Mathematical Symbols With Name Pdf

This is a list of symbols found within all branches of mathematics to express a formula or to replace a constant.

When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, "≡" may represent congruence or a definition. Further, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. In short, convention dictates the meaning.

Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and in TeX, as an image.

Symbol
in HTML Symbol
in TeX Name Explanation Examples Read as Category

is equal to;
equals

everywhere

means

and

represent the same thing or value. 2 = 2

≠

$\ne$

inequality

is not equal to;
does not equal

everywhere

$x \ne y$ means that

and

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.)

$2 + 2 \ne 5$

strict inequality

is less than,
is greater than

order theory

means

is less than

x > y means is greater than .

proper subgroup

is a proper subgroup of

group theory

means

is a proper subgroup of

$\ll \!\,$

$\gg \!\,$

(very) strict inequality

is much less than,
is much greater than

order theory

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 ≪ e^x $\le \!\,$

$\ge \!\,$

inequality

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

order theory

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

group theory

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

reduction

is reducible to

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. If

$\exists f \in F \mbox{ . } \forall x \in \mathbb{N} \mbox{ . } x \in A \Leftrightarrow f(x) \in B$

then

$A \leq_{F} B$

$\leqq \!\,$

$\geqq \!\,$

...is less than ... is greater than...

7k ≡ 28 (mod 2) is only true if k is an even integer. Assume that the problem requires k to be non-negative; the domain is defined as 0 ≦ k ≦ ∞. 10a ≡ 5 (mod 5) for 1 ≦ a ≦ 10

vector inequality

... is less than or equal... is greater than or equal...

order theory

x ≦ y means that each component of vector x is less than or equal to each corresponding component of vector y.

x ≧ y means that each component of vector x is greater than or equal to each corresponding component of vector y.

It is important to note that x ≦ y remains true if every element is equal. However, if the operator is changed, x ≤ y is true if and only if x ≠ y is also true.

≺

$\prec \!\,$

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

L ₁ ≺ L ₂ means that the problem L ₁ is Karp reducible to L ₂.^[1] If L ₁ ≺ L ₂ and L ₂ ∈ P, then L ₁ ∈ P.

∝

$\propto \!\,$

proportionality

is proportional to;
varies as

everywhere

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

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

A ∝ B means the problem A can be polynomially reduced to the problem B. If L ₁ ∝ L ₂ and L ₂ ∈ P, then L ₁ ∈ P. $+ \!\,$ 4 + 6 means the sum of 4 and 6. 2 + 7 = 9

disjoint union

the disjoint union of ... and ...

set theory

A ₁ + A ₂ means the disjoint union of sets A ₁ and A ₂. A ₁ = {3, 4, 5, 6} ∧ A ₂ = {7, 8, 9, 10} ⇒
A ₁ + A ₂ = {(3,1), (4,1), (5,1), (6,1), (7,2), (8,2), (9,2), (10,2)}

−

$- \!\,$ 9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5

negative;
minus;
the opposite of

−3 means the negative of the number 3. −(−5) = 5

minus;
without

set theory

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}

$\pm \!\,$

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.

∓

$\mp \!\,$

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).

$\times \!\,$

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 ASCII text is preferred.)

7 × 8 = 56

Cartesian product

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

set theory

X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}

cross product

cross

u × v means the cross product of vectors u 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, or U(R).

$\begin{align} (\mathbb{Z} / 5\mathbb{Z})^\times & = \{ [1], [2], [3], [4] \} \ & \cong C_4 \ \end{align}$ $* \!\,$

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

f *g means the convolution of f and g. $(f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau)\, d\tau$ .

complex conjugate

conjugate

z* means the complex conjugate of z.

( $\bar{z}$ can also be used for the conjugate of z, as described below.)

$(3+4i)^\ast = 3-4i$ .

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, or U(R).

$\begin{align} (\mathbb{Z} / 5\mathbb{Z})^\ast & = \{ [1], [2], [3], [4] \} \ & \cong C_4 \ \end{align}$

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 hypernatural numbers.

Hodge dual

Hodge dual;
Hodge star

*v means the Hodge dual of a vector v. If v is a k-vector within an n-dimensional oriented inner product space, then *v is an (n−k)-vector. If $\{e_i\}$ are the standard basis vectors of $\mathbb{R}^5$ , $*(e_1\wedge e_2\wedge e_3)= e_4\wedge e_5$

$\cdot \!\,$

times;
multiplied by

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

dot product

dot

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

placeholder

(silent)

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. $\|\cdot\|$

⊗

$\otimes \!\,$

tensor product, tensor product of modules

tensor product of

$V \otimes U$ means the tensor product of V and U.^[3] $V \otimes_R U$ means the tensor product of modules V and U over the ring R. {1, 2, 3, 4}⊗ {1, 1, 2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} ${\,\wedge\!\!\!\!\!\!\bigcirc\,}$

Kulkarni–Nomizu product

tensor algebra

Derived from the tensor product of two symmetric type (0,2) tensors; it has the algebraic symmetries of the Riemann tensor. $f=g{\,\wedge\!\!\!\!\!\!\bigcirc\,}h$ has components $f_{\alpha\beta\gamma\delta}=g_{\alpha\gamma}h_{\beta\delta}+g_{\beta\delta}h_{\alpha\gamma}-g_{\alpha\delta}h_{\beta\gamma}-g_{\beta\gamma}h_{\alpha\delta}$ . $\div \!\,$

$/ \!\,$

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

group theory

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

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) }

√

$\surd \!\,$

$\sqrt{\ } \!\,$

the (principal) square root of

real numbers

$\sqrt{x}$ means the nonnegative number whose square is

. $\sqrt{4}=2$

the (complex) square root of

if $z=r\,\exp(i\phi)$ is represented in polar coordinates with $-\pi < \phi \le \pi$ , then $\sqrt{z} = \sqrt{r} \exp(i \phi/2)$ . $\sqrt{-1}=i$ $\bar{x} \!\,$

mean

overbar;
… bar

$\bar{x}$ (often read as "x bar") is the mean (average value of x_i

). $x = \{1,2,3,4,5\}; \bar{x} = 3$ .

complex conjugate

conjugate

$\overline{z}$ means the complex conjugate of z.

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

$\overline{3+4i} = 3-4i$ .

finite sequence, tuple

$\overline{a}$ means the finite sequence/tuple (a_1,a_2, ... ,a_n).

. $\overline{a}:=(a_1,a_2, ... ,a_n)$ .

algebraic closure

algebraic closure of

field theory

$\overline{F}$ is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as $\overline{\mathbb{Q}}$ because it is the algebraic closure of the rational numbers ${\mathbb{Q}}$ .

topological closure

(topological) closure of

$\overline{S}$ 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, $\overline{\mathbb{Q}} = \mathbb{R}$ (the rational numbers are dense in the real numbers).

$\hat a$

unit vector

hat

$\mathbf{\hat a}$ (pronounced "a hat") is the normalized version of vector $\mathbf a$ , having length 1.

|…|

$| \ldots | \!\,$

absolute value;
modulus

absolute value of; modulus of

numbers

|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 vector x. For x = (3,-4)
$|\textbf{x}| = \sqrt{3^2 + (-4)^2} = 5$

determinant

determinant of

|A| means the determinant of the matrix A $\begin{vmatrix} 1&2 \ 2&9 \\end{vmatrix} = 5$

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.

||…||

$\| \ldots \| \!\,$

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

numbers

||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 $\mid \!\,$

$\nmid \!\,$

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

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

set theory

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. The function f :R →R defined by f(x) = x ² 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.

is incomparable to

order theory

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

exact divisibility

exactly divides

p ^a ||n means p ^a exactly divides n (i.e. p ^a divides n but p ^a+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

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#S ^m is homeomorphic to A, for any manifold A, and the sphere S ^m.

primorial

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

ℵ

$\aleph \!\,$

aleph number

aleph

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

ℶ

$\beth \!\,$

beth number

beth

ℶ_α represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ). $\beth_1 = |P(\mathbb{N})| = 2^{\aleph_0}.$

𝔠

$\mathfrak c \!\,$

cardinality of the continuum

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

The cardinality of $\mathbb R$ is denoted by $|\mathbb R|$ or by the symbol $\mathfrak c$ (a lowercase Fraktur letter C). $\mathfrak c = {\beth}_{1}$

$: \!\,$

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.

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 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 ⟨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.

$A:B = \sum_{i,j} A_{ij}B_{ij}$

index of a subgroup

index of subgroup

group theory

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 $|G:H| = \frac{|G|}{|H|}$

$! \!\,$

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)

factorial

n! means the product 1 × 2 × ... × n. $4! = 1\times2\times3\times4 = 24$ ${\ \choose\ }$

combination;
binomial coefficent

n choose k

$\begin{pmatrix} n \ k \end{pmatrix}=\frac{n!/(n-k)!}{k!}= \frac{(n-k+1)\cdots(n-2)\cdot(n-1)\cdot n}{k!}$
means (in the case of n = positive integer) the number of combinations of k elements drawn from a set of n elements.

(This may also be written as C(n, k), _n C _k or ⁿ C _k.)

$\begin{pmatrix} 73 \ 5 \end{pmatrix} = \frac{73!/(73-5)!}{5!}=\frac{69\cdot 70\cdot 71\cdot 72\cdot 73}{1\cdot2\cdot3\cdot4\cdot5}=15020334$

$\begin{pmatrix} .5 \ 7 \end{pmatrix}=\frac{-5.5\cdot-4.5\cdot-3.5\cdot-2.5\cdot-1.5\cdot-.5\cdot.5}{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7}=\frac{33}{2048}\,\!$

$\left(\!\!{\ \choose\ }\!\!\right)$

multiset coefficient

u multichoose k

$\left(\!\!{u\choose k}\!\!\right)={u+k-1\choose k}=\frac{(u+k-1)!/(u-1)!}{k!}$
(when u is positive integer)
means reverse or rising binomial coefficient. $\left(\!\!{-5.5\choose 7}\!\!\right)=\frac{-5.5\cdot-4.5\cdot-3.5\cdot-2.5\cdot-1.5\cdot-.5\cdot.5}{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7}={.5\choose 7}=\frac{33}{2048}\,\!$

$\sim \!\,$

probability distribution

has distribution

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

A~B means that B can be generated by using a series of elementary row operations on A $\begin{bmatrix} 1&2 \ 2&4 \\end{bmatrix} \sim \begin{bmatrix} 1&2 \ 0&0 \\end{bmatrix}$

approximation theory

m ~n means the quantities m and n have the same order of magnitude, or general size.

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

2 ~ 5

8 × 9 ~ 100

but π² ≈ 10

asymptotically equivalent

is asymptotically equivalent to

asymptotic analysis

f ~g means $\lim_{n\to\infty} \frac{f(n)}{g(n)} = 1$ . x ~ x+1

equivalence relation

are in the same equivalence class

everywhere

a ~b means $b \in [a]$ (and equivalently $a \in [b]$ ). 1 ~ 5 mod 4

≈

$\approx \!\,$

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

is isomorphic to

group theory

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.

≀

$\wr \!\,$

wreath product

wreath product of … by …

group theory

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

This may also be written A _wr H.

$S_n \wr Z_2$ is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices. $\triangleleft \!\,$

$\triangleright \!\,$

normal subgroup

is a normal subgroup of

group theory

N◅G means that N is a normal subgroup of group G. 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 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 $\triangleright$ S = R - R $\ltimes$ S $\ltimes \!\,$

$\rtimes \!\,$

semidirect product

the semidirect product of

group theory

N ⋊_φH 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 ×.)

$D_{2n} \cong C_n \rtimes C_2$

semijoin

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 $\ltimes$ S = $\Pi$ _{a₁ ,..,a_n}(R $\bowtie$ S)

⋈

$\bowtie \!\,$

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

therefore;
so;
hence

everywhere

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

∵

$\because \!\,$

because

because;
since

everywhere

Sometimes used in proofs before reasoning. 11 is prime ∵ it has no positive integer factors other than itself and one. $\blacksquare \!\,$

$\Box \!\,$

$\blacktriangleright \!\,$

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

vector calculus

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. $\square = \frac{1}{c^2}{\partial^2 \over \partial t^2 } - {\partial^2 \over \partial x^2 } - {\partial^2 \over \partial y^2 } - {\partial^2 \over \partial z^2 }$ $\Rightarrow \!\,$

$\rightarrow \!\,$

$\supset \!\,$

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 ⇒,^[5] or it may have the meaning for superset given below.)

x = 2 ⇒ x ² = 4 is true, but x ² = 4 ⇒ x = 2 is in general false (since x could be −2). $\Leftrightarrow \!\,$

$\leftrightarrow \!\,$

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 $\neg \!\,$

$\sim \!\,$

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)

∧

$\and \!\,$

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 multivectors u and v. In three dimensional Euclidean space the wedge product and the cross product of two vectors are each other's Hodge dual. $u \wedge v = *(u \times v)\ \text{ if } u, v \in \mathbb{R}^3$

… (raised) to the power of …

everywhere

a ^ b means a raised to the power of b

(a ^ b is more commonly written a ^b. The symbol ^ is generally used in programming languages where ease of typing and use of plain ASCII text is preferred.)

2^3 = 2³ = 8

∨

$\or \!\,$

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. $\oplus \!\,$

$\veebar \!\,$

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

direct sum of

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

(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 ∩ W = {0})

∀

$\forall \!\,$

for all;
for any;
for each

predicate logic

∀x: P(x) means P(x) is true for all x. ∀n ∈ℕ: n ² ≥ n.

∃

$\exists \!\,$

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.

∃!

$\exists! \!\,$

there exists exactly one

predicate logic

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

$:= \!\,$

$\equiv \!\,$

$:\Leftrightarrow \!\,$

$\triangleq \!\,$

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

$\doteq \!\,$

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 equivalent to Q.

$\cosh x := \frac{e^x + e^{-x}}{2}$

≅

$\cong \!\,$

congruence

is congruent to

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

isomorphic

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.)

$\mathbb{R}^2 \cong \mathbb{C}$ .

≡

$\equiv \!\,$

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

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

{ , }

${\{\ ,\!\ \}} \!\,$

the set of …

set theory

{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

set theory

{x : P(x)} means the set of all x for which P(x) is true.^[6] {x | P(x)} is the same as {x : P(x)}. {n ∈ℕ : n ² < 20} = { 1, 2, 3, 4} $\empty \!\,$

$\varnothing \!\,$

$\{\} \!\,$

empty set

the empty set

set theory

∅ means the set with no elements.^[6] { } means the same. {n ∈ℕ : 1 <n ² < 4} = ∅ $\in \!\,$

$\notin \!\,$

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

2⁻¹∉ℕ

∋

$\ni$

such that symbol

such that

often abbreviated as "s.t."; : and | are also used to abbreviate "such that". The use of ∋ goes back to early mathematical logic and its usage in this sense is declining. Choose

∋ 2|

and 3|

. (Here | is used in the sense of "divides".)

set membership

contains as an element

S∋

means the same thing as

∈S, where S is a set and

is an element of S.

∌

$\not\ni$

set membership

does not contain as an element

S∌

means the same thing as

∉S, where S is a set and

is not an element of S. $\subseteq \!\,$

$\subset \!\,$

is a subset of

set theory

(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

$\supseteq \!\,$

$\supset \!\,$

is a superset of

set theory

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

∪

$\cup \!\,$

the union of … or …;
union

set theory

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

∩

$\cap \!\,$

intersected with;
intersect

set theory

A ∩B means the set that contains all those elements that A and B have in common.^[7] {x ∈ℝ : x ² = 1} ∩ℕ = {1}

∆

$\vartriangle \!\,$

symmetric difference

set theory

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

(Not to be confused with delta, Δ, described below.)

{1,5,6,8} ∆ {2,5,8} = {1,2,6}

∖

$\setminus \!\,$

minus;
without

set theory

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}

→

$\to \!\,$

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) := x ².

↦

$\mapsto \!\,$

maps to

set theory

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

∘

$\circ \!\,$

function composition

composed with

set theory

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).

$\circ \!\,$

Hadamard product

entrywise product

For two matrices (or vectors) of the same dimensions $A, B \in {\mathbb R}^{m \times n}$ the Hadamard product is a matrix of the same dimensions $A \circ B \in {\mathbb R}^{m \times n}$ with elements given by $(A \circ B)_{i,j} = (A)_{i,j} \cdot (B)_{i,j}$ . This is often used in matrix based programming such as MATLAB where the operation is done by A.*B $\begin{bmatrix} 1&2 \ 2&4 \\end{bmatrix} \circ \begin{bmatrix} 1&2 \ 0&0 \\end{bmatrix} = \begin{bmatrix} 1&4 \ 0&0 \\end{bmatrix}$ $\mathbb{N} \!\,$

$\mathbf{N} \!\,$

N;
the (set of) natural numbers

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

$\mathbf{Z} \!\,$

Z;
the (set of) integers

numbers

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

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

$\mathbf{Z}_n \!\,$

$\mathbf{Z}_p \!\,$

Z_n;
the (set of) integers modulo n

numbers

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 ℤ/p ℤ or ℤ/(p) instead.

ℤ ₃ = {[0], [1], [2]}

p-adic integers

the (set of) p-adic integers

numbers

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

$\mathbb{P} \!\,$

$\mathbf{P} \!\,$

projective space

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

ℙ means a space with a point at infinity. $\mathbb{P}^1$ , $\mathbb{P}^2$

the probability of

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

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

$\mathbf{Q} \!\,$

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

numbers

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

π∉ ℚ

$\mathbb{R} \!\,$

$\mathbf{R} \!\,$

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

numbers

ℝ means the set of real numbers. π ∈ ℝ

√(−1)∉ ℝ

$\mathbb{C} \!\,$

$\mathbf{C} \!\,$

C;
the (set of) complex numbers

numbers

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

$\mathbf{H} \!\,$

quaternions or Hamiltonian quaternions

H;
the (set of) quaternions

numbers

ℍ means {a +bi +cj +dk : a,b,c,d ∈ℝ}.

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 $f(x)=O(g(x))\mbox{ as }x\to\infty\,$

∞

$\infty \!\,$

infinity

numbers

∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. $\lim_{x\to 0} \frac{1}{|x|} = \infty$

⌊…⌋

$\lfloor \ldots \rfloor \!\,$

floor

floor;
greatest integer;
entier

numbers

⌊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

⌈…⌉

$\lceil \ldots \rceil \!\,$

ceiling

numbers

⌈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

⌊…⌉

$\lfloor \ldots \rceil \!\,$

nearest integer function

nearest integer to

numbers

⌊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

[ : ]

$[\ :\ ] \!\,$

degree of a field extension

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

[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

floor;
greatest integer;
entier

numbers

[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

numbers

[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

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 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 Imf, the image of f under its domain.)

$\sin [\mathbb{R}] = [-1, 1]$

closed interval

order theory

$[a,b] = \{x \in \mathbb{R} : a \le x \le b \}$ . 0 and 1/2 are in the interval [0,1].

commutator

the commutator of

group theory, ring theory

[g,h] = g ⁻¹ h ⁻¹ gh (or ghg ⁻¹ h ⁻¹), if g, h ∈ G (a group).

[a,b] = ab − ba, if a, b ∈ R (a ring or commutative algebra).

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

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

triple scalar product

the triple scalar product of

vector calculus

[a,b,c] = a × b · c, the scalar product of a ×b with c. [a,b,c] = [b,c,a] = [c,a,b]. $(\ ) \!\,$

$(\ ,\ ) \!\,$

set theory

f(x) means the value of the function f at the element x. If f(x) := x ², then f(3) = 3² = 9.

image

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 Imf, the image of f under its domain.)

$\sin (\mathbb{R}) = [-1, 1]$

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

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. $(\ ,\ ) \!\,$

$]\ ,\ [ \!\,$

open interval

order theory

$(a,b) = \{x \in \mathbb{R} : a < x < b \}$ .

(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.)

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

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

$(\ ,\ ] \!\,$

$]\ ,\ ] \!\,$

left-open interval

half-open interval;
left-open interval

order theory

$(a,b] = \{x \in \mathbb{R} : a < x \le b \}$ . (−1, 7] and (−∞, −1] $[\ ,\ ) \!\,$

$[\ ,\ [ \!\,$

right-open interval

half-open interval;
right-open interval

order theory

$[a,b) = \{x \in \mathbb{R} : a \le x < b \}$ . [4, 18) and [1, +∞) $\langle\ \rangle \!\,$

$\langle\ ,\ \rangle \!\,$

inner product

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

average

average of

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

we can define the structure functions S_q ( $\tau$ ):

$S_q = \langle |g(t + \tau) - g(t)|^q \rangle_t$

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.
⟨u ₁,u ₂, …⟩is shorthand for ⟨{u ₁,u ₂, …}⟩.

Note that the notation ⟨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).

$\left\lang \left( \begin{smallmatrix} 1 \ 0 \ 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} 0 \ 1 \ 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} 0 \ 0 \ 1 \end{smallmatrix} \right) \right\rang = \mathbb{R}^3$ .

subgroup generated by a set

the subgroup generated by

group theory

$\langle S \rangle$ means the smallest subgroup of G (where S ⊆ G, a group) containing every element of S.
$\langle g_1, g_2, \ldots, \rangle$ is shorthand for $\langle g_1, g_2, \ldots \rangle$ . In S₃, $\langle(1 \; 2) \rangle = \{id,\; (1 \; 2)\}$ and $\langle (1 \; 2 \; 3) \rangle = \{id, \; (1 \; 2 \; 3),(1 \; 2 \; 3))\}$ .

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.)

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

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

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

$\langle\ |\ \rangle \!\,$

$(\ |\ ) \!\,$

inner product

inner product of

⟨u |v⟩ means the inner product of u and v, where u and 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 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.

|⟩

$|\ \rangle \!\,$

ket vector

the ket …;
the vector …

Dirac notation

|φ⟩ means the vector with label φ, which is in a Hilbert space. 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.

⟨|

$\langle\ | \!\,$

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⟩.

∑

$\sum$

summation

sum over … from … to … of

$\sum_{k=1}^{n}{a_k}$ means $a_1 + a_2 + \cdots + a_n$ . $\sum_{k=1}^{4}{k^2} = 1^2 + 2^2 + 3^2 + 4^2 ::= 1 + 4 + 9 + 16 = 30$

∏

$\prod$

product over … from … to … of

$\prod_{k=1}^na_k$ means $a_1 a_2 \dots a_n$ . $\prod_{k=1}^4(k+2) = (1+2)(2+2)(3+2)(4+2) ::= 3 \times 4 \times 5 \times 6 = 360$

Cartesian product

the Cartesian product of;
the direct product of

set theory

$\prod_{i=0}^{n}{Y_i}$ means the set of all (n+1)-tuples

(y ₀, …, y _n).

$\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3$

∐

$\coprod \!\,$

coproduct

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.

$\Delta \!\,$

delta

delta;
change in

Δx 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.)

$\tfrac{\Delta y}{\Delta x}$ is the gradient of a straight line

Laplacian

Laplace operator

vector calculus

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 $\Delta f = \nabla^2 f = \nabla \cdot \nabla f$

$\delta \!\,$

Dirac delta function

Dirac delta of

hyperfunction

$\delta(x) = \begin{cases} \infty, & x = 0 \ 0, & x \ne 0 \end{cases}$ δ(x)

Kronecker delta

Kronecker delta of

hyperfunction

$\delta_{ij} = \begin{cases} 1, & i = j \ 0, & i \ne j \end{cases}$ δ_ij

Functional derivative

Functional derivative of

Differential operators

$\begin{align}\left\langle \frac{\delta F[\varphi(x)]}{\delta\varphi(x)}, f(x) \right\rangle &= \int \frac{\delta F[\varphi(x)]}{\delta\varphi(x')} f(x')dx' \&= \lim_{\varepsilon\to 0}\frac{F[\varphi(x)+\varepsilon f(x)]-F[\varphi(x)]}{\varepsilon} \&= \left.\frac{d}{d\epsilon}F[\varphi+\epsilon f]\right|_{\epsilon=0}.\end{align}$ $\frac{\delta V(r)}{\delta \rho(r')} = \frac{1}{4\pi\epsilon_0|r-r'|}.$

∂

$\partial \!\,$

partial derivative

partial;
d

∂f/∂x _i means the partial derivative of f with respect to x _i, where f is a function on (x ₁, …, x _n). If f(x,y) := x ² y, then ∂f/∂x = 2xy

boundary

boundary of

∂M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}

degree of a polynomial

degree of

∂f means the degree of the polynomial f.

(This may also be written deg f.)

∂(x ² − 1) = 2

∇

$\nabla \!\,$

vector calculus

∇f (x₁, …, x_n) is the vector of partial derivatives (∂f / ∂x ₁, …, ∂f / ∂x _n). If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)

divergence

del dot;
divergence of

vector calculus

$\nabla \cdot \vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z}$ If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$ , then $\nabla \cdot \vec v = 3y + 2yz$ .

curl

curl of

vector calculus

$\nabla \times \vec v = \left( {\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \mathbf{i}$
$+ \left( {\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \mathbf{j} + \left( {\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \mathbf{k}$ If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$ , then $\nabla\times\vec v = -y^2\mathbf{i} - 3x\mathbf{k}$ .

′

$' \!\,$

… 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 single-quote character ' is sometimes used instead, especially in ASCII text.)

If f(x) :=x ², then f ′(x) = 2x

^•

$\dot{\,} \!\,$

… dot;
time derivative of

$\dot{x}$ means the derivative of x with respect to time. That is $\dot{x}(t)=\frac{\partial}{\partial t}x(t)$ . If x(t) :=t ², then $\dot{x}(t)=2t$ .

∫

$\int \!\,$

indefinite integral or antiderivative

indefinite integral of
the antiderivative of

∫f(x) dx means a function whose derivative is f. ∫x ² dx = x ³/3 + C

definite integral

integral from … to … of … with respect to

∫_a ^bf(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. $\int_{a}^{b} x^2 dx= \frac{b^3 - a^3}{3}$

line integral

line/ path/ curve/ integral of… along…

∫_Cf ds means the integral of f along the curve C, $\textstyle \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt$ , where r is a parametrization of C.

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

∮

$\oint \!\,$

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

If C is a Jordan curve about 0, then $\oint_C {1 \over z}\,dz = 2\pi i$ .

$\pi \!\,$

projection

Projection of

$\pi_{a_1, \ldots,a_n}( R )$ restricts

to the $\{a_1,\ldots,a_n\}$ attribute set. $\pi_{\text{Age,Weight}}(\text{Person})$

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 the circumference to the diameter of a circle. A=πR²=314.16→R=10 $\sigma \!\,$

selection

Selection of

The selection $\sigma_{a \theta b}( R )$ selects all those tuples in

for which $\theta$ holds between the

and the

attribute. The selection $\sigma_{a \theta v}( R )$ selects all those tuples in

for which $\theta$ holds between the

attribute and the value

. $\sigma_{Age \ge 34}( Person )$
$\sigma_{Age = Weight}( Person )$ $<: \!\,$

${<}{\cdot} \!\,$

cover

is covered by

order theory

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

T ₁ <:T ₂ means that T ₁ is a subtype of T ₂. If S <:T and T <:U then S <:U (transitivity).

^†

${}^\dagger \!\,$

conjugate transpose

conjugate transpose;
adjoint;
Hermitian adjoint/conjugate/transpose

matrix operations

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

This may also be written A ^*T, A ^T*, A ^*, A ^T or A ^T .

If A = (a _ij) then A ^† = ( a _ji ).

${}^{\mathsf{T}} \!\,$

transpose

matrix operations

A ^T means A, but with its rows swapped for columns.

This may also be written A ^' , A ^t or A ^tr.

If A = (a _ij) then A ^T = (a _ji).

⊤

$\top \!\,$

top element

the top element

lattice theory

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

the top type; top

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

⊥

$\bot \!\,$

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 (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 $\mathbb{R}^3$ , $(\mathbb{R}^2)^{\perp} \cong \mathbb{R}$ .

coprime

is coprime to

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

is independent of

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 ∧ ⊥ = ⊥

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

order theory

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

⊧

$\vDash \!\,$

entailment

entails

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

⊢

$\vdash \!\,$

inference

infers;
is derived from

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

partition

is a partition of

p⊢n means that p is a partition of n. (4,3,1,1)⊢ 9, $\sum_{\lambda \vdash n} (f_\lambda)^2 = n!$ .

$\vdots \!\,$

vertical ellipsis

everywhere

Denotes that certain constants and terms are missing out (e.g. for clarity) and that only the important terms are being listed. $P(r,t) = \chi \vdots E(r,t_1)E(r,t_2)E(r,t_3)$