Mathematics:Symbol

Α, α

Alpha (ALPHA)

Β, β

Beta (BETA)

Γ, γ

Gamma (GAMMA)

Δ, δ

Delta (DELTA

Ε, ε

Epsilon (EPSILON)

The lowercase letter is used as the symbol for permittivity.

Ζ, ζ

Zeta (ZETA)

Η, η

Eta (ETA)

Used as the symbol for efficiency.

Θ, θ

Theta (THETA)

The lowercase letter denotes the size of an angle.

Ι, ι

Iota (IOTA)

Κ, κ

Kappa (KAPPA)

Λ, λ

Lambda (LAMBDA)

Μ, μ

Mu (MU)

The lowercase letter is used as the symbol for permeability.

Μ, μ

Micro or micron

One millionth

Ν, ν

Nu (NU)

Ξ, ξ

Xi (XI)

Ο, ο

Omicron (OMICRON)

Π, π

Pi (PI)

The uppercase pi is used as the symbol for 'product (cumulative multiplication)', and the lowercase as the symbol for the ratio of a circle's circumference to its diameter (3.14).

Ρ, ρ

Rho (RHO)

The lowercase letter is used as the unit of resistivity.

Σ, σ

Sigma (SIGMA)

The uppercase is used as the symbol for 'summation (cumulative addition)', and the lowercase as the symbol for conductivity.

Τ, τ

Tau (TAU)

Υ, υ

Upsilon (UPSILON)

Φ, φ

Phi (PHI)

The uppercase is used as the symbol for magnetic flux.

Χ, χ

Chi (CHI)

Ψ, ψ

Psi (PSI)

Ω, ω

Omega (OMEGA)

The uppercase is used as the unit of resistance (read as "ohm"), and the lowercase as the symbol for angular velocity.

∂

Round D

Partial derivative symbol

∇

Nabla

Differential operator symbol

∬

Double integral

Multiple integral symbol (integration must be done twice.)

∮

Circular integral

Line integral symbol (also called a closed contour integral (周回積分 – an integral taken along a closed curve))

σ

Sigma (lowercase)

Symbol for the standard deviation.

Σ

Sigma (uppercase)

Used as a symbol for sums, with a subscript and a superscript written on it.

i

Eye.

The imaginary unit. The number whose square is -1.

√

Root

Square root.

∫

Integral

Integral symbol.

∴

Therefore, or hence

∵

Because

≒

Approximate value.

dθ

D-theta

Symbol used in differentiation.

≡

Symbol for congruence or modulo (mod) = symbol for congruence of figures.

∈

(The left is an) element (of the right).

∋

(The right is an) element (of the left).

⊂

(The left is a) subset (of the right). (The right set) contains (the left set).

⊃

(The right is a) subset (of the left). (The right set) contains (the left set).

∪

Union.

∩

Intersection.

∀

For any.

∃

There exists. exist.

∞

Infinity

!

Factorial

WLOG

Without Loss Of Generality

Without loss of generality

Mainly used in mathematics when distinguishing the size or order of the letters does not affect the proof.

s.t.

such that

Such that

Mainly used when adding a condition.

iff

if and only if

Necessary and sufficient condition

States that two propositions are equivalent.

TFAE

The Following Are Equivalent

The following are equivalent

Used to state that several propositions are equivalent. The difference from iff is that there can be three or more propositions.

ETS

Enough To Show

It is enough to show the following

When proving a proposition, proving another proposition equivalent to it completes the proof. This abbreviation is used in such cases.

Q.E.D.

Quod Erat Demonstrandum

End of proof

An abbreviation used at the end of a proof, but many mathematicians think using it is showing off, so ■ or □ is used more often.

cyc

cyclic

Cyclic

When the indices in a sum or product are cyclic, writing just cyc is enough.

sym

symmetric

Symmetric

Used when the indices in a sum or product are symmetric.

Def

Definition

Definition

Lemma

Lemma

Lemma (auxiliary theorem)

Thm

Theorem

Theorem

Cor

Corollary

Corollary

e.g.

exemplī grātiā

For example

ex is also used. Sometimes read as "for example".

i.e.

id est

That is

Sometimes read as "that is".

Eq

Equation

The given equation

Equation itself means an equation, but it is sometimes used in a proof, when continuing after explaining an equality, to avoid writing the same expression again.

C

Constant of integration

The most common letter used for the constant of integration.

c

Constant

constant; the most common letter used for a constant.

D

Domain

Domain; the most common letter used for a domain.

Diagonal matrix

Diagonal matrix

d

Distance

distane; letter used for distance in measure theory or geometry.

Diameter

diameter

E

Expected value

Expectation value; letter used for the expected value in statistics. Usually written as E(X).

Elementary matrix

In linear algebra, denotes a matrix obtained by applying an elementary operation to the identity matrix.

e

Euler's number

Identity element

Letter for the identity element in abstract algebra.

F

Field

Field; letter used for a general field in abstract algebra.

Antiderivative

In analysis, the letter mainly used for the indefinite integral of a function f.

f

Function

function; the most common letter used for a general function.

G

Group

Group; letter for a general group in abstract algebra.

\(\mathbb{H}\)

\mathbb{H}

Set of quaternions

Denotes the set of quaternions. The H comes from Hamilton's name.

H

Subgroup

Common letter for a subgroup of G.

h

Height

height; common letter for height in geometry.

I

Ideal

Ideal; letter for an ideal in abstract algebra.

Identity function

Identity function; the usual letter for the identity function in analysis.

Identity matrix

Identity matrix; letter for the identity matrix in linear algebra. A _n is sometimes appended to indicate the size of the matrix.

i

Imaginary unit

The usual letter for the imaginary unit in algebra. Usually defined as \(\sqrt{-1}\).

Index

index; often used for an index in sums or products.

Quaternion

One of the basic elements of the quaternions.

J

Jordan matrix

In linear algebra, a matrix whose entries above the diagonal are all 1, whose diagonal entries are eigenvalues, and whose remaining entries are all 0.

j

Quaternion

One of the basic elements of the quaternions.

k

Field

Used together with F to denote a field. Note that it is lowercase.

Quaternion

One of the basic elements of the quaternions.

\(\mathcal{L}\)

\mathcal{L}

Laplace transform

Letter for the Laplace transform.

L

Lower triangular matrix

Lower triangular matrix; in linear algebra, a matrix whose entries above the diagonal are all 0.

l

Length

length; letter for length in geometry.

M

Matrix

Matrix; letter for a matrix in linear algebra.

m

Mean

mean; letter for the arithmetic mean.

\(\mathbb{N}\)

\mathbb{N}

Set of natural numbers

Depending on the book, 0 may be included or excluded.

n

Natural number

Letter for an arbitrary natural number.

O

Asymptotic notation

Big O

P

Polynomial

Notation for a general polynomial.

p

Prime

prime; the most common letter for a prime number.

\(\mathbb{Q}\)

\mathbb{Q}

Set of rational numbers

Q

Quotient polynomial

Quotient; letter for the quotient in polynomial division.

q

Quotient

quotient; letter for the quotient in division of numbers.

\(\mathbb{R}\)

\mathbb{R}

Set of real numbers

R

Remainder polynomial

Remainder; letter for the remainder in polynomial division.

Ring

Ring; the usual letter for a ring in abstract algebra.

r

Remainder

remainder; letter for the quotient in division of numbers.

Radius

radius

S

Sum

Sum; letter mainly used to denote a sum.

T

Transpose

Transposition; placed at the top of a matrix to denote the transposed matrix.

t

Time

time; the most common letter for time.

U

Universal set

Universal set

Upper triangular matrix

Upper triangular matrix; in linear algebra, a matrix whose entries below the diagonal are all 0.

V

Klein four-group

In abstract algebra, the letter denoting \(\left\{e,\,\left(12\right)\left(34\right),\,\left(13\right)\left(24\right),\,\left(14\right)\left(23\right)\right\}\).

v

Vector

The most common letter for a vector.

W

Wronskian

A determinant that checks whether finitely many functions are linearly independent

X

Set

Letter often used for an arbitrary set.

Domain

Letter often used, together with D, for the domain of a function.

x

Unknown

Independent variable

Y

Codomain

Letter often used for the codomain of a function.

y

Function

Together with f, the most common letter for a function.

Dependent variable

\(\mathbb{Z}\)

\mathbb{Z}

Set of integers

z

Complex number

The most common letter for an arbitrary complex number.

ℵ

\aleph

Aleph

Cardinality of an infinite set

For reference, this is not a Greek letter but a Hebrew letter.

α

\alpha

Alpha

Root

Often used to denote a root of an equation.

Β

\Beta

Beta

Beta function

\(\Beta\left(x,y\right)=\int_0^1t^{x-1}\left(1-t\right)^{y-1}\mathrm{d}t\)

Γ

\Gamma

Gamma

Gamma function

\(\Gamma\left(x\right)=\int_0^{\infty}x^{t-1}e^{-x}\mathrm{d}x\)

Roots of unity

\(\Gamma_n=\left\{e^{\frac{2\pi ik}{n}}\mid k\in\mathbb{Z}\right\}\)

ɣ

\gamma

Euler–Mascheroni constant

\(\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{1}{k}-\ln n\right)\)

∆

\Delta

Delta

Difference

Used to denote the difference between two values. An increment is one example.

Laplacian

\(\Delta=\nabla^2\)

δ

\delta

Dirac delta function

\(\delta\left(x\right)=\begin{cases}\infty,&x=0\\0,&x\neq0\end{cases}\)

Kronecker delta

\(\delta_{ij}=\begin{cases}1,&i=j\\0,&i\neq j\end{cases}\)

Very small positive number

In analysis, it denotes a small positive number.

∇

\nabla

Del
Nabla

Gradient

\(\nabla=\left(\partial/\partial x_1,\ldots,\partial/\partial x_n\right)\)

Divergence

\(\nabla\cdot\vec{v}=\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}\)

Curl

\(\nabla\times\vec{v}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\\partial x&\partial y&\partial z\\v_x&v_y&v_z\end{vmatrix}\)

ε

\varepsilon

Epsilon

Very small positive number

In analysis, it denotes a small positive number.

ζ

\zeta

Zeta

Riemann zeta function

\(\zeta\left(s\right)=\sum_{i=1}^{\infty}\frac{1}{n^s}\)

θ

\theta

Theta

Angle

Mainly denotes an angle in radians.

ι

\iota

Iota

Imaginary unit

Very occasionally used as the imaginary unit in place of i.

κ

\kappa

Kappa

Curvature

\(\kappa=\left\|\frac{\mathrm{d}\mathbf{T}}{\mathrm{d}s}\right\|\)

Λ

\Lambda

Lambda

Diagonalizing matrix

\(M=\Lambda D\Lambda^{-1}\)

λ

\lambda

Eigenvalue

\(\det\left(X-\lambda I\right)=0\)

Lagrange multiplier

μ

\mu

Mu

Integrating factor

A function multiplied in to solve a differential equation

Mean

In statistics, denotes the mean together with m.

ξ

\xi

Xi

Eigenvector

Π

\Pi
\prod

Pi

Product

\(\prod_{i=1}^nx_i=x_1x_2\cdots x_n\)

π

\pi

Ratio of a circle's circumference to its diameter

∐

\coprod

Coproduct

Disjoint union

ρ

\rho

Rho

Radius

Denotes the radius in spherical coordinates.

Σ

\Sigma
\sum

Sigma

Sum

\(\sum_{i=1}^nx_i=x_1+x_2+\cdots+x_n\)

Symmetry group

Means the symmetry group in geometry.

σ

\sigma

Standard deviation

\(\sigma=\sqrt{E\left[\left(X-\mu\right)^2\right]}\)

Divisor function

\(\sigma_x\left(n\right)=\sum_{d\mid n}d^x\)

Permutation

Symbol for a general permutation in abstract algebra

τ

\tau

Tau

Number of positive positive numbers

\(\tau\left(n\right)=\sigma_0\left(n\right)\)

Diameter

\(\tau=2\pi\)

Torsion

\(\tau=-\mathbf{n}\cdot\mathbf{b}'\)

Transposition

Symbol for a permutation of two elements in abstract algebra

Φ

\Phi

Phi

Cyclotomic polynomial

\(\Phi_n\left(x\right)=\prod_{\underset{\gcd\left(k,n\right)=1}{1\leq k\leq n}}\left(x-e^{2i\pi\frac{k}{n}}\right)\)

φ

\phi

Euler's totient function

\(\phi\left(n\right)=n\) or less: the number of natural numbers among them that are coprime to \(n\)

Golden ratio

\(\phi=\frac{1+\sqrt5}{2}\)

χ

\chi

Chi

Chi-squared test

Used in statistics.

ω

\omega

Omega

Cube root of unity

\(\omega=\frac{1\pm\sqrt{3}i}{2}\)

+

Addition

−

Subtraction

±

\pm

Plus or minus

In a case like ±1 it means positive or negative; in a case like 2±2 it means addition or subtraction.

∓

\mp

Minus or plus

Same as above, only with the order reversed. Used to keep the two apart when the signs must be taken in corresponding order.

×

\times

Multiplication

Cross product

Symbol for the cross product of vectors.

Cartesian product

·

\cdot

Multiplication

Dot product

Symbol for the dot product of vectors.

Placeholder

Used when you do not want to use a letter variable inside another symbol.

÷

\div

Division

⁄

Division

Also used to write fractions.

Quotient group
Quotient ring

Strictly speaking, it denotes the left cosets.

Negation

Drawn over another symbol to reverse that symbol's meaning. Note that its meaning is entirely different from ∖.

√

\sqrt[]{}

Radical sign

:

Proportion

Index of a subgroup

∴

\therefore

Therefore

∵

\because

Because

∃

\exists

There exists

∀

\forall

For all

!

Factorial

Unique

Logical negation

In programming it means negation, but in mathematics it mostly means uniqueness, so be careful.

¬

\lnot

Logical negation

~

\sim

Logical negation

Because ~ has many other meanings, ¬ is usually used for logical negation.

Similarity

Similarity in geometry

Equivalence relation

Approximately equal

Likewise, because ~ has many other meanings, ≈ is used more often.

∝

\propto

Is proportional to

∞

\infty

Infinity

■
□
▶

\blacksquare
\square
\blacktriangleright

End of proof

Squares are mostly used.

=

Equals sign

Means the left and right sides are equivalent.

≠

\neq

Not equal

Means the left and right sides are different.

≈

\approx

Approximately equal

Means the values of the left and right sides are close.

=:
:=
\(\overset{\text{def}}{=}\)

\overset{\text{def}}{=}

Definition

Used when giving a mathematical definition. The symbol used varies widely from book to book.

≡

\equiv

Definition

Sometimes used for mathematical definitions, but it is usually used in the meaning below.

Congruence

Covers both congruence in geometry and congruence in number theory.

≅

\cong

Congruence

Means only congruence in geometry.

Isomorphism

Means an isomorphism in abstract algebra.

↔
⇔

\leftrightarrow
\Leftrightarrow

Equivalence

The single-line arrow means only that the two propositions are true; the double-line arrow means that the equivalence itself is true.

<
>

Comparison of magnitude

Means the side the sign opens toward is larger.

Proper subgroup

Means a subgroup that is not equal to the original group.

≪
≫

\ll
\gg

Comparison of magnitude

Means the side the sign opens toward is much larger.

≤
≥

\leq
\geq

Comparison of magnitude

Means the side the sign opens toward is greater than or equal to the other side.

Subgroup

◅
▻

\triangleleft
\triangleright

Normal subgroup

The side the triangle points to is the subgroup.

Ideal

Used less often than for normal subgroups.

→
⇒

\rightarrow
\Rightarrow

Sufficient condition

The single-line arrow means only a sufficient condition; the double-line arrow means a sufficient condition that is true.

←
⇐

\leftarrow
\Leftarrow

Necessary condition

The single-line arrow means only a necessary condition; the double-line arrow means a necessary condition that is true.

⊃
⊂

\supset
\subset

(Proper) subset

In some books it means just a subset.

⊇
⊆

\supseteq
\subseteq

Subset

\(\supsetneq\)
\(\subsetneq\)

\supsetneq
\subsetneq

Proper subset

→

\rightarrow

Function mapping

The arrow starts at the domain and points to the codomain.

↪

\hookrightarrow

Injective function

Indicates that the function is one-to-one.

↠

\twoheadrightarrow

Surjective function

Indicates that the function is surjective.

↦

\mapsto

Function mapping

The arrow starts at an element of the domain and points to an element of the codomain.

\(\binom{\phantom{n}}{\phantom{k}}\)

\binom{}{}

Combination

\(\left(\binom{\phantom{n}}{\phantom{k}}\right)\)

\left(\binom{}{}\right)

Combination with repetition

|…|

\left| \right|

Absolute value

Norm

Denotes only the Euclidean distance.

Determinant

Cardinality

Size of a set

‖…‖

\left\| right\|

Norm

Includes the Euclidean distance.

{…}

\left\{ \right\}

Sequence

Fractional part of a real number

{ , }

\left\{ , \right\}

Set

Means a set written in roster notation, listing its elements.

{ : }
{ | }
{ ; }

\left\{ : }\right\}
\left\{ \mid\ }right\}
\left\{ ; \right\}

Set

Means a set written in set-builder notation.

⌊…⌋

\left\lfloor \right\rfloor

Floor function

\(\left\lfloor x\right\rfloor\) means the largest of the integers less than \(x\).

⌈…⌉

\left\lceil \right\rceil

Ceiling function

\(\left\lceil x\right\rceil\) means the smallest of the integers greater than \(x\).

[…]

\left[ \right]

Equivalence class

Floor function

Polynomial ring

\(R\left[x\right]\) is the ring of polynomials whose coefficients are elements of the ring \(R\).

[ : ]

\left[ : \right]

Index of a group

The subgroup comes after the colon.

[ , ]

\left[ , \right]

Closed interval

Least common multiple

Sometimes written with lcm in front, sometimes without.

(…)

\left( \right)

Function evaluation

\(f\left(x\right)\) means evaluating the function \(f\) at \(x\).

Image

\(f\left(X\right)\) denotes the image of the function \(f\) whose domain is \(X\).

Sequence

Rarely used, since sequences are mostly written with {…}.

Principal ideal

( , )

\left( , \right)

Open interval

Ordered pair

Greatest common divisor

Sometimes written with gcd in front, sometimes without.

( , ]
[ , )

\left( \right]
\left[ \right)

Half-open interval
Half-closed interval

The ( or ) side is the open end.

<…>

\left< \right>

Subgroup

\(\left< X\right>\) means the subgroup generated by the set \(X\).

Cyclic group

\(\left< g\right>\) means the cyclic group whose generator is \(g\).

< , >

\left< , \right>

Inner product

*

Convolution

\(\left(f*g\right)\left(t\right)=\int_{-\infty}^{\infty}f\left(\tau\right)g\left(t-\tau\right)\mathrm{d}\tau\)

Set of units

\(R^*\) is the set of all units (invertible elements) of the ring \(R\). × is sometimes used instead of *.

∖

\setminus

Set difference

− is also used.

Quotient group
Quotient ring

Strictly speaking, it denotes the right cosets.

|

Conditional probability

\(P\left(A|B\right)\) is the probability that \(A\) occurs given \(B\).

Restriction

\(f|_{D'}\) means restricting the domain of the function \(f\) to \(D'\).

Function evaluation

\(f|_{x=x_0}\) means evaluating the function \(f\) at \(x_0\).

Evaluating an integral

\(F|_a^b\) means computing the value \(F\left(b\right)-\left(a\right)\) for the antiderivative \(F\).

∣

\mid

Divides

\(a\mid b\) means \(b\) is divisible by \(a\).

∤

\nmid

Does not divide

∣∣

\mid\mid

Exactly divides

\(a^n\mid\mid b\) means \(a^n\) divides \(b\) but \(a^{n+1}\) does not.

∥

\parallel

Parallel

A slightly slanted // is also used.

∦

\nparallel

Not parallel

⊕

\oplus

Exclusive or

\(\bar{\phantom{a}}\)

\bar{}

Mean

Algebraic closure

Complex conjugate

\(\overline{\phantom{abc}}\)

\overline{}

Number notation

To denote the three-digit number \(abc\), you can write it as \(\overline{abc}\). The advantage is that it is distinguished from the product \(abc\).

Line segment

\(\vec{\phantom{v}}\)

\vec{}

Vector

Sometimes written in bold instead of with an arrow.

align=center|\(\overrightarrow{\phantom{AB}}\)
\(\overleftarrow{\phantom{AB}}\)

\overrightarrow{}
\overleftarrow{}

Ray

\(\overrightarrow{AB}\) means the ray that starts at \(A\) and extends toward \(B\). If the arrow points the other way, so does the ray.

Vector

\(\overrightarrow{AB}\) denotes the vector with initial point \(A\) and terminal point \(B\).

\(\overleftrightarrow{\phantom{AB}}\)

\overleftrightarrow{}

Line

\(\hat{\phantom{a}}\)

\hat{}

Omission

\(a_1a_2\cdots\hat{a_i}\cdots a_n\) means that only \(a_i\) is left out.

'

Derivative

\(\dot{\phantom{a}}\)

\dot{}

Derivative

∅

\emptyset

Empty set

∪

\cup

Union

∩

\cap

Intersection

\(\triangle\)

\triangle

Symmetric difference

\(A\triangle B=\left(A\setminus B\right)\cup\left(B\setminus A\right)\)

\(\uplus\)
⊔

\uplus
\sqcup

Disjoint union

\(\phantom{A}^{\mathrm{C}}\)
\(\phantom{A}'\)

\mathrm{C}

Complement

\(\mathcal{P}\)

\mathcal{P}

Power set

∈
∋

\in
\ni

Element

The side the symbol points toward is the set.

∉
∌

\notin
\not\ni

Not an element

\(\mathbb{Z}^\cdot\)
\(\mathbb{Q}^\cdot\)
\(\mathbb{R}^\cdot\)

\mathbb{Z}^{}
\mathbb{Q}^{}
\mathbb{R}^{}

Sets of numbers

When • is replaced with +, the set contains only positive numbers; with −, only negative numbers; with ≥0, only 0 and positive numbers; with ≤0, only 0 and negative numbers; with × or *, all numbers other than 0.

\(\mathbb{Z}^\times\)

\mathbb{Z}^\times

Set of units

It can mean the integers other than 0, but it can also mean the set of only the units among the integers. In that case, \(\mathbb{Z}^\times=\left\{\pm1\right\}\).

n(…)

Number of elements

Means the number of elements of a finite set. A symbol of unknown origin used only in Korea.

∧

\land
\wedge

Logical conjunction

Same meaning as AND.

∨

\lor
\vee

Logical disjunction

Same meaning as OR.

⊕
\(\veebar\)
\(\dot{\vee}\)

\oplus
\veebar
\dot{\vee}

Exclusive disjunction

Same meaning as XOR.

↑

\uparrow

Negated conjunction

Same meaning as NAND.

↓

\downarrow

Negated disjunction

Same meaning as NOR.

├

\vdash

Logical derivation

Same meaning as →.

\(\top\)

\top

True

Denotes a proposition that is always true.

⊥

\bot

Contradiction

Denotes a proposition that is always false.

\(\sum_{i=1}^n\)
\(\sum_{i\in I}\)

\sum</code>

Sum

The former adds over a finite or countably infinite set of indices; the latter can also be used for uncountably infinite indices.

\(\prod_{i=1}^n\)
\(\prod_{i\in I}\)

\prod

Product

The former multiplies over a finite or countably infinite set of indices; the latter can also be used for uncountably infinite indices.

\(\left\{a_n\right\}_{n=1}^{\infty}\)
\(\left(a_n\right)_{n=1}^{\infty}\)

\left\{ \right\}
\left( \right)

Sequence

The subscript denotes the starting index and the superscript the ending index. Usually omitted unless it is a special case.

∞

\infty

Infinity

Positive infinity does not need a sign, but negative infinity must always have a − sign.

max

\max

Maximum

min

\min

Minimum

\(\Re\)

\Re

Real part of a complex number

\(\Im\)

\Im

Imaginary part of a complex number

\(f:X\to Y\)
\(X\overset{f}{\to}Y\)

\to
\overset{}{\to}

Function

Function notation that shows the domain and codomain

\(f:x\mapsto y\)
\(x\overset{f}{\mapsto}y\)

\mapsto
\overset{}{\mapsto}

Function

Function notation that shows the element of the codomain corresponding to an element of the domain

\(\phantom{f}^{-1}\)

^{-1}

Inverse function

\(\circ\)

\circ

Composite function

\(f^n\left(x\right)\)

^n

Composite function

Denotes the same function composed \(n\) times. Note that it differs from \(\left(f\left(x\right)\right)^n\).

sin

\sin

Sine

cos

\cos

Cosine

tan

\tan

Tangent

csc

\csc

Cosecant

sec

\sec

Secant

cot

\cot

Cotangent

arcsin
arccos
arctan

\arcsin
\arccos</br>\arctan

Inverse trigonometric functions

sinh
cosh
tanh

\sinh
\cosh
\tanh

Hyperbolic functions

exp

\exp

Exponential function with base \(e\)

\(\log_nx\)

\log_{}{}

Logarithmic function with base \(n\)

log

\log

Common logarithm

Logarithm with base 10. However, it usually means the natural logarithm.

Natural logarithm

ln

\ln

Natural logarithm

sgn

Sign function

\(\lim_{x\to a}f\left(x\right)\)

\lim

Limit

A sequence can also appear in place of the function.

→

\to

Limit

Means the value on the left approaches the value on the right.

\(x\to a^+\)

\to

Right-hand limit

\(x\to a^-\)

\to

Left-hand limit

sup

\sup

Supremum

inf

\inf

Infimum

lim sup

\limsup

Limit superior

lim inf

\liminf

Limit inferior

\(N_{\varepsilon}\left(x_0\right)\)

N_{}\left( \right)

Neighborhood

\(\varepsilon\) is the distance and \(x_0\) is the center point.

\(\mathcal{D}_f\)

\mathcal{D}_{}

Set of discontinuities of a function

'

Derivative

Means the function has been differentiated as many times as there are ' marks. Usually at most three are written; from four on, a different notation is used.

\(f'_+\)

{}'_+

Right derivative

\(f'_-\)

{}'_-

Left derivative

\(f^{\left(n\right)}\left(x\right)\)

^{\left( \right)}

Derivative

Denotes the function differentiated \(n\) times.

\(\mathrm{d}\)

\mathrm{d}

Derivative

Leibniz notation

\(\partial\)

\partial

Partial differentiation

Boundary

Refers to the topological boundary.

\(f_x\)

{}_{}

Partial derivative

Denotes the function partially differentiated with respect to the subscript variable. Partial differentiation is applied from left to right.

\(\mathit{P}\)

\mathit{P}

Partition of an interval

\(\mathit{P}^*\)

\mathit{P}^*

Refinement

\(\mathit{U}\left(\mathit{P},f\right)\)

\mathit{U}

Upper Darboux sum

\(\mathit{P}\) denotes the partition and \(f\) the function.

\(\mathit{L}\left(\mathit{P},f\right)\)

\mathit{L}

Lower Darboux sum

\(\overline{\int}\)

\overline{\int}

Upper integral

\(\underline{\int}\)

\underline{\int}

Lower integral

\(S\left(\mathit{P},f,\xi\right)\)

S\left(\mathit{}, ,\right)

Riemann sum

\(\mathit{P}\) denotes the partition, \(f\) the function, and \(\xi\) an arbitrary point in each subinterval.

\(\int\)

\int

Indefinite integral

\(\int_a^b\)

\int_{}^{}

Definite integral

\(\oint\)

\oint

Line integral

\(\iint\)

\iint

Multiple integral

Surface integral

\(\iiint\)

\iiint

Triple integral

Volume integral

\(\mathcal{B}\)

\mathcal{B}

Set of bounded functions

The interval must also be written after it.

\(\mathcal{C}\)

\mathcal{C}

Set of continuous functions

The interval must also be written after it.

\(C^0\)

C^0

Set of continuous functions

The functions must be continuous at every point of the given domain.

\(C^k\)

C^k

Set of smooth functions

The set of functions that are \(k\) times differentiable and whose derivatives are all continuous.

\(C^\infty\)

C^\infty

Set of smooth functions

The set of functions that are infinitely differentiable and whose derivatives are all continuous.

\(\mathcal{R}\)

\mathcal{R}

Set of Riemann-integrable functions

The interval must also be written after it.

\(\begin{pmatrix}&\\&\end{pmatrix}\)
\(\begin{bmatrix}&\\&\end{bmatrix}\)
\(\left(a_{ij}\right)\)

\begin{pmatrix} & \\ & \end{pmatrix}
\begin{bmatrix} & \\ & \end{bmatrix}
\left({}_{ij}\right)

Matrix

The top two show every entry; the bottom one is a shorthand notation.

\(a_{ij}\)
\(\left(A\right)_{ij}\)

{}_{ij}
\left( \right)_{ij}

Matrix entry

Denotes the entry in row \(i\), column \(j\) of the matrix \(A\).

\(A^{-1}\)

{}^{-1}

Inverse matrix

Never write it as 1/\(A\).

\(A^T\)

{}^T

Transpose matrix

\(\bar{A}\)

\bar{}

Conjugate matrix

\(A^*\)

{}^*

Conjugate transpose

Transpose + conjugate

tr

Trace

det
|A|
\(\begin{vmatrix}&\\&\end{vmatrix}\)

\det
\left| \right|
\begin{vmatrix} & \\ & \end{vmatrix}

Determinant

\(\mathcal{M}_{n\times m}\left(\mathbb{F}\right)\)

\mathcal{M}_{ \times }\left( \right)

Set of matrices

The set of \(n\times m\) matrices whose entries are elements of the field \(\mathbb{F}\).

[A I I]

\left[ \mid \right]

Augmented matrix

O

Zero matrix

E

Identity matrix

I is usually used for the identity matrix.

Elementary matrix

I

Identity matrix

ERO

Elementary row operation

Nothing erotic about it.

ECO

Elementary column operation

\(A_{ij}\)

{}_{ij}

Minor matrix

The matrix obtained by deleting the \(i\)-th row and the \(j\)-th column of the original matrix.

adj

Adjugate matrix

C

Cofactor matrix

\(c_{ij}=\left(-1\right)^{i+j}\det\left(A_{ij}\right)\)

\(\left[T\right]_\alpha^\beta\)

\left[T\right]_{}^{}

Linear transformation

\(\alpha\) and \(\beta\) are bases.

\(\mathcal{L}\left(\mathcal{V},\mathcal{W}\right)\)

\mathcal{L}

Vector space of linear transformations

The set of all \(T:\mathcal{V}\to\mathcal{W}\).

dim

\dim

Dimension

The number of elements in a basis.

span

Span

ker

\ker

Kernel

nul

Nullity

The dimension of the kernel.

range
im

Image

Row

Row space

The vector space made up of the row vectors of a matrix.

Col

Column space

The vector space made up of the column vectors of a matrix.

rank
rk

Rank

The dimension of the image.

S(n)

Proposition

Denotes a proposition about the natural number \(n\).

\(F_n\)

Fibonacci sequence

\(F\) generally denotes a field, but when it is described as a sequence, it usually means this.

\(\Phi_d\left(x\right)\)

\Phi

Cyclotomic polynomial

\(\Phi_d\left(x\right)=\prod\left(x-\xi\right)\), where \(\xi\) is a primitive \(d\)-th root of unity.

\(\phi\left(n\right)\)

\phi

Euler's totient function

\(\phi\left(n\right)=\deg\left(\Phi_n\left(x\right)\right)\)

gcd

\gcd

Greatest common divisor

As a letter, it is usually written \(d\).

lcm

Least common multiple

\(\mathcal{F}\left(X\right)\)

\mathcal{F}

Set of functions

The set of functions whose domain and codomain are both \(X\).

\(S_X\)

S_X

Set of permutations

The set of permutations of \(X\).

\(S_n\)

S_n

Set of permutations

The set of permutations of the natural numbers from 1 to \(n\).

\(A_n\)

A_n

Alternating group

\(\left(i_1\,i_2\,\ldots\,i_r\right)\)

Permutation

A permutation of length \(r\).

\(\left(i\,j\right)\)

Transposition

A permutation of length 2. Usually denoted by the symbol \(\tau\).

e
1

Identity element

Denotes the identity element of a group. Its meaning is slightly different from the number 1.

\(S^1\)

S^1

Circle group

\(S^1=\left\{z\in\mathbb{C}\mid\left|z\right|=1\right\}\)

\(\Gamma_n\)

\Gamma_n

Roots of unity

\(\Gamma_n=\left\{\xi^k\mid0\leq k< n\right\}\). \(\xi\) is an \(n\)-th root of unity.

\(\mathcal{P}\)

\mathcal{P}

Parity group

The group whose elements are the two parities.

\(\mathcal{B}\)

\mathcal{B}

Boolean group

\(GL\left(n,\mathbb{F}\right)\)

General linear group

\(n\) is the size of the square matrices, and \(\mathbb{F}\) is the field.

\(SL\left(n,\mathbb{F}\right)\)

Special linear group

\(SO\left(2,\mathbb{R}\right)\)

Special orthogonal group

The group formed by the matrices that represent rotations in the plane.

\(Aff\left(n,\mathbb{F}\right)\)

Affine group

\(n\) is the dimension, and \(\mathbb{F}\) is the field.

\(Isom\left(\mathbb{R}^n\right)\)

Isometry group

\(n\) denotes the dimension.

\(O\left(n,\mathbb{F}\right)\)

Orthogonal group

\(\Sigma\left(\Omega\right)\)

\Sigma

Symmetry group

Refers to the symmetry group in geometry.

\(D_{2n}\)

D_{2n}

Dihedral group

\(\left< a\right>\)

\left< \right>

Cyclic group

\(a\) is the generator.

\(\left|G\right|\)
\(\left|g\right|\)

\left| \right|

Order

[G : H]

Index of a group

Here \(H\leq G\).

\(\cong\)

\cong

Isomorphism

\(\gamma_g\)

\gamma_g

Conjugation

\(\gamma_g:G\to G,\quad\gamma_g\left(a\right)=gag{^-1}\)

\(\triangleleft\)

\triangleleft

Normal subgroup

Z(G)

Center

\(Z\left(G\right)=\left\{z\in G\mid zg=gz,\,\forall g\in G\right\}\)

\(Q_8\)

Q_8

Quaternion group

\(\mathbb{I}_m\)

\mathbb{I}_m

Complete residue system

The complete residue system modulo \(m\).

\(\mathit{U}\left(\mathbb{I}_m\right)\)

\mathit{U}\left(\mathbb{I}_m\right)

Reduced residue system

The reduced residue system modulo \(m\).

Sub(G; K)

Set of subgroups

When \(K\triangleleft G\), the set of subgroups of \(G\) that contain \(K\).

Sub(G/K)

Set of subgroups of a quotient group

When \(K\triangleleft G\), the set of subgroups of \(G/K\).

\(\mathcal{O}\left(x\right)\)

\mathcal{O}

Orbit

When \(G\) is a group acting on \(X\), \(\mathcal{O}\left(x\right)=\left\{gx\mid g\in G\right\}\).

\(G_x\)

G_x

Stabilizer subgroup

When \(G\) is a group acting on \(X\), \(G_x=\left\{g\in G\mid gx=x\right\}\).

\(C_G\left(x\right)\)
\(x^G\)

C_G
x^G

Centralizer

\(C_G\left(x\right)=\left\{g\in G\mid gxg^{-1}=x\right\}\)

\(\mathbb{Z}\left[i\right]\)

\mathbb{Z}\left[i\right]

Gaussian integers

The set of numbers of the form \(a+bi,\quad a,b\in\mathbb{Z}\).

{0}
(0)

Zero ring

A ring in which 0=1 holds.

u

Unit

In a ring, the common symbol for an element that has a multiplicative inverse.

U(R)

Set of units

The set of all units of the ring R.

\(\mathbb{F}_p\)

\mathbb{F}_p

Field

When \(p\) is prime, \(\mathbb{I}_p\) becomes a field; this symbol is used to denote it.

Frac(R)

Field of fractions

R is a domain.

deg

\deg

Degree

R[x]

Polynomial ring

R is a commutative ring.

k(x)

Field of fractions

When k is a field, \(Frac\left(k\left[x\right]\right):=k\left(x\right)\).

\(\chi_A\left(x\right)\)

\chi

Characteristic function

When \(A\) is a subset of \(X\), \(\chi_A:X\to\mathbb{F}_2\) is defined as \(\chi_A\left(x\right)=\begin{cases}1,&\text{if }x\in A\\0,&\text{if }x\notin A\end{cases}\).

(b)

Principal ideal

b denotes the generator.

LT(f)

Leading term

Denotes the leading term of the function \(f\).

PID

Principal ideal domain

PID is short for Principal Ideal Domain.

\(\partial\)

\partial

Degree function

For a commutative ring R, it suffices that \(\partial:R^\times\to\mathbb{N}\) satisfies the same properties as the degree of a polynomial.

ED

Euclidean domain

ED is short for Euclidean Domain.

UFD

Unique factorization domain

UFD is short for Unique Factorization Domain.

\(\angle\)

\angle

Angle

R

Right angle

The R comes from Right angle.

\(\perp\)

\perp

Perpendicular

\(\phantom{a}^\circ\)

^\circ

Unit of angle

Read as "degree".

'

Unit of angle

Read as "minute"; 60 minutes make 1 degree.

''

Unit of angle

Read as "second"; 60 seconds make 1 minute.

rad

Unit of angle

Read as "radian"; \(\pi\,\mathrm{rad}=180^\circ\). The symbol is usually omitted.

\(\triangle\)

\triangle

Triangle

\(\square\)

\square

Quadrilateral

\(\overset{\frown}{\phantom{a}}\)

\overarc{}
\overset{\frown}{}

Arc

\(_nP_r\)

_{}P_{}

Permutation

The nuance is slightly different from a permutation in abstract algebra.

\(\binom{\phantom{n}}{\phantom{r}}\)
\(_nC_r\)

\binom{}{}
_{}C_{}

Combination

The former is used more often.

\(_n\Pi_r\)

_{}\Pi_{}

Permutation with repetition

A symbol of unknown origin used only in Korea.

\(\left(\binom{\phantom{n}}{\phantom{r}}\right)\)
\(_nH_r\)

\left(\binom{}{}\right)
_{}H_{}

Combination with repetition

The former is used more often. The latter is a symbol of unknown origin used only in Korea.

P

Probability

|

Conditional probability

E

Expected value

V
var

Variance

\(\sigma\)
std
SD

\sigma

Standard deviation

\(\rho\)

\rho

Correlation coefficient

\(\bar{\phantom{x}}\)

\bar{}

Mean

Mainly used for the mean of several indexed variables. The general mean is \(\mu\).

\(\tilde{\phantom{x}}\)

\tilde{}

Median

Mo

Mode

\(\hat{\phantom{p}}\)

\hat{}

Estimator

Q

Quartile

\(Q_1\) denotes the lower 25%, \(Q_2\) the middle, and \(Q_3\) the upper 25%.

B(n,p)

Binomial distribution

\(n\) is the number of trials, and \(p\) is the probability.

\(N\left(\mu,\sigma^{2}\right)\)

Normal distribution

\(\mu\) is the mean, and \(\sigma\) is the standard deviation.

z

Standard score

\(z=\left(x-\mu\right)/\sigma\)