbol * is generally used in programming languages, where ease of typing and use of ASCIItext is preferred.) 7 × 8 = 56 Cartesian product

the Cartesian product of ... and ...;
the direct product of ... and ...

X×Y means the set of all ordered pairswith 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)} cross product

cross

u × v means the cross product of vectorsu and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2) group of units

the group of units of

ring theory

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, orU(R). *

multiplication

times;
multiplied by

a * b means the product of a and b.

(Multiplication can also be denoted with × or ⋅, or even simple juxtaposition. * is generally used where ease of typing and use of ASCII text is preferred, such as programming languages.) 4 * 3 means the product of 4 and 3, or 12. convolution

convolution;
convolved with

functional analysis

f * g means the convolution of f and g. . complex conjugate

conjugate

z* means the complex conjugate of z.

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

the group of units of

ring theory

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, orU(R). hyperreal numbers

the (set of) hyperreals

non-standard analysis

*R means the set of hyperreal numbers. Other sets can be used in place of R. *N is the hypernaturalnumbers. Hodge dual

Hodge dual;
Hodge star

*v means the Hodge dual of a vector v. If vis a k-vectorwithin an n-dimensional oriented inner product space, then *v is an (n−k)-vector. If are the standard basis vectors of , ·

multiplication

times;
multiplied by

3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56 dot product

dot

tensor product, tensor product of modules

u · v means the dot product of vectorsu and v (1,2,5) · (3,4,−1) = 6 placeholder

(silent)

functional analysis

A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. ⊗

tensor product of

means the tensor product of V and U.[3]means the tensor product of modules Vand U over the ringR. {1, 2, 3, 4} ⊗ {1, 1, 2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} ÷

⁄

division(Obelus)

divided by;
over

6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = 0.5

12 ⁄ 4 = 3 quotient group

mod

G / H means the quotient of group Gmodulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a,b+2a}} quotient set

mod

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) } √

square root

the (principal) square root of

real numbers

means the nonnegative number whose square is . complex square root

the (complex) square root of

if is represented in polar coordinates with , then . x

mean

overbar;
… bar

statistics

(often read as "x bar") is the mean (average value of ). . complex conjugate

conjugate

means the complex conjugate of z.

(z* can also be used for the conjugate of z, as described above.) . finite sequence, tuple

finite sequence, tuple

model theory

means the finite sequence/tuple . . algebraic closure

algebraic closure of

field theory

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

(topological) closure of

topology

is the topological closure of the set S.

This may also be denoted as cl(S) orCl(S). In the space of the real numbers, (the rational numbers are dense in the real numbers). |…|

absolute value;
modulus

absolute value of; modulus of

|x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3

|-5| = |5| = 5

| i | = 1

| 3 + 4i | = 5 Euclidean norm or Euclidean length or magnitude

Euclidean norm of

|x| means the (Euclidean) length of vectorx. For x = (3,-4)
determinant

determinant of

matrix theory

|A| means the determinant of the matrix A cardinality

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. …

norm

norm of;
length of

x means the norm of the element x of a normed vector space.[4]x + y ≤ x + y nearest integer function

nearest integer to

x means the nearest integer to x.

(This may also be written [x], ⌊x⌉, nint(x) orRound(x).) 1 = 1, 1.6 = 2, −2.4 = −2, 3.49 = 3 ∣

∤

divisor, divides

divides

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.) Since 15 = 3×5, it is true that 3|15 and 5|15. conditional probability

given

probability

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

restriction of … to …;
restricted to

f|A means the function f restricted to the set A, that is, it is the function with domainA ∩ dom(f) that agrees with f. The function f : R → R defined by f(x) = x2 is not injective, but f|R+ is injective. such that

such that;
so that

everywhere

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

parallel

is parallel to

x y means x is parallel to y. If l m and m ⊥ n then l ⊥ n. incomparability

is incomparable to

x y means x is incomparable to y. {1,2} {2,3} under set containment. exact divisibility

exactly divides

pa n means pa exactly divides n (i.e. pa divides nbut pa+1 does not). 23 360. #

cardinality

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

connected sum of;
knot sum of;
knot composition of

topology, knot theory

A#B is the connected sum of the manifolds Aand B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition. A#Sm is homeomorphicto A, for any manifold A, and the sphere Sm. primorial

primorial

n# is product of all prime numbers less than or equal to n. 12# = 2 × 3 × 5 × 7 × 11 = 2310 ℵ

aleph number

aleph

ℵα represents an infinite cardinality (specifically, theα-th one, where α is an ordinal). |ℕ| = ℵ0, which is called aleph-null. ℶ

beth number

beth

cardinality of the continuum

ℶα represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ). ?

cardinality of the continuum;
c;
cardinality of the real numbers

The cardinality of is denoted by or by the symbol (a lowercase Frakturletter C). :

such that

such that;
so that

everywhere

: means "such that", and is used in proofs and theset-builder notation (described below). ∃ n ∈ ℕ: n is even. field extension

extends;
over

field theory

K : F means the field K extends the field F.

This may also be written as K ≥ F. ℝ : ℚ inner productof matrices

inner product of

A : B means the Frobenius inner product of the matrices A and B.

The general inner product is denoted by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as described below. For spatial vectors, the dot product notation, x·y is common.See also Bra-ket notation. index of a subgroup

index of subgroup

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

factorial

combinatorics

n! means the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24 logical negation

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¬Ais preferred.) !(!A) ⇔ A
x ≠ y ⇔ !(x = y) ~

probability distribution

has distribution

statistics

X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution row equivalence

is row equivalent to

matrix theory

A~B means that B can be generated by using a series of elementary row operations on A same order of magnitude

roughly similar;
poorly approximates

approximation theory

m ~ n means the quantities m and nhave the sameorder 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 asymptotically equivalent

is asymptotically equivalent to

asymptotic analysis

f ~ g means . x ~ x+1 equivalence relation

are in the same equivalence class

everywhere

a ~ b means (and equivalently ). 1 ~ 5 mod 4 ≈

approximately equal

is approximately equal to

everywhere

x ≈ y means x is approximately equal to y.

This may also be written ≃, ≅, ~, ♎ (Libra Symbol),or≒. π ≈ 3.14159 isomorphism

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. ≀

wreath product

wreath product of … by …

A ≀ H means the wreath product of the group A by the group H.

This may also be written A wr H. is isomorphic to the automorphismgroup of thecomplete bipartite graph on (n,n) vertices. ◅

▻

normal subgroup

is a normal subgroup of

N ◅ G means that N is a normal subgroup of groupG. Z(G) ◅ G ideal

is an ideal of

ring theory

I ◅ R means that I is an ideal of ring R. (2) ◅ Z antijoin

the antijoin of

R ▻ S means the antijoin of the relations Rand S, the tuples in R for which there is not a tuple in S that is equal on their common attribute names. RS = R - R S ⋉

⋊

semidirect product

the semidirect product of

N ⋊φ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ. Also, if G = N ⋊φ H, then G is said to split over N.

(⋊ may also be written the other way round, as ⋉, or as ×.) semijoin

the semijoin of

R ⋉ S is the semijoin of the relations Rand S, the set of all tuples in R for which there is a tuple in Sthat is equal on their common attribute names. R S = a1,..,an(R S) ⋈

natural join

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. ∴

therefore

therefore;
so;
hence

everywhere

Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal. ∵

because

because;
since

everywhere

Sometimes used in proofs before reasoning. 3331 is prime ∵ it has no positive integer factors other than itself and one. ■

□

∎

▮

‣

end of proof

QED;
tombstone;
Halmos symbol

everywhere

Used to mark the end of a proof.

(May also be written Q.E.D.) D'Alembertian

non-Euclidean Laplacian

propositional logic, Heyting algebra

It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions. ⇒

→

⊃

material implication

implies;
if … then

A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.

(→ may mean the same as ⇒, or it may have the meaning for functionsgiven below.)

(⊃ may mean the same as ⇒,[5]or it may have the meaning for supersetgiven below.) x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since xcould be −2). ⇔

↔

material equivalence

if and only if;
iff

A ⇔ B means A is true if B is true and A is false if Bis false. x + 5 = y+ 2 ⇔ x + 3 = y ¬

˜

logical negation

not

propositional logic, lattice theory

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) ∧

logical conjunction or meetin a lattice

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

wedge product;
exterior product

exterior algebra

u ∧ v means the wedge product of any multivectorsuand v. In three dimensional Euclidean space the wedge product and the cross product of two vectorsare each other's Hodge dual. exponentiation

… (raised) to the power of …

everywhere

a ^ b means a raised to the power of b

(a ^ b is more commonly writtenab. The symbol ^ is generally used in programming languages where ease of typing and use of plain ASCII text is preferred.) 2^3 = 23 = 8 ∨

logical disjunction or joinin a lattice

or;
max;
join

propositional logic, lattice theory

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. ⊕

⊻

exclusive or

xor

propositional logic, Boolean algebra

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

direct sum of

abstract algebra

The direct sum is a special way of combining several objects into one general object.

(The bun symbol ⊕, or the coproductsymbol ∐, is used; ⊻ is only for logic.) Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0}) ∀

universal quantification

for all;
for any;
for each

predicate logic

∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n. ∃

existential quantification

there exists;
there is;
there are

predicate logic

∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even. ∃!

uniqueness quantification

there exists exactly one

predicate logic

∃! x: P(x) means there is exactly one x such thatP(x) is true. ∃! n ∈ ℕ: n + 5 = 2n. =:

:=

≡

:⇔

≜

≝

≐

definition

is defined as;
is equal by definition to

everywhere

x := y, y =: x 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 equivalentto Q. ≅

congruence

is congruent to

△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. isomorphic

is isomorphic to

abstract algebra

G ≅ H means that group G is isomorphic (structurally identical) to group H.

(≈ can also be used for isomorphic, as described above.) . ≡

congruence relation

... is congruent to ... modulo ...

modular arithmetic

a ≡ b (mod n) means a − b is divisible by n 5 ≡ 2 (mod 3) { , }

setbrackets

the set of …

{a,b,c} means the set consisting of a, b, and c.[6]ℕ = { 1, 2, 3, …} { : }

{ | }

{ ; }

set builder notation

the set of … such that

{x : P(x)} means the set of all xfor which P(x) is true.[6]{x | P(x)} is the same as {x : P(x)}. {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4} ∅

{ }

empty set

the empty set

∅ means the set with no elements.[6]{ } means the same. {n ∈ ℕ : 1 < n2 < 4} = ∅ ∈

∉

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;[6]a ∉ Smeans a is not an element of S.[6](1/2)−1 ∈ ℕ

2−1 ∉ ℕ ⊆

⊂

subset

is a subset of

(subset) A ⊆ B means every element of A is also an element of B.[7]

(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 ⊇

⊃

superset

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

ℝ ⊃ ℚ ∪

set-theoretic union

the union of … or …;
union

set-theoretic intersection

A ∪ B means the set of those elements which are either in A, or in B, or in both.[7]A ⊆ B ⇔ (A ∪ B) = B ∩

intersected with;
intersect

A ∩ B means the set that contains all those elements that A and B have in common.[7]{x ∈ ℝ : x2 = 1} ∩ ℕ = {1} ∆

symmetric difference

A ∆ B means the set of elements in exactly one of Aor B.

(Not to be confused with delta, Δ, described below.) {1,5,6,8} ∆ {2,5,8} = {1,2,6} ∖

set-theoretic complement

minus;
without

A ∖ B means the set that contains all those elements of A that are not in B.[7]

(− can also be used for set-theoretic complement as described above.) {1,2,3,4} ∖ {3,4,5,6} = {1,2} →

functionarrow

from … to

set theory, type theory

f: X → Y means the function f maps the set X into the set Y. Let f: ℤ → ℕ∪{0} be defined by f(x) := x2. ↦

functionarrow

maps to

f: a ↦ b means the function f maps the element a to the element b. Let f: x ↦ x+1 (the successor function). ∘

function composition

composed with

f∘g is the function, such that (f∘g)(x) = f(g(x)).[8]if f(x) := 2x, and g(x) := x + 3, then (f∘g)(x) = 2(x + 3). o

Hadamard product

entrywise product

For two matrices (or vectors) of the same dimensions the Hadamard product is a matrix of the same dimensions with elements given by . This is often used in matrix based programming such as MATLABwhere the operation is done by A.*B ℕ

natural numbers

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 theoristsand computer scientists prefer the former. To avoid confusion, always check an author's definition of N.

Set theorists often use the notation ω (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation ≤. ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| > 0: a ∈ ℤ} ℤ

Z

integers

Z;
the (set of) integers

ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}.

ℤ+ or ℤ> means {1, 2, 3, ...} . ℤ* or ℤ≥ means {0, 1, 2, 3, ...} .

ℤ = {p, −p : p ∈ ℕ ∪ {0}} ℤn

ℤp

Zn

Zp

integers mod n

Zn;
the (set of) integers modulon

ℤn 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, useℤ/pℤ or ℤ/(p) instead. ℤ3 = {[0], [1], [2]} p-adic integers

the (set of) p-adic integers

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

projective space

P;
the projective space;
the projective line;
the projective plane

topology

ℙ means a space with a point at infinity. , probability

the probability of

probability theory

ℙ(X) means the probability of the event Xoccurring.

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

Q

rational numbers

Q;
the (set of) rational numbers;
the rationals

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

π ∉ ℚ ℝ

R

real numbers

R;
the (set of) real numbers;
the reals

ℝ means the set of real numbers. π ∈ ℝ

√(−1) ∉ ℝ ℂ

C

C;
the (set of) complex numbers

ℂ means {a + b i : a,b ∈ ℝ}. i = √(−1) ∈ ℂ ℍ

H

quaternionsor Hamiltonian quaternions

H;
the (set of) quaternions

Computational complexity theory

ℍ means {a + b i + c j + d k : a,b,c,d ∈ ℝ}. O

Big O notation

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 ∞

infinity

∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. ⌊…⌋

floor

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 ⌈…⌉

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) orceiling(x).) ⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2 ⌊…⌉

nearest integer function

nearest integer to

degree of a field extension

⌊x⌉ means the nearest integer to x.

(This may also be written [x], x, nint(x) orRound(x).) ⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊-3.4⌉ = -3, ⌊4.49⌉ = 4 [ : ]

the degree of

field theory

[K : F] means the degree of the extension K: F. [ℚ(√2) : ℚ] = 2

[ℂ : ℝ] = 2

[ℝ : ℚ] = ∞ [ ]

[ , ]

[ , , ]

equivalence class

the equivalence class of

abstract algebra

[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 iffa ≡ b (mod 5).

Then [2] = {…, −8, −3, 2, 7, …}.

floor

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 function

nearest integer to

[x] means the nearest integer to x.

(This may also be written ⌊x⌉, x, nint(x) orRound(x). Not to be confused with the floor function, as described above.) [2] = 2, [2.6] = 3, [-3.4] = -3, [4.49] = 4 Iverson bracket

1 if true, 0 otherwise

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

image of … under …

everywhere

f[X] means { f(x) : x ∈ X }, the image of the function funder the set X ⊆ dom(f).

(This may also be written as f(X) if there is no risk of confusing the image of f underX with the function application f ofX. Another notation is Im f, the image of f under its domain.) closed interval

closed interval

group theory, ring theory

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

the commutator of

[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). xy = x[x, y] (group theory).

[AB, C] = A[B, C] + [A, C]B (ring theory). triple scalar product

the triple scalar product of

[a, b, c] = a × b · c, the scalar product of a ×b withc. [a, b, c] = [b, c, a] = [c, a, b].

( , )

functionapplication

f(x) means the value of the function f at the elementx. If f(x) := x2, then f(3) = 32 = 9. image

image of … under …

everywhere

f(X) means { f(x) : x ∈ X }, the image of the function funder the set X ⊆ dom(f).

(This may also be written as f[X] if there is a risk of confusing the image of f underX with the function application f ofX. Another notation is Im f, the image of f under its domain.) combinations

(from) n choose r

combinatorics

means the number of combinations of relements drawn from a set of n elements.

(This may also be written as nCr.) 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

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

highest common factor;
greatest common divisor; hcf; gcd

number theory

(a, b) means the highest common factor of aand b.

(This may also be written hcf(a, b) orgcd(a, b).) (3, 7) = 1 (they are coprime); (15, 25) = 5. ( , )

] , [

open interval

(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation]a,b[ can be used instead.)

4 is not in the interval (4, 18).

(0, +∞) equals the set of positive real numbers.

(( ))

multichoose

combinatorics

means n multichoose k. ( , ]

] , ]

left-open interval

half-open interval;
left-open interval

. (−1, 7] and (−∞, −1] [ , )

[ , [

right-open interval

half-open interval;
right-open interval

. [4, 18) and [1, +∞) ⟨⟩

⟨,⟩

inner product

inner product of

⟨u,v⟩ means the inner product of u and v, where uand v are members of an inner product space.

Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the inner product or the linear span.

There are many variants of the notation, such as⟨u | v⟩ and (u | v), which are described below. For spatial vectors, the dot product notation, x·y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts. The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is:
⟨x, y⟩ = 2 × −1 + 3 × 5 = 13 average

average of

statistics

let S be a subset of N for example, represents the average of all the element in S. for a time series :g(t) (t = 1, 2,...)

we can define the structurefunctions Sq():

linear span

(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.
⟨u1, u2, …⟩is shorthand for ⟨{u1, u2, …}⟩.

Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the inner productor the linear span.

The span of S may also be written asSp(S).

. subgroup generated by a set

the subgroup generated by

means the smallest subgroup of G (where S⊆G, a group) containing every element of S.
is shorthand for . In S3, and . tuple

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.)

is an ordered pair (or 2-tuple).

is an ordered triple (or 3-tuple).

is the empty tuple(or 0-tuple).

⟨|⟩

(|)

inner product

inner product of

⟨u | v⟩ means the inner product of u and v, where uand v are members of an inner product space.[9](u| v) means the same.

Another variant of the notation is ⟨u, v⟩ which is described above. For spatial vectors, the dot productnotation, 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. |⟩

ket vector

the ket …;
the vector …

Dirac notation

|φ⟩ means the vector with label φ, which is in aHilbert space. A qubit's state can be represented as α|0⟩+ β|1⟩, where αand β are complex numbers s.t. |α|2 + |β|2 = 1. ⟨|

bra vector

the bra …;
the dual of …

Dirac notation

⟨φ| means the dual of the vector |φ⟩, a linear functional which maps a ket |ψ⟩ onto the inner product ⟨φ|ψ⟩. ∑

summation

sum over … from … to … of

means a1 + a2 + … + an. = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30 ∏

product

product over … from … to … of

means a1a2···an. = (1+2)(2+2)(3+2)(4+2) = 3 × 4 × 5 × 6 = 360 Cartesian product

the Cartesian product of;
the direct product of

means the set of all (n+1)-tuples(y0, …, yn). ∐

coproduct

coproduct over … from … to … of

category theory

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. Δ

delta

delta;
change in

Δx means a (non-infinitesimal) change in x.

(If the change becomes infinitesimal, δ and evendare used instead. Not to be confused with the symmetric difference, written ∆, above.) is the gradient of a straight line Laplacian

Laplace operator

The Laplace operator is a second order differential operator in n-dimensional Euclidean space If ƒ is a twice-differentiable real-valued function, then the Laplacian of ƒ is defined by δ

Dirac delta function

Dirac delta of

hyperfunction

δ(x) Kronecker delta

Kronecker delta of

hyperfunction

δij ∂

partial derivative

partial;
d

∂f/∂xi means the partial derivative of f with respect toxi, where f is a function on (x1, …, xn). If f(x,y) := x2y, then ∂f/∂x = 2xy boundary

boundary of

topology

∂M means the boundary of M ∂{x : x ≤ 2} = {x : x = 2} degree of a polynomial

degree of

algebra

∂f means the degree of the polynomial f.

(This may also be written deg f.) ∂(x2 − 1) = 2 ∇

gradient

del;
nabla;
gradientof

∇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) divergence

del dot;
divergence of

If , then . curl

curl of

If , then . ′

derivative

… prime;
derivative of

f ′(x) means the derivative of the function f at the pointx, i.e., the slope of the tangentto f at x.

(The single-quote character ' is sometimes used instead, especially in ASCII text.) If f(x) := x2, then f ′(x) = 2x •

derivative

… dot;
time derivative of

indefinite integral orantiderivative

means the derivative of x with respect to time. That is . If x(t) := t2, then . ∫

indefinite integral of
the antiderivative of

∫ f(x) dx means a function whose derivative is f. ∫x2 dx = x3/3 + C definite integral

integral from … to … of … with respect to

∫ab f(x) dx means the signed area between the x-axis and the graph of the functionf between x = a and x =b. ∫ab x2 dx = b3/3 − a3/3; line integral

line/ path/ curve/ integral of… along…

∫C f ds means the integral of falong the curve C, , where r is a parametrization of C.

(If the curve is closed, the symbol ∮ may be used instead, as described below.) ∮

Contour integral;
closed line integral

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 capitalS, ∮S, is used to denote that the integration is over a closed surface.

If C is a Jordan curveabout 0, then . π

projection

Projection of

restricts to the attribute set. Pi

pi;
3.1415926;
≈22÷7

mathematical constant

Used in various formulas involving circles; π is equivalent to the amount of area a circle would take up in a square of equal width with an area of 4 square units, roughly 3.14/4. It is also the ratio of thecircumferenceto the diameter of a circle. A=πR 2=314.16→R=10 σ

selection

Selection of

The selection selects all those tuples in for which holds between the and the attribute. The selection selects all those tuples in for which holds between the attribute and the value .
<:

<·

cover

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. subtype

is a subtype of

type theory

T1 <: T2 means that T1 is a subtype of T2. If S <: T and T <: Uthen S <: U (transitivity). †

conjugate transpose

conjugate transpose;
adjoint;
Hermitian adjoint/conjugate/transpose

matrix operations

A† means the transpose of the complex conjugate ofA.[10]

This may also be written A*T, AT*, A*, AT or AT. If A = (aij) then A† = (aji). T

transpose

matrix operations

AT means A, but with its rows swapped for columns.

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

top element

the top element

lattice theory

⊤ means the largest element of a lattice. ∀x : x ∨ ⊤ = ⊤ top type

the top type; top

type theory

⊤ means the top or universal type; every type in thetype system of interest is a subtype of top. ∀ types T, T <: ⊤ ⊥

perpendicular

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 complement

orthogonal/ perpendicular complement of;
perp

W⊥ means the orthogonal complement of W(whereW is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W. Within , . coprime

is coprime to

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

is independent of

probability

A ⊥ B means A is an event whose probability is independent of event B. If A ⊥ B, then P(A|B)= P(A). bottom element

the bottom element

lattice theory

⊥ means the smallest element of a lattice. ∀x : x∧ ⊥ = ⊥ bottom type

the bottom type;
bot

type theory

⊥ 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 comparability

is comparable to

propositional logic,predicate logic

x ⊥ y means that x is comparable to y. {e, π} ⊥ {1, 2, e, 3, π} under set containment. ⊧

entailment

entails

model theory

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 ⊢

inference

infers;
is derived from

x ⊢ y means y is derivable from x. A → B ⊢ ¬B → ¬A. partition

is a partition of

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