NationMaster - Encyclopedia: Table of mathematical symbols

disjoint union

A₁ + A₂ means the disjoint union of sets A₁ and A₂.

A₁ = {1, 2, 3, 4} ∧ A₂ = {2, 4, 5, 7} ⇒
A₁ + A₂ = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}

the disjoint union of ... and ...

−

subtraction

9 − 4 means the subtraction of 4 from 9.

8 − 3 = 5

minus

negative sign

−3 means the negative of the number 3.

−(−5) = 5

negative; minus

set-theoretic complement

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}

minus; without

multiplication

3 × 4 means the multiplication of 3 by 4.

7 × 8 = 56

times

Cartesian product

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)}

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

cross product

u × v means the cross product of vectors u and v

(1,2,5) × (3,4,−1) =
(−22, 16, − 2)

cross

vector algebra

multiplication

3 · 4 means the multiplication of 3 by 4.

7 · 8 = 56

times

dot product

u · v means the dot product of vectors u and v

(1,2,5) · (3,4,−1) = 6

dot

vector algebra

÷

⁄

division

6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.

2 ÷ 4 = .5

12 ⁄ 4 = 3

divided by

plus-minus

6 ± 3 means both 6 + 3 and 6 - 3.

The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.

plus or minus

plus-minus

10 ± 2 or eqivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.

If a = 100 ± 1 mm, then a is ≥ 99 mm and ≤ 101 mm.

plus or minus

measurement

∓

minus-plus

6 ± (3 ∓ 5) means both 6 + (3 - 5) and 6 - (3 + 5).

cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).

minus or plus

√

square root

√x means the positive number whose square is x.

√4 = 2

the principal square root of; square root

real numbers

complex square root

if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(i φ/2).

√(-1) = i

the complex square root of …

square root

complex numbers

|…|

absolute value or modulus

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

|3| = 3

|–5| = |5|

| i | = 1

| 3 + 4i | = 5

absolute value (modulus) of

propositional logic, lattice theory

Euclidean distance

|x – y| means the Euclidean distance between x and y.

For x = (1,1), and y = (4,5),
|x – y| = √([1–4]² + [1–5]²) = 5

Euclidean distance between; Euclidean norm of

Geometry

Determinant

|A| means the determinant of the matrix A

$begin{vmatrix} 1&2 2&4 end{vmatrix} = 0$

determinant of

Matrix theory

divides

A single vertical bar is used to denote divisibility.
a|b means a divides b.

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

divides

Number Theory

factorial

n ! is the product 1 × 2× ... × n.

4! = 1 × 2 × 3 × 4 = 24

factorial

combinatorics

transpose

Swap rows for columns

A i j = (A T) j i

transpose

matrix operations

probability distribution

X ~ D, means the random variable X has the probability distribution D.

X ~ N(0,1), the standard normal distribution

has distribution

statistics

Row equivalence

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

$begin{bmatrix} 1&2 2&4 end{bmatrix} sim begin{bmatrix} 1&2 0&0 end{bmatrix}$

is row equivalent to

Matrix theory

⇒

→

⊃

material implication

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 ⇒, or it may have the meaning for superset given below.

x = 2 ⇒ x² = 4 is true, but x² = 4 ⇒ x = 2 is in general false (since x could be −2).

implies; if … then

propositional logic, Heyting algebra

⇔

↔

material equivalence

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

if and only if; iff

propositional logic

¬

˜

logical negation

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.)

¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y)

not

propositional logic

∧

logical conjunction or meet in a lattice

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.

and; min

∨

logical disjunction or join in a lattice

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.

or; max

propositional logic, lattice theory

⊕

⊻

exclusive or

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.

xor

propositional logic, Boolean algebra

direct sum

The direct sum is a special way of combining several modules into one general module (the 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 = ∅)

direct sum of

Abstract algebra

∀

universal quantification

∀ x: P(x) means P(x) is true for all x.

∀ n ∈ ℕ: n² ≥ n.

for all; for any; for each

predicate logic

∃

existential quantification

∃ x: P(x) means there is at least one x such that P(x) is true.

∃ n ∈ ℕ: n is even.

there exists

predicate logic

∃!

uniqueness quantification

∃! x: P(x) means there is exactly one x such that P(x) is true.

∃! n ∈ ℕ: n + 5 = 2n.

there exists exactly one

predicate logic

:=

≡

:⇔

definition

x := y or x ≡ y means x is defined to be another name for y

(Some writers use ≡ to mean congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.

cosh x := (1/2)(exp x + exp (−x))

A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B)

is defined as

everywhere

≅

congruence

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

is congruent to

geometry

≡

congruence relation

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

5 ≡ 11 (mod 3)

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

modular arithmetic

{ , }

set brackets

{a,b,c} means the set consisting of a, b, and c.

ℕ = { 1, 2, 3, …}

the set of …

{ : }

{ | }

set builder notation

{x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}.

{n ∈ ℕ : n² < 20} = { 1, 2, 3, 4}

the set of … such that

∅

{ }

empty set

∅ means the set with no elements. { } means the same.

{n ∈ ℕ : 1 < n² < 4} = ∅

the empty set

∈

∉

set membership

a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S.

(1/2)⁻¹ ∈ ℕ

2⁻¹ ∉ ℕ

is an element of; is not an element of

everywhere, set theory

⊆

⊂

subset

(subset) A ⊆ B means every element of A is also element of B.

(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

is a subset of

⊇

⊃

superset

A ⊇ B means every element of B is also 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

is a superset of

∪

set-theoretic union

(exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both.
"A or B, but not both."

(inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B.
"A or B or both".

A ⊆ B ⇔ (A ∪ B) = B (inclusive)

the union of … and …

union

set-theoretic intersection

∩

A ∩ B means the set that contains all those elements that A and B have in common.

{x ∈ ℝ : x² = 1} ∩ ℕ = {1}

intersected with; intersect

Δ

symmetric difference

A Δ B

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

{1,5,6,8}

Δ

{2,5,8} = {1,2,6}

symmetric difference

∖

set-theoretic complement

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

− can also be used for set-theoretic complement as described above.

{1,2,3,4} ∖ {3,4,5,6} = {1,2}

minus; without

( )

function application

f(x) means the value of the function f at the element x.

If f(x) := x², then f(3) = 3² = 9.

precedence grouping

Perform the operations inside the parentheses first.

(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.

parentheses

everywhere

f:X→Y

function arrow

f: X → Y means the function f maps the set X into the set Y.

Let f: ℤ → ℕ be defined by f(x) := x².

from … to

set theory,type theory

function composition

fog is the function, such that (fog)(x) = f(g(x)).

if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).

composed with

ℕ

N

natural numbers

N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.

ℕ = {|a| : a ∈ ℤ, a ≠ 0}

ℤ

Z

integers

ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ⁺ means {1, 2, 3, ...} = ℕ.

ℤ = {p, -p : p ∈ ℕ} ∪ {0}

ℚ

Q

rational numbers

ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.

3.14000... ∈ ℚ

π ∉ ℚ

ℝ

R

real numbers

ℝ means the set of real numbers.

π ∈ ℝ

√(−1) ∉ ℝ

ℂ

C

complex numbers

ℂ means {a + b i : a,b ∈ ℝ}.

i = √(−1) ∈ ℂ

arbitrary constant

C can be any number, most likely unknown; usually occurs when calculating antiderivatives.

if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C, where F'(x) = f(x)

integral calculus

𝕂

K

real or complex numbers

K means the statement holds substituting K for R and also for C.

$x^2inmathbb{C},forall xin mathbb{K}$

because
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Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... The plus and minus signs (+ and âˆ’) are used to represent the notions of positive and negative as well as the operations of addition and subtraction. ... 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In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... Intuitionistic Type Theory, or Constructive Type Theory, or Martin-LÃ¶f Type Theory or just Type Theory (with capital letters) is at the same time a functional programming language, a logic and a set theory based on the principles of mathematical constructivism. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... A number is an abstract idea used in counting and measuring. ... The integers are commonly denoted by the above symbol. ... A number is an abstract idea used in counting and measuring. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... A number is an abstract idea used in counting and measuring. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A number is an abstract idea used in counting and measuring. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... A number is an abstract idea used in counting and measuring. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... This article deals with the concept of an integral in calculus. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

$x^2inmathbb{C},forall xin mathbb{R}$

and

$x^2inmathbb{C},forall xin mathbb{C}$ .

∞

infinity

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

lim_x→0 1/|x| = ∞

infinity

||…||

norm

|| x || is the norm of the element x of a normed vector space.

|| x + y || ≤ || x || + || y ||

norm of

length of

∑

summation

$sum_{k=1}^{n}{a_k}$ means a₁ + a₂ + … + a_n. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... The infinity symbol âˆž in several typefaces. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as... A number is an abstract idea used in counting and measuring. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... For evaluation of sums in closed form see evaluating sums. ...

$sum_{k=1}^{4}{k^2}$ = 1² + 2² + 3² + 4²

= 1 + 4 + 9 + 16 = 30

sum over … from … to … of

∏

product

$prod_{k=1}^na_k$ means a₁a₂···a_n. Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...

$prod_{k=1}^4(k+2)$ = (1+2)(2+2)(3+2)(4+2)

= 3 × 4 × 5 × 6 = 360

product over … from … to … of

Cartesian product

$prod_{i=0}^{n}{Y_i}$ means the set of all (n+1)-tuples Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...

(y₀, …, y_n).

$prod_{n=1}^{3}{mathbb{R}} = mathbb{R}timesmathbb{R}timesmathbb{R} = mathbb{R}^3$

the Cartesian product of; the direct product of

∐

coproduct

coproduct over … from … to … of

category theory

′

^•

derivative

f ′(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.

The dot notation indicates a time derivative. That is $dot{x}(t)=frac{partial}{partial t}x(t)$ . In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... For a non-technical overview of the subject, see Calculus. ... Look up Slope in Wiktionary, the free dictionary. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

If f(x) := x², then f ′(x) = 2x

… prime

derivative of

∫

indefinite integral or antiderivative

∫ f(x) dx means a function whose derivative is f.

∫x² dx = x³/3 + C

indefinite integral of

the antiderivative of

definite integral

∫_a^b f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.

∫₀^b x² dx = b³/3;

integral from … to … of … with respect to

∮

contour integral or closed line integral

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 this formula involves a closed surface integral, the representation describes only the first or initial 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 ∰.

This symbol can also frequently be found with a subscript capital letter C, ∮_C, denoting that the closed loop integral is 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. Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ... This article deals with the concept of an integral in calculus. ... Area is a physical quantity expressing the size of a part of a surface. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ... In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ... In mathematics â€” in particular, in multivariable calculus â€” a volume integral refers to an integral over a 3-dimensional domain. ...

contour integral of

∇

gradient

∇f (x₁, …, x_n) is the vector of partial derivatives (∂f / ∂x₁, …, ∂f / ∂x_n).

If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)

del, nabla, gradient of

vector calculus

divergence

$nabla cdot vec v = {partial v_x over partial x} + {partial v_y over partial y} + {partial v_z over partial z}$

If $vec v := 3xymathbf{i}+y^2 zmathbf{j}+5mathbf{k}$ , then $nabla cdot vec v = 3y + 2yz$ .

del dot, divergence of

vector calculus

curl

$nabla times vec v = left( {partial v_z over partial y} - {partial v_y over partial z} right) mathbf{i} + left( {partial v_x over partial z} - {partial v_z over partial x} right) mathbf{j} + left( {partial v_y over partial x} - {partial v_x over partial y} right) mathbf{k}$

If $vec v := 3xymathbf{i}+y^2 zmathbf{j}+5mathbf{k}$ , then $nablatimesvec v = -y^2mathbf{i} - 3xmathbf{k}$ .

curl of

vector calculus

∂

partial differential

With f (x₁, …, x_n), ∂f/∂x_i is the derivative of f with respect to x_i, with all other variables kept constant.

If f(x,y) := x²y, then ∂f/∂x = 2xy

partial, d

propositional logic, predicate logic

boundary

∂M means the boundary of M

∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}

boundary of

topology

⊥

perpendicular

x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.

If l ⊥ m and m ⊥ n then l || n.

is perpendicular to

geometry

bottom element

x = ⊥ means x is the smallest element.

∀x : x ∧ ⊥ = ⊥

the bottom element

lattice theory

parallel

x || y means x is parallel to y.

If l || m and m ⊥ n then l ⊥ n.

is parallel to

geometry

⊧

entailment

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

entails

model theory

⊢

inference

x ⊢ y means y is derived from x.

A → B ⊢ ¬B → ¬A

infers or is derived from

◅

normal subgroup

N ◅ G means that N is a normal subgroup of group G.

Z(G) ◅ G

is a normal subgroup of

group theory

quotient group

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}}

mod

group theory

quotient set

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]}

mod

≈

isomorphism

G ≈ H means that group G is isomorphic to group H

Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group.

is isomorphic to

group theory

approximately equal

x ≈ y means x is approximately equal to y

π ≈ 3.14159

is approximately equal to

everywhere

same order of magnitude

m ~ n, means the quantities m and n have the general size.

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

2 ~ 5

8 × 9 ~ 100

but π² ≈ 10

roughly similar

poorly approximates

Approximation theory

〈,〉

( | )

< , >

·

:

inner product

〈x,y〉 means the inner product of x and y as defined in an inner product space.

For spatial vectors, the dot product notation, x·y is common.
For matricies, the colon notation may be used. Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ... For other uses, see Gradient (disambiguation). ... In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ... Nabla is a symbol, shown as . ... For other uses, see Gradient (disambiguation). ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Fig. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Implication or entailment is used in propositional logic and predicate logic to describe a relationship between two sentences or sets of sentences. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Inference is the act or process of deriving a conclusion based solely on what one already knows. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ... This article is about the mathematical group. ... Group theory is that branch of mathematics concerned with the study of groups. ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ... It has been suggested that this article or section be merged with estimation. ... In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...

The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is:
〈x, y〉 = 2×−1 + 3×5 = 13

A:B =	∑	A_ijB_ij
	i,j

inner product of

⊗

tensor product

V ⊗ U means the tensor product of V and U.

{1, 2, 3, 4} ⊗ {1,1,2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}

tensor product of