Cac Ky Hieu Toan Hoc

7/31/2019 Cac Ky Hieu Toan Hoc

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Symbols

Symbolin

HTML

Symbolin T E X

NameExplanation ExamplesRead as

Category

=

equality

is equal to;equals

everywhere

x =y meansx andy represent the same thing orvalue.

2 = 21 + 1 = 2

inequality

is not equal to;does not equal

everywhere

x y means thatx andy do not represent the samething or value.

(The forms !=, /= or are generally used inprogramming languages where ease of typing

and use ofASCIItext is preferred.)

2 + 2 5

strict inequality

is less than,is greater than

order theory

x y meansx is greater thany.

3 < 45 > 4

proper subgroup

is a proper subgroup of

group theory

H< G meansHis a proper subgroup ofG.5Z < ZA3 < S3
http://en.wikipedia.org/wiki/HTMLhttp://en.wikipedia.org/wiki/TeXhttp://en.wikipedia.org/wiki/Equals_signhttp://en.wikipedia.org/wiki/Equality_(mathematics)http://en.wikipedia.org/wiki/Not_equals_signhttp://en.wikipedia.org/wiki/Inequality_(mathematics)http://en.wikipedia.org/wiki/ASCIIhttp://en.wikipedia.org/wiki/Less-than_signhttp://en.wikipedia.org/wiki/Greater-than_signhttp://en.wikipedia.org/wiki/Inequality_(mathematics)http://en.wikipedia.org/wiki/Order_theoryhttp://en.wikipedia.org/wiki/Proper_subgrouphttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/HTMLhttp://en.wikipedia.org/wiki/TeXhttp://en.wikipedia.org/wiki/Equals_signhttp://en.wikipedia.org/wiki/Equality_(mathematics)http://en.wikipedia.org/wiki/Not_equals_signhttp://en.wikipedia.org/wiki/Inequality_(mathematics)http://en.wikipedia.org/wiki/ASCIIhttp://en.wikipedia.org/wiki/Less-than_signhttp://en.wikipedia.org/wiki/Greater-than_signhttp://en.wikipedia.org/wiki/Inequality_(mathematics)http://en.wikipedia.org/wiki/Order_theoryhttp://en.wikipedia.org/wiki/Proper_subgrouphttp://en.wikipedia.org/wiki/Group_theory


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Karp reduction

is Karp reducible to;is polynomial-time many-

one reducible to

computational complexitytheory

L1L2 means that the problemL1 is Karpreducible toL2.[1]

IfL1L2 andL2 P, thenL1 P.

proportionality

is proportional to;varies as

everywhere

yx means thaty = kx for some constant k. ify = 2x, thenyx.

Karp reduction[2]

is Karp reducible to;is polynomial-time many-

one reducible to

computational complexitytheory

AB means theproblemA can be polynomiallyreduced to the problemB.

IfL1L2 andL2 P, thenL1 P.

+ additionplus;add

arithmetic

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

disjoint union A1 +A2 means the disjoint union of setsA1 andA2.A1 = {3, 4, 5, 6} A2 = {7, 8, 9, 10}
http://en.wikipedia.org/wiki/Karp_reductionhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-0http://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-0http://en.wikipedia.org/wiki/P_(complexity)http://en.wikipedia.org/wiki/P_(complexity)http://en.wikipedia.org/wiki/Alpha_(letter)http://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Karp_reductionhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-1http://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-1http://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_problemhttp://en.wikipedia.org/wiki/P_(complexity)http://en.wikipedia.org/wiki/Plus_signhttp://en.wikipedia.org/wiki/Additionhttp://en.wikipedia.org/wiki/Plus_and_minus_signshttp://en.wikipedia.org/wiki/Plus_and_minus_signshttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Disjoint_unionhttp://en.wikipedia.org/wiki/Karp_reductionhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-0http://en.wikipedia.org/wiki/P_(complexity)http://en.wikipedia.org/wiki/Alpha_(letter)http://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Karp_reductionhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-1http://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_problemhttp://en.wikipedia.org/wiki/P_(complexity)http://en.wikipedia.org/wiki/Plus_signhttp://en.wikipedia.org/wiki/Additionhttp://en.wikipedia.org/wiki/Plus_and_minus_signshttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Disjoint_union


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the disjoint union of ...

and ...

set theory

A1 +A2 = {(3,1), (4,1), (5,1), (6,1), (7,2), (8,2), (9,2),

(10,2)}

subtraction

minus;take;

subtract

arithmetic

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

negative sign

negative;minus;

the opposite of

arithmetic

3 means thenegative of the number 3. (5) = 5

set-theoretic complement

minus;without

set theory

A B means the set that contains all the elementsofA that are not inB.

(can also be used for set-theoretic complementas described below.)

{1,2,4} {1,3,4} = {2}

plus-minusplus or minus

6 3 means both 6 + 3 and 6 3. The equationx = 5 4, has two solutions,x = 7 andx = 3.
http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Minus_signhttp://en.wikipedia.org/wiki/Subtractionhttp://en.wikipedia.org/wiki/Plus_and_minus_signshttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Plus_and_minus_signshttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Negative_numberhttp://en.wikipedia.org/wiki/Negative_numberhttp://en.wikipedia.org/wiki/Complement_(set_theory)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Plus-minus_signhttp://en.wikipedia.org/wiki/Plus-minus_signhttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Minus_signhttp://en.wikipedia.org/wiki/Subtractionhttp://en.wikipedia.org/wiki/Plus_and_minus_signshttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Plus_and_minus_signshttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Negative_numberhttp://en.wikipedia.org/wiki/Complement_(set_theory)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Plus-minus_signhttp://en.wikipedia.org/wiki/Plus-minus_sign


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arithmeticplus-minus

plus or minus

measurement

10 2 or equivalently 10 20% means the rangefrom 10 2 to 10 + 2.

Ifa = 100 1mm, then a 99 mm and a 101 mm.

minus-plus

minus or plus

arithmetic

6 (3 5) means both 6 + (3 5) and 6 (3 +5).

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

multiplication

times;multiplied by

arithmetic

3 4 means the multiplication of 3 by 4.

(The symbol * is generally used in programminglanguages, where ease of typing and use ofASCIItext is preferred.)

7 8 = 56

Cartesian product

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

the direct product of ...and ...

set theory

XYmeans the set of all ordered pairs with thefirst 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 ofvectorsu and v (1,2,5) (3,4,1) =(22, 16, 2)
http://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Plus-minus_signhttp://en.wikipedia.org/wiki/Measurementhttp://en.wikipedia.org/wiki/Millimetrehttp://en.wikipedia.org/wiki/Millimetrehttp://en.wikipedia.org/wiki/Minus-plus_signhttp://en.wikipedia.org/wiki/Minus-plus_signhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Multiplication_signhttp://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/ASCIIhttp://en.wikipedia.org/wiki/ASCIIhttp://en.wikipedia.org/wiki/Cartesian_producthttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Ordered_pairshttp://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Plus-minus_signhttp://en.wikipedia.org/wiki/Measurementhttp://en.wikipedia.org/wiki/Millimetrehttp://en.wikipedia.org/wiki/Minus-plus_signhttp://en.wikipedia.org/wiki/Minus-plus_signhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Multiplication_signhttp://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/ASCIIhttp://en.wikipedia.org/wiki/Cartesian_producthttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Ordered_pairshttp://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Vector_(geometry)


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linear algebragroup 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 writtenR* as described below,orU(R).

* multiplicationtimes;

multiplied by

arithmetic

a * b means the product ofa and b.

(Multiplication can also be denoted with or,or even simple juxtaposition. * is generally usedwhere ease of typing and use ofASCIItext is

preferred, such as programming languages.)

4 * 3 means the product of 4 and 3, or 12.

convolution

convolution;convolved with

functional analysis

f*gmeans the convolution offandg..

complex conjugate

conjugate

complex numbers

z* means the complex conjugate ofz.

( can also be used for the conjugate of z, as

described below.)

.

group of units

the group of units of

ring theory

R* consists of the set of units of the ring R, alongwith the operation of multiplication.

This may also be writtenRas described above,orU(R).

hyperreal numbers *Rmeans the set of hyperreal numbers. Other *N is the hypernatural numbers.
http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Group_of_unitshttp://en.wikipedia.org/wiki/Ring_theoryhttp://en.wikipedia.org/wiki/Asteriskhttp://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/ASCIIhttp://en.wikipedia.org/wiki/Convolutionhttp://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Group_of_unitshttp://en.wikipedia.org/wiki/Ring_theoryhttp://en.wikipedia.org/wiki/Hyperreal_numberhttp://en.wikipedia.org/wiki/Hypernaturalhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Group_of_unitshttp://en.wikipedia.org/wiki/Ring_theoryhttp://en.wikipedia.org/wiki/Asteriskhttp://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/ASCIIhttp://en.wikipedia.org/wiki/Convolutionhttp://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Group_of_unitshttp://en.wikipedia.org/wiki/Ring_theoryhttp://en.wikipedia.org/wiki/Hyperreal_numberhttp://en.wikipedia.org/wiki/Hypernatural


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the (set of) hyperreals

non-standard analysis

sets can be used in place ofR.

Hodge dual

Hodge dual;Hodge star

linear algebra

*v means the Hodge dual of a vectorv. Ifv is a k-vectorwithin an n-dimensional orientedinnerproductspace, then *v is an (nk)-vector.

If are thestandard basis vectors of ,

multiplication

times;multiplied by

arithmetic

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

dot product

dot

linear algebra

u v means the dot product ofvectorsu and v

(1,2,5) (3,4,1) = 6

placeholder

(silent)

functional analysis

A means a placeholder for an argument of afunction. Indicates the functional nature of anexpression without assigning a specific symbolfor an argument.

tensor product,tensorproduct of modulesmeans the tensor product ofVand U.[3]

means the tensor product of modules Vand Uover the ringR.

{1, 2, 3, 4} {1, 1, 2} ={{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
http://en.wikipedia.org/wiki/Non-standard_analysishttp://en.wikipedia.org/wiki/Hodge_dualhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/P-vectorhttp://en.wikipedia.org/wiki/P-vectorhttp://en.wikipedia.org/wiki/P-vectorhttp://en.wikipedia.org/wiki/P-vectorhttp://en.wikipedia.org/wiki/Dimension_(vector_space)http://en.wikipedia.org/wiki/Dimension_(vector_space)http://en.wikipedia.org/wiki/Dimension_(vector_space)http://en.wikipedia.org/wiki/Orientation_(mathematics)http://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Vector_spacehttp://en.wikipedia.org/wiki/Vector_spacehttp://en.wikipedia.org/wiki/Standard_basishttp://en.wikipedia.org/wiki/Standard_basishttp://en.wikipedia.org/wiki/Middle_dothttp://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Tensor_producthttp://en.wikipedia.org/wiki/Tensor_product_of_moduleshttp://en.wikipedia.org/wiki/Tensor_product_of_moduleshttp://en.wikipedia.org/wiki/Tensor_product_of_moduleshttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-m-nielsen-quantum-71-72-2http://en.wikipedia.org/wiki/Ring_(mathematics)http://en.wikipedia.org/wiki/Non-standard_analysishttp://en.wikipedia.org/wiki/Hodge_dualhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/P-vectorhttp://en.wikipedia.org/wiki/P-vectorhttp://en.wikipedia.org/wiki/Dimension_(vector_space)http://en.wikipedia.org/wiki/Orientation_(mathematics)http://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Vector_spacehttp://en.wikipedia.org/wiki/Standard_basishttp://en.wikipedia.org/wiki/Middle_dothttp://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Tensor_producthttp://en.wikipedia.org/wiki/Tensor_product_of_moduleshttp://en.wikipedia.org/wiki/Tensor_product_of_moduleshttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-m-nielsen-quantum-71-72-2http://en.wikipedia.org/wiki/Ring_(mathematics)


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tensor product of

linear algebra

division (Obelus)

divided by;over

arithmetic

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 /Hmeans the quotient of group Gmodulo its

subgroupH.

{0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a,

b+2a}}

quotient set

mod

set theory

A/~ means the set of all ~equivalence classesinA.

If we define ~ by x ~ y x y , then/~ = { {x + n : n } : x [0,1) }

square rootthe (principal) square root

of

real numbers

means the nonnegative number whose squareis .

complex square root

the (complex) square rootof

if is represented inpolarcoordinates with , then

.
http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Division_signhttp://en.wikipedia.org/wiki/Fraction_slashhttp://en.wikipedia.org/wiki/Division_(mathematics)http://en.wikipedia.org/wiki/Obelushttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Quotient_grouphttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Ideal_(ring_theory)http://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Equivalence_classhttp://en.wikipedia.org/wiki/Equivalence_classhttp://en.wikipedia.org/wiki/Equivalence_classhttp://en.wikipedia.org/wiki/Radical_symbolhttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Square_root#Square_roots_of_complex_numbershttp://en.wikipedia.org/wiki/Polar_coordinate_systemhttp://en.wikipedia.org/wiki/Polar_coordinate_systemhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Division_signhttp://en.wikipedia.org/wiki/Fraction_slashhttp://en.wikipedia.org/wiki/Division_(mathematics)http://en.wikipedia.org/wiki/Obelushttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Quotient_grouphttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Ideal_(ring_theory)http://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Equivalence_classhttp://en.wikipedia.org/wiki/Radical_symbolhttp://en.wikipedia.org/wiki/Square_roothttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Square_root#Square_roots_of_complex_numbershttp://en.wikipedia.org/wiki/Polar_coordinate_systemhttp://en.wikipedia.org/wiki/Polar_coordinate_system


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complex numbers

x

mean

overbar; bar

statistics

(often read as x bar) is the mean (averagevalue of ). .

complex conjugate

conjugate

complex numbers

means the complex conjugate ofz.

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

.

algebraic closure

algebraic closure of

field theory

is the algebraic closure of the fieldF.The field ofalgebraic numbers is sometimes denoted asbecause it is the algebraic closure of the rational numbers

.

topological closure

(topological) closure of

topology

is the topological closure of the set S.

This may also be denoted as cl(S) orCl(S).

In the space of the real numbers, (the rationalnumbers are dense in the real numbers).

|| absolute value;modulusabsolute value of; modulus

of

numbers

|x| means the distance along the real line (oracross the complex plane) betweenx and zero.

|3| = 3

|5| = |5| = 5

| i | = 1

| 3 + 4i | = 5
http://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Overlinehttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Algebraic_closurehttp://en.wikipedia.org/wiki/Field_theory_(mathematics)http://en.wikipedia.org/wiki/Algebraic_numberhttp://en.wikipedia.org/wiki/Rational_numbershttp://en.wikipedia.org/wiki/Topological_closurehttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Dense_(topology)http://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/0_(number)http://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Overlinehttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Algebraic_closurehttp://en.wikipedia.org/wiki/Field_theory_(mathematics)http://en.wikipedia.org/wiki/Algebraic_numberhttp://en.wikipedia.org/wiki/Rational_numbershttp://en.wikipedia.org/wiki/Topological_closurehttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Dense_(topology)http://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Real_linehttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/0_(number)


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Euclidean norm orEuclidean length ormagnitude

Euclidean norm of

geometry

|x| means the (Euclidean) length ofvectorx.

Forx = (3,-4)

determinant

determinant of

matrix theory

|A| means the determinant of the matrix A

cardinality

cardinality of;size of;order of

set theory

|X| means the cardinality of the setX.

(# may be used instead as described below.)|{3, 5, 7, 9}| = 4.

||||

norm

norm of;length of

linear algebra

||x || means thenormof the elementx of anormed vector space.[4]

||x +y || ||x || + ||y ||

nearest integer function

nearest integer to

numbers

||x|| means the nearest integer tox.

(This may also be written [x], x, nint(x) orRound(x).)

||1|| = 1, ||1.6|| = 2, ||2.4|| = 2, ||3.49|| = 3
http://en.wikipedia.org/wiki/Euclidean_normhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Euclidean_vectorhttp://en.wikipedia.org/wiki/Euclidean_vectorhttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Matrix_theoryhttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Norm_(mathematics)http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Norm_(mathematics)http://en.wikipedia.org/wiki/Norm_(mathematics)http://en.wikipedia.org/wiki/Norm_(mathematics)http://en.wikipedia.org/wiki/Normed_vector_spacehttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-m-nielsen-quantum-66-3http://en.wikipedia.org/wiki/Nearest_integer_functionhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Euclidean_normhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Euclidean_vectorhttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Matrix_theoryhttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Norm_(mathematics)http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Norm_(mathematics)http://en.wikipedia.org/wiki/Normed_vector_spacehttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-m-nielsen-quantum-66-3http://en.wikipedia.org/wiki/Nearest_integer_functionhttp://en.wikipedia.org/wiki/Number


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is incomparable to

order theoryexact divisibility

exactly divides

number theory

pa || n meanspa exactly divides n (i.e.pa divides nbutpa+1 does not).

23 || 360.

#

cardinality

cardinality of;size of;

order of

set theory

#Xmeans the cardinality of the setX.

(|| may be used instead as described above.)

#{4, 6, 8} = 3

connected sum

connected sum of;knot sum of;

knot composition of

topology,knot theory

A#B is the connected sum of the manifoldsA andB. IfA andB are knots, then this denotes the knotsum, which has a slightly stronger condition.

A#Sm is homeomorphic toA, for any manifoldA, and thesphere Sm.

aleph number

aleph

set theory

represents an infinite cardinality (specifically,the -th one, where is an ordinal).

|| =0, which is called aleph-null.

beth number represents an infinite cardinality (similar to

, but does not necessarily index all of the
http://en.wikipedia.org/wiki/Order_theoryhttp://en.wikipedia.org/wiki/Divisibilityhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Number_signhttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Connected_sumhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Knot_theoryhttp://en.wikipedia.org/wiki/Homeomorphichttp://en.wikipedia.org/wiki/Aleph_(letter)http://en.wikipedia.org/wiki/Aleph_numberhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Beth_(letter)http://en.wikipedia.org/wiki/Beth_numberhttp://en.wikipedia.org/wiki/Order_theoryhttp://en.wikipedia.org/wiki/Divisibilityhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Number_signhttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Connected_sumhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Knot_theoryhttp://en.wikipedia.org/wiki/Homeomorphichttp://en.wikipedia.org/wiki/Aleph_(letter)http://en.wikipedia.org/wiki/Aleph_numberhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Beth_(letter)http://en.wikipedia.org/wiki/Beth_number


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beth

set theory

numbers indexed by . ).

cardinality of the continuum

cardinality of thecontinuum;

c;cardinality of the real

numbers

set theory

The cardinality of is denoted by or by thesymbol (a lowercaseFrakturletter C).

: such thatsuch that;

so that

everywhere

: means such that, and is used in proofs and theset-builder notation (described below).

n : n is even.

field extension

extends;over

field theory

K:Fmeans the fieldKextends the fieldF.

This may also be written asKF.

:

inner product of matrices

inner product of

linear algebra

A :B means the Frobenius inner product of thematricesA andB.

The general inner product is denoted byu, v ,u | v or(u | v), as described below. For
http://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Cardinality_of_the_continuumhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Fraktur_(script)http://en.wikipedia.org/wiki/Fraktur_(script)http://en.wikipedia.org/wiki/Colon_(punctuation)http://en.wikipedia.org/wiki/Set-builder_notationhttp://en.wikipedia.org/wiki/Field_extensionhttp://en.wikipedia.org/wiki/Field_theory_(mathematics)http://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Cardinality_of_the_continuumhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Fraktur_(script)http://en.wikipedia.org/wiki/Colon_(punctuation)http://en.wikipedia.org/wiki/Set-builder_notationhttp://en.wikipedia.org/wiki/Field_extensionhttp://en.wikipedia.org/wiki/Field_theory_(mathematics)http://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Linear_algebra


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spatial vectors, the dot productnotation,xyiscommon. See also Bra-ket notation.

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, thenumber of "copies" (cosets) of H that fill up G

!

factorial

factorial

combinatorics

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

logical negation

not

propositional logic

The statement !A is true if and only ifA is false.

A slash placed through another operator is thesame as "!" placed in front.

(The symbol! is primarily from computerscience. It is avoided in mathematical texts,where the notation Ais preferred.)

!(!A) Ax y !(x =y)

~ probability distributionhas distribution

statistics

X ~ D, means the random variableXhas the

probability distributionD.

X~N(0,1), thestandard normal distribution

row equivalence

is row equivalent to

matrix theory

A~B means thatB can be generated by using aseries ofelementary row operations onA
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same order of magnitude

roughly similar;

poorly approximates

approximation theory

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

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

2 ~ 5

8 9 ~ 100

but 2 10

asymptotically equivalent

is asymptotically equivalentto

asymptotic analysis

f~gmeans .x ~ x+1

equivalence relation

are in the same equivalenceclass

everywhere

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

approximately equal

is approximately equal to

everywhere

x y meansx is approximately equal toy.

This may also be written, , ~ or . 3.14159

isomorphism

is isomorphic to

group theory

G Hmeans that group G is isomorphic(structurally identical) to groupH.

(can also be used for isomorphic, as describedbelow.)

Q / {1, 1} V,where Q is the quaternion group and Vis theKlein four-group.

wreath product AHmeans the wreath product of the groupA bythe groupH.is isomorphic to the automorphism group of the

complete bipartite graph on (n,n) vertices.
http://en.wikipedia.org/wiki/Order_of_magnitudehttp://en.wikipedia.org/wiki/Approximationhttp://en.wikipedia.org/wiki/Approximation_theoryhttp://en.wikipedia.org/wiki/Order_of_magnitudehttp://en.wikipedia.org/wiki/Asymptotic_analysishttp://en.wikipedia.org/wiki/Asymptotic_analysishttp://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Equals_sign#Approximately_equalhttp://en.wikipedia.org/wiki/Isomorphismhttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Quaternion_grouphttp://en.wikipedia.org/wiki/Klein_four-grouphttp://en.wikipedia.org/wiki/Klein_four-grouphttp://en.wikipedia.org/wiki/Klein_four-grouphttp://en.wikipedia.org/wiki/Wreath_producthttp://en.wikipedia.org/wiki/Graph_automorphismhttp://en.wikipedia.org/wiki/Complete_bipartite_graphhttp://en.wikipedia.org/wiki/Order_of_magnitudehttp://en.wikipedia.org/wiki/Approximationhttp://en.wikipedia.org/wiki/Approximation_theoryhttp://en.wikipedia.org/wiki/Order_of_magnitudehttp://en.wikipedia.org/wiki/Asymptotic_analysishttp://en.wikipedia.org/wiki/Asymptotic_analysishttp://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Equals_sign#Approximately_equalhttp://en.wikipedia.org/wiki/Isomorphismhttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Quaternion_grouphttp://en.wikipedia.org/wiki/Klein_four-grouphttp://en.wikipedia.org/wiki/Klein_four-grouphttp://en.wikipedia.org/wiki/Wreath_producthttp://en.wikipedia.org/wiki/Graph_automorphismhttp://en.wikipedia.org/wiki/Complete_bipartite_graph


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wreath product of by

group theory

This may also be writtenA wrH.

normal subgroup

is a normal subgroup of

group theory

NG means thatNis a normal subgroup ofgroup G.

Z(G) G

ideal

is an ideal of

ring theory

IR means thatIis an ideal of ringR. (2) Z

antijoin

the antijoin of

relational algebra

RSmeans the antijoin of the relationsR and S,the tuples inR for which there is not a tuple in Sthat is equal on their common attribute names.

R S=R -R S

semidirect product

the semidirect product of

group theory

NHis the semidirect product ofN(a normalsubgroup) andH(a subgroup), with respect to .Also, ifG =NH, then G is said to split overN.

(may also be written the other way round, as, or as .)

semijoin

the semijoin of

RSis the semijoin of the relationsR and S, theset of all tuples inR for which there is a tuple in Sthat is equal on their common attribute names.

R S= a1,..,an(R S)
http://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Normal_subgrouphttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Ideal_of_a_ringhttp://en.wikipedia.org/wiki/Ring_theoryhttp://en.wikipedia.org/wiki/Antijoinhttp://en.wikipedia.org/wiki/Relational_algebrahttp://en.wikipedia.org/wiki/Semidirect_producthttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Semijoinhttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Normal_subgrouphttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Ideal_of_a_ringhttp://en.wikipedia.org/wiki/Ring_theoryhttp://en.wikipedia.org/wiki/Antijoinhttp://en.wikipedia.org/wiki/Relational_algebrahttp://en.wikipedia.org/wiki/Semidirect_producthttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Semijoin


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relational algebra

natural join

the natural join of

relational algebra

RSis the natural join of the relationsR and S,the set of all combinations of tuples inR and Sthat are equal on their common attribute names.

therefore

therefore;so;

hence

everywhere

Sometimes used in proofs beforelogicalconsequences.

All humans are mortal. Socrates is a human. Socrates is mortal.

because

because;since

everywhere

Sometimes used in proofs before reasoning.3331 isprime it has no positive integer factors other than itself and one.

end of proof

QED;

tombstone;Halmos symbol

everywhere

Used to mark the end of a proof.

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

D'Alembertian It is the generalisation of the Laplace operatorinthe sense that it is the differential operator whichis invariant under the isometry group of the
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non-Euclidean Laplacian

vector calculus

underlying space and it reduces to the Laplaceoperator if restricted to time independentfunctions.

material implication

implies;if then

propositional logic,Heytingalgebra

AB means ifA is true thenB is also true; ifAis false then nothing is said aboutB.

( may mean the same as, or it may have the

meaning forfunctions given below.)

( may mean the same as,[5]or it may havethe meaning forsupersetgiven below.)

x = 2 x2 = 4 is true, butx2 = 4 x = 2 is in general

false (sincex could be 2).

material equivalence

if and only if;iff

propositional logic

AB meansA is true ifB is true andA is false ifB is false.

x + 5 =y + 2 x + 3 =y

logical negation

not

propositional logic

The statement A is true if and only ifA is false.

A slash placed through another operator is thesame as "" placed in front.

(The symbol~ has many other uses, so or the

(A) Ax y (x = y)
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slash notation is preferred. Computer scientistswill often use ! but this is avoided inmathematical texts.)

logical conjunction ormeetin alattice

and;min;meet

propositional logic, latticetheory

The statementAB is true ifA andB are bothtrue; else it is false.

For functionsA(x) andB(x),A(x) B(x) is usedto mean min(A(x), B(x)).

n < 4 n >2 n = 3 when n is a natural number.

wedge product

wedge product;exterior product

linear algebra

u v means the wedge product ofvectorsu andv. This generalizes the cross product to higherdimensions.

(For vectors inR3, can also be used.)

exponentiation

(raised) to the power of

everywhere

a ^ b means a raised to the power ofb

(a ^ bis more commonly writtenab. The symbol^is generally used in programming languages

where ease of typing and use of plain ASCII text

is preferred.)

2^3 = 23 = 8

logical disjunctionorjoinin alattice

or;max;

The statementAB is true ifA orB (or both)are true; if both are false, the statement is false.

For functionsA(x) andB(x),A(x) B(x) is usedto mean max(A(x), B(x)).

n 4 n 2 n 3 when n is a natural number.
http://en.wikipedia.org/wiki/Logical_conjunctionhttp://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Propositional_logichttp://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Wedge_producthttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Logical_disjunctionhttp://en.wikipedia.org/wiki/Logical_disjunctionhttp://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Logical_conjunctionhttp://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Propositional_logichttp://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Wedge_producthttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Vector_(geometry)http://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Logical_disjunctionhttp://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Natural_number


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join

propositional logic, latticetheory

exclusive or

xor

propositional logic, Booleanalgebra

The statementAB is true when either A or B,but not both, are true.AB means the same. (A) A is always true,AA is always false.

direct sum

direct sum of

abstract algebra

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

(The bun symbol , or thecoproductsymbol,is used; is only for logic.)

Most commonly, for vector spaces U, V, and W, the

following consequence is used:U= V W ( U= V+ W) ( V W= {0})

universal quantification

for all;for any;for each

predicate logic

x:P(x) meansP(x) is true for allx. n: n2 n.

existential quantification

there exists;there is;there are

predicate logic

x:P(x) means there is at least onex such thatP(x) is true.

n: n is even.
http://en.wikipedia.org/wiki/Propositional_logichttp://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Exclusive_orhttp://en.wikipedia.org/wiki/Propositional_logichttp://en.wikipedia.org/wiki/Boolean_algebra_(logic)http://en.wikipedia.org/wiki/Boolean_algebra_(logic)http://en.wikipedia.org/wiki/Direct_sumhttp://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Coproducthttp://en.wikipedia.org/wiki/Coproducthttp://en.wikipedia.org/wiki/Turned_ahttp://en.wikipedia.org/wiki/Universal_quantificationhttp://en.wikipedia.org/wiki/Predicate_logichttp://en.wikipedia.org/wiki/Existential_quantificationhttp://en.wikipedia.org/wiki/Predicate_logichttp://en.wikipedia.org/wiki/Propositional_logichttp://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Lattice_(order)http://en.wikipedia.org/wiki/Exclusive_orhttp://en.wikipedia.org/wiki/Propositional_logichttp://en.wikipedia.org/wiki/Boolean_algebra_(logic)http://en.wikipedia.org/wiki/Boolean_algebra_(logic)http://en.wikipedia.org/wiki/Direct_sumhttp://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Coproducthttp://en.wikipedia.org/wiki/Turned_ahttp://en.wikipedia.org/wiki/Universal_quantificationhttp://en.wikipedia.org/wiki/Predicate_logichttp://en.wikipedia.org/wiki/Existential_quantificationhttp://en.wikipedia.org/wiki/Predicate_logic


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!uniqueness quantification

there exists exactly one

predicate logic

! x:P(x) means there is exactly onex such thatP(x) is true.

! n: n + 5 = 2n.

=:

:=

:

definition

is defined as;is equal by definition to

everywhere

x :=y,y =:x orx y meansx is defined to beanother name fory, under certain assumptionstaken in context.

(Some writers use to mean congruence).

P: Q meansPis defined to be logicallyequivalent to Q.
http://en.wikipedia.org/wiki/Uniqueness_quantificationhttp://en.wikipedia.org/wiki/Predicate_logichttp://en.wikipedia.org/wiki/Triple_barhttp://en.wikipedia.org/wiki/Definitionhttp://en.wikipedia.org/wiki/Congruence_(geometry)http://en.wikipedia.org/wiki/Logical_equivalencehttp://en.wikipedia.org/wiki/Logical_equivalencehttp://en.wikipedia.org/wiki/Uniqueness_quantificationhttp://en.wikipedia.org/wiki/Predicate_logichttp://en.wikipedia.org/wiki/Triple_barhttp://en.wikipedia.org/wiki/Definitionhttp://en.wikipedia.org/wiki/Congruence_(geometry)http://en.wikipedia.org/wiki/Logical_equivalencehttp://en.wikipedia.org/wiki/Logical_equivalence


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congruence

is congruent to

geometry

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

isomorphic

is isomorphic to

abstract algebra

GHmeans that group G is isomorphic(structurally identical) to groupH.

( can also be used for isomorphic, as describedabove.)

.

congruence relation

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

modular arithmetic

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

{ , }set brackets

the set of

set theory

{a,b,c} means the set consisting ofa, b, and c.[6] = { 1, 2, 3, }

{ : }

{ | }

set builder notation

the set of such that

set theory

{x :P(x)} means the set of allx for whichP(x) istrue.[6] {x |P(x)} is the same as {x :P(x)}.

{n : n2 < 20} = { 1, 2, 3, 4}
http://en.wikipedia.org/wiki/Congruence_(geometry)http://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Isomorphichttp://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Triple_barhttp://en.wikipedia.org/wiki/Congruence_relationhttp://en.wikipedia.org/wiki/Modular_arithmetichttp://en.wikipedia.org/wiki/Curly_bracketshttp://en.wikipedia.org/wiki/Curly_bracketshttp://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-3-5http://en.wikipedia.org/wiki/Set_builder_notationhttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-3-5http://en.wikipedia.org/wiki/Congruence_(geometry)http://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Isomorphichttp://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Triple_barhttp://en.wikipedia.org/wiki/Congruence_relationhttp://en.wikipedia.org/wiki/Modular_arithmetichttp://en.wikipedia.org/wiki/Curly_bracketshttp://en.wikipedia.org/wiki/Curly_bracketshttp://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-3-5http://en.wikipedia.org/wiki/Set_builder_notationhttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-3-5


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{ }

empty set

the empty set

set theory

means the set with no elements.[6] { } means

the same.

{n : 1 < n2 < 4} =

set membership

is an element of;is not an element of

everywhere, set theory

a Smeans a is an element of the set S;[6]aSmeans a is not an element ofS.[6]

(1/2)1

21

subset

is a subset of

set theory

(subset)AB means every element ofA is also

an element ofB.[7]

(proper subset)AB meansAB butA B.

(Some writers use the symbol as if it were thesame as .)

(A B) A

superset

is a superset of

set theory

AB means every element ofB is also anelement ofA.

AB meansAB butA B.

(Some writers use the symbol as if it were thesame as.)

(AB) B

set-theoretic union

the union of or ;

AB means the set of those elements which areeither inA, or inB, or in both.[7]

AB (AB) =B
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union

set theory

set-theoretic intersection

intersected with;intersect

set theory

A B means the set that contains all thoseelements thatA andB have in common.[7]

{x :x2 = 1} = {1}

symmetric difference

symmetric difference

set theory

A B means the set of elements in exactly one ofA orB.

(Not to be confused with delta, , describedbelow.)

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

set-theoretic complement

minus;without

set theory

AB means the set that contains all thoseelements ofA that are not inB.[7]

( can also be used for set-theoretic complementas described above.)

{1,2,3,4} {3,4,5,6} = {1,2}

functionarrow

from to

set theory, type theory

f:X Ymeans the functionfmaps the setXintothe set Y. Letf: {0} be defined by f(x) :=x2.

functionarrow

maps to

f: ab means the functionfmaps the element ato the element b.

Letf:xx+1 (the successor function).
http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Intersection_(set_theory)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-4-6http://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-4-6http://en.wikipedia.org/wiki/Symmetric_differencehttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Complement_(set_theory)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-4-6http://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-4-6http://en.wikipedia.org/wiki/Arrow_(symbol)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Intuitionistic_type_theoryhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Intersection_(set_theory)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-4-6http://en.wikipedia.org/wiki/Symmetric_differencehttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Complement_(set_theory)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-4-6http://en.wikipedia.org/wiki/Arrow_(symbol)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Intuitionistic_type_theoryhttp://en.wikipedia.org/wiki/Function_(mathematics)


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set theory

function composition

composed with

set theory

fgis the function, such that (fg)(x) =f(g(x)).[8] iff(x) := 2x, andg(x) :=x + 3, then (fg)(x) = 2(x + 3).

N

natural numbers

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 mathematicsbeing studied; e.g.number theorists prefer the

latter; analysts,set theorists andcomputerscientists prefer the former. To avoid confusion,

always check an author's definition ofN.

Set theorists often use the notation (forleastinfinite ordinal) to denote the set of naturalnumbers (including zero), along with the

standard ordering relation .

= {|a| : a } or = {|a| > 0: a }

Z

integers

Z;the (set of) integers

numbers

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

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

1, 2, 3, ...} .

= {p, p :p {0}}

nintegers mod n

Zn;the (set of) integers modulo

n

n means {[0], [1], [2], ...[n1]} with additionand multiplication modulo n.

Note that any letter may be used instead ofn,such asp. To avoid confusion with p-adic

3 = {[0], [1], [2]}
http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-5-7http://en.wikipedia.org/wiki/Nhttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Analysis_(mathematics)http://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Least_infinite_ordinalhttp://en.wikipedia.org/wiki/Least_infinite_ordinalhttp://en.wikipedia.org/wiki/Least_infinite_ordinalhttp://en.wikipedia.org/wiki/Zhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Modular_arithmetichttp://en.wikipedia.org/wiki/Modular_arithmetichttp://en.wikipedia.org/wiki/Modular_arithmetichttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-d-goldrei-set-5-7http://en.wikipedia.org/wiki/Nhttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Analysis_(mathematics)http://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Least_infinite_ordinalhttp://en.wikipedia.org/wiki/Least_infinite_ordinalhttp://en.wikipedia.org/wiki/Zhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Modular_arithmetic


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p

Zn

Zp

numbersnumbers, use/por/(p) instead.

p-adic integers

the (set of)p-adic integers

numbers

Note that any letter may be used instead ofp,such asnorl.

P

projective space

P;

the projective space;the projective line;the projective plane

topology

means a space with a point at infinity. ,

probability

the probability of

probability theory

(X) means the probability of the eventXoccurring.

This may also be written as P(X), Pr(X), P[X] orPr[X].

If a fair coin is flipped, (Heads) = (Tails) = 0.5.

Q

rational numbers

Q;the (set of) rational

numbers;the rationals

means {p/q :p, q}. 3.14000...
http://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/P-adic_integershttp://en.wikipedia.org/wiki/P-adic_integershttp://en.wikipedia.org/wiki/P-adic_integershttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Phttp://en.wikipedia.org/wiki/Projective_planehttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Qhttp://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/P-adic_integershttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Phttp://en.wikipedia.org/wiki/Projective_planehttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Qhttp://en.wikipedia.org/wiki/Rational_number


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numbers

R

real numbers

R;the (set of) real numbers;

the reals

numbers

means the set of real numbers.

(1)

C

complex numbers

C;the (set of) complex

numbers

numbers

means {a + bi : a,b}. i = (1)

H

quaternions or Hamiltonianquaternions

H;the (set of) quaternions

numbers

means {a + bi + cj + dk: a,b,c,d}.

OBig O notation

big-oh of

Computational complexitytheory

The Big O notation describes the limitingbehaviorof a function, when the argument tendstowards a particular value orinfinity.

If f(x) = 6x4 2x3 + 5 and g(x) = x4 , then
http://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Rhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Chttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Hhttp://en.wikipedia.org/wiki/Quaternionhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Ohttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Asymptotic_analysishttp://en.wikipedia.org/wiki/Asymptotic_analysishttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Infinityhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Rhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Chttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Hhttp://en.wikipedia.org/wiki/Quaternionhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Ohttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Asymptotic_analysishttp://en.wikipedia.org/wiki/Asymptotic_analysishttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Infinity


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infinity

infinity

numbers

is an element of the extended number line thatis greater than all real numbers; it often occurs in

limits.

floor

floor;greatest integer;

entier

numbers

x means the floor ofx, i.e. the largest integerless than or equal tox.

(This may also be written [x], floor(x) orint(x).)

4 = 4, 2.1 = 2, 2.9 = 2, 2.6 = 3

ceiling

ceiling

numbers

x means the ceiling ofx, i.e. the smallest integergreater than or equal tox.

(This may also be written ceil(x) orceiling(x).)

4 = 4, 2.1 = 3, 2.9 = 3, 2.6 = 2


nearest integer to

numbers

x means the nearest integer tox.

(This may also be written [x], ||x||, nint(x) orRound(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 extensionK:F.

[(2) : ] = 2

[ : ] = 2

[ : ] =

[ ] equivalence class [a] means the equivalence class ofa, i.e. {x :x ~a}, where ~ is an equivalence relation.Let a ~ b be trueiffa b (mod 5).

Then [2] = {, 8, 3, 2, 7, }.
http://en.wikipedia.org/wiki/Infinityhttp://en.wikipedia.org/wiki/Infinityhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Extended_real_number_linehttp://en.wikipedia.org/wiki/Limit_(mathematics)http://en.wikipedia.org/wiki/Floor_and_ceiling_functionshttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Floor_and_ceiling_functionshttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Nearest_integer_functionhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Degree_of_a_field_extensionhttp://en.wikipedia.org/wiki/Field_theory_(mathematics)http://en.wikipedia.org/wiki/Brackethttp://en.wikipedia.org/wiki/Equivalence_classhttp://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Iffhttp://en.wikipedia.org/wiki/Iffhttp://en.wikipedia.org/wiki/Modular_arithmetichttp://en.wikipedia.org/wiki/Infinityhttp://en.wikipedia.org/wiki/Infinityhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Extended_real_number_linehttp://en.wikipedia.org/wiki/Limit_(mathematics)http://en.wikipedia.org/wiki/Floor_and_ceiling_functionshttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Floor_and_ceiling_functionshttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Nearest_integer_functionhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Degree_of_a_field_extensionhttp://en.wikipedia.org/wiki/Field_theory_(mathematics)http://en.wikipedia.org/wiki/Brackethttp://en.wikipedia.org/wiki/Equivalence_classhttp://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Iffhttp://en.wikipedia.org/wiki/Modular_arithmetic


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[ , ][ , , ]

the equivalence class of

abstract algebra

[a]R means the same, but withR as theequivalence relation.

floor

floor;greatest integer;

entier

numbers

[x] means the floor ofx, i.e. the largest integerless than or equal tox.

(This may also be writtenx, floor(x) orint(x).Not to be confused with the nearest integer

function, as described below.)

[3] = 3, [3.5] = 3, [3.99] = 3, [3.7] = 4


nearest integer to

numbers

[x] means the nearest integer tox.

(This may also be writtenx, ||x||, nint(x) orRound(x).Not to be confused with the floorfunction, as described above.)

[2] = 2, [2.6] = 3, [-3.4] = -3, [4.49] = 4

Iverson bracket

1 if true, 0 otherwise

propositional logic

[S] maps a true statement Sto 1 and a falsestatement Sto 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) :xX}, the image of thefunctionfunder the setX dom(f).

(This may also be written asf(X) if there is norisk of confusing the image offunderXwith thefunction applicationfofX.Another notation isImf, the image offunder its domain.)

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

order theorycommutator

the commutator of

group theory, ring theory

[g, h] =g1h1gh (orghg1h1), ifg, h G (agroup).

[a, b] = ab ba, ifa, bR (a ring orcommutative algebra).

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

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

triple scalar product

the triple scalar product of

vector calculus

[a, b, c] = a b c, the scalar product ofabwith c.

[a, b, c] = [b, c, a] = [c, a, b].

( )

( , )

functionapplication

of

set theory

f(x) means the value of the functionfat theelementx.

Iff(x) :=x2, thenf(3) = 32 = 9.

image

image of under

everywhere

f(X) means {f(x) :xX}, the image of thefunctionfunder the setX dom(f).

(This may also be written asf[X] if there is a risk

of confusing the image offunderXwith thefunction applicationfofX.Another notation isImf, the image offunder its domain.)

combinations

(from) n choose r means the number of combinations ofrelements drawn from a set ofn elements.
http://en.wikipedia.org/wiki/Order_theoryhttp://en.wikipedia.org/wiki/Commutatorhttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Ring_theoryhttp://en.wikipedia.org/wiki/Group_(mathematics)http://en.wikipedia.org/wiki/Ring_(algebra)http://en.wikipedia.org/wiki/Commutative_algebrahttp://en.wikipedia.org/wiki/Triple_scalar_producthttp://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Scalar_producthttp://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Brackethttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Domain_of_a_functionhttp://en.wikipedia.org/wiki/Combinationhttp://en.wikipedia.org/wiki/Order_theoryhttp://en.wikipedia.org/wiki/Commutatorhttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Ring_theoryhttp://en.wikipedia.org/wiki/Group_(mathematics)http://en.wikipedia.org/wiki/Ring_(algebra)http://en.wikipedia.org/wiki/Commutative_algebrahttp://en.wikipedia.org/wiki/Triple_scalar_producthttp://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Scalar_producthttp://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Brackethttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Domain_of_a_functionhttp://en.wikipedia.org/wiki/Combination


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combinatorics(This may also be written asnCr.)precedence grouping

parentheses

everywhere

Perform the operations inside the parenthesesfirst.

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

tuple

tuple; n-tuple;ordered pair/triple/etc;row vector; sequence

everywhere

An ordered list (or sequence, or horizontal vector,or row vector) of values.

(Note that the notation (a,b) is ambiguous: itcould be an ordered pair or an open interval. Set

theorists and computer scientists often use angle

bracketsinstead 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 ofa andb.

(This may also be written hcf(a, b) orgcd(a, b).)

(3, 7) = 1 (they are coprime); (15, 25) = 5.

( , )

] , [

open interval

open interval

order theory

.

(Note that the notation (a,b) is ambiguous: itcould be an ordered pair or an open interval. The

notation ]a,b[ can be used instead.)

4 is not in the interval (4, 18).(0, +) equals the set of positive real numbers.

(( )) multichoose means n multichoosex.
http://en.wikipedia.org/wiki/Combinatoricshttp://en.wikipedia.org/wiki/Tuplehttp://en.wikipedia.org/wiki/Empty_tuplehttp://en.wikipedia.org/wiki/Highest_common_factorhttp://en.wikipedia.org/wiki/Open_intervalhttp://en.wikipedia.org/wiki/Order_theoryhttp://en.wikipedia.org/wiki/Multichoosehttp://en.wikipedia.org/wiki/Combinatoricshttp://en.wikipedia.org/wiki/Tuplehttp://en.wikipedia.org/wiki/Empty_tuplehttp://en.wikipedia.org/wiki/Highest_common_factorhttp://en.wikipedia.org/wiki/Open_intervalhttp://en.wikipedia.org/wiki/Order_theoryhttp://en.wikipedia.org/wiki/Multichoose


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multichoose

combinatorics( , ]

] , ]

left-open interval

half-open interval;left-open interval

order theory

. (1, 7] and (, 1]

[ , )

[ , [

right-open interval

half-open interval;

right-open interval

order theory

.[4, 18) and [1, +)

,

inner product

inner product of

linear algebra

u,v means the inner product of u and v, whereu and v are members of an inner product space.

Note that the notationu, v may beambiguous: it could mean the inner product or

the linear span.

There are many variants of the notation, such asu | v and(u | v), which are described below.For spatial vectors, thedot productnotation,xyis common. For matrices, the colon notation

A :Bmay be used. As andcan be hard totype, the more keyboard friendly forms < and>are sometimes seen. These are avoided in

The standard inner product between two vectorsx = (2, 3)andy = (1, 5) is:

x, y= 2 1 + 3 5 = 13
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mathematical texts.

average

average of

statistics

let S be a subset of N for example, representsthe average of all the element in S.

for a time series :g(t) (t= 1, 2,...)

we can define the structurefunctions Sq( ):

linear span

(linear) span of;linear hull of

linear algebra

S means the span of S V. That is, it is theintersection of all subspaces ofVwhich contain S.u1, u2, is shorthand for { u1, u2, } .

Note that the notationu, v may beambiguous: it could mean theinner productor

the linear span.

The span ofSmay also be written as Sp(S).

.

subgroup generated by a set

the subgroup generated by

group theory

means the smallest subgroup ofG (where S G, a group) containing every element ofS.

is shorthand for .

In S3, and

.

tuple

tuple; n-tuple;ordered pair/triple/etc;row vector; sequence

everywhere

An ordered list (or sequence, or horizontal vector,or row vector) of values.

(The notation (a,b) is often used as well.)

is an ordered pair (or 2-tuple).

is an ordered triple (or 3-tuple).

is theempty tuple (or 0-tuple).

| inner product u | v means the inner product of u and v,where u and v are members of aninner product
http://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Algebraic_structurehttp://en.wikipedia.org/wiki/Algebraic_structurehttp://en.wikipedia.org/wiki/Linear_spanhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Generating_set_of_a_grouphttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Dihedral_group_of_order_6http://en.wikipedia.org/wiki/Dihedral_group_of_order_6http://en.wikipedia.org/wiki/Dihedral_group_of_order_6http://en.wikipedia.org/wiki/Tuplehttp://en.wikipedia.org/wiki/Empty_tuplehttp://en.wikipedia.org/wiki/Empty_tuplehttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Inner_product_spacehttp://en.wikipedia.org/wiki/Inner_product_spacehttp://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Algebraic_structurehttp://en.wikipedia.org/wiki/Linear_spanhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Generating_set_of_a_grouphttp://en.wikipedia.org/wiki/Group_theoryhttp://en.wikipedia.org/wiki/Dihedral_group_of_order_6http://en.wikipedia.org/wiki/Tuplehttp://en.wikipedia.org/wiki/Empty_tuplehttp://en.wikipedia.org/wiki/Inner_producthttp://en.wikipedia.org/wiki/Inner_product_space


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(|)inner product of

linear algebra

space.[9] (u | v) means the same.

Another variant of the notation isu, v which

is described above. For spatial vectors, the dotproductnotation,xyis common. For matrices,the colon notationA :Bmay be used. Asandcan be hard to type, the more keyboardfriendly forms < and> are sometimes seen.These are avoided in mathematical texts.

|

ket vector

the ket ;the vector

Dirac notation

| means the vector with label , which is in aHilbert space.

A qubit's state can be represented as |0 + |1 , where andare complex numbers s.t. ||2 + ||2 = 1.

|

bra vector

the bra ;the dual of

Dirac notation

| means the dual of the vector | , a linearfunctional which maps a ket | onto the inner product | .

summation

sum over from to of

arithmetic

means a1 + a2 + + an.= 12 + 22 + 32 + 42

= 1 + 4 + 9 + 16 = 30

productproduct over from to means a1a2an. = (1+2)(2+2)(3+2)(4+2)
http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Inner_product_spacehttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-m-nielsen-quantum-62-8http://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Ket_vectorhttp://en.wikipedia.org/wiki/Dirac_notationhttp://en.wikipedia.org/wiki/Hilbert_spacehttp://en.wikipedia.org/wiki/Qubithttp://en.wikipedia.org/wiki/Bra_vectorhttp://en.wikipedia.org/wiki/Dirac_notationhttp://en.wikipedia.org/wiki/Linear_functionalhttp://en.wikipedia.org/wiki/Linear_functionalhttp://en.wikipedia.org/wiki/Linear_functionalhttp://en.wikipedia.org/wiki/Sigma_(letter)http://en.wikipedia.org/wiki/Summationhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Pi_(letter)http://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Inner_product_spacehttp://en.wikipedia.org/wiki/List_of_mathematical_symbols#cite_note-m-nielsen-quantum-62-8http://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Ket_vectorhttp://en.wikipedia.org/wiki/Dirac_notationhttp://en.wikipedia.org/wiki/Hilbert_spacehttp://en.wikipedia.org/wiki/Qubithttp://en.wikipedia.org/wiki/Bra_vectorhttp://en.wikipedia.org/wiki/Dirac_notationhttp://en.wikipedia.org/wiki/Linear_functionalhttp://en.wikipedia.org/wiki/Linear_functionalhttp://en.wikipedia.org/wiki/Sigma_(letter)http://en.wikipedia.org/wiki/Summationhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Pi_(letter)http://en.wikipedia.org/wiki/Multiplication


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of

arithmetic

= 3 4 5 6 = 360

Cartesian product

the Cartesian product of;the direct product of

set theory

means the set of all (n+1)-tuples

(y0, ,yn).

coproduct

coproduct over from to of

category theory

A general construction which subsumes thedisjoint union of sets and of topological spaces,the free product of groups, and the direct sumof

modules and vector spaces. The coproduct of afamily of objects is essentially the "least specific"object to which each object in the family admits amorphism.

delta

delta;change in

calculus

x means a (non-infinitesimal) change inx.

(If the change becomes infinitesimal, and even dare used instead. Not to be confused with the

symmetric difference, written , above.)

is the gradient of a straight line

Laplacian

Laplace operator

vector calculus

The Laplace operator is a second orderdifferential operator in n-dimensional Euclideanspace

If is a twice-differentiablereal-valued function, then the

Laplacian of is defined by
http://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Cartesian_producthttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/N-tuplehttp://en.wikipedia.org/wiki/Coproducthttp://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Disjoint_unionhttp://en.wikipedia.org/wiki/Disjoint_union_(topology)http://en.wikipedia.org/wiki/Disjoint_union_(topology)http://en.wikipedia.org/wiki/Free_producthttp://en.wikipedia.org/wiki/Direct_sumhttp://en.wikipedia.org/wiki/Direct_sumhttp://en.wikipedia.org/wiki/Morphismhttp://en.wikipedia.org/wiki/Delta_(letter)http://en.wikipedia.org/wiki/Delta_(letter)http://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Laplacianhttp://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Real-valuedhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Cartesian_producthttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/N-tuplehttp://en.wikipedia.org/wiki/Coproducthttp://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Disjoint_unionhttp://en.wikipedia.org/wiki/Disjoint_union_(topology)http://en.wikipedia.org/wiki/Free_producthttp://en.wikipedia.org/wiki/Direct_sumhttp://en.wikipedia.org/wiki/Morphismhttp://en.wikipedia.org/wiki/Delta_(letter)http://en.wikipedia.org/wiki/Delta_(letter)http://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Laplacianhttp://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Real-valued


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Dirac delta function

Dirac delta of

hyperfunction

(x)

Kronecker delta

Kronecker delta of

hyperfunction

ij

partial derivative

partial;

d

calculus

f/xi means the partial derivative offwith

respect toxi, wherefis a function on (x1, ,xn).Iff(x,y) :=x2y, then f/x = 2xy

boundary

boundary of

topology

Mmeans the boundary ofM {x : ||x|| 2} = {x : ||x|| = 2}

degree of a polynomial

degree of

algebra

fmeans the degree of the polynomialf.

(This may also be written degf.)(x2 1) = 2

gradient

del;nabla;

f(x1, , xn) is the vector of partial derivatives(f/ x1, , f/ xn).

Iff(x,y,z) := 3xy +z, then f= (3y, 3x, 2z)
http://en.wikipedia.org/wiki/Delta_(letter)http://en.wikipedia.org/wiki/Dirac_delta_functionhttp://en.wikipedia.org/wiki/Hyperfunctionhttp://en.wikipedia.org/wiki/Kronecker_deltahttp://en.wikipedia.org/wiki/Hyperfunctionhttp://en.wikipedia.org/wiki/Rounded_dhttp://en.wikipedia.org/wiki/Partial_derivativehttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Boundary_(topology)http://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Degree_of_a_polynomialhttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Nabla_symbolhttp://en.wikipedia.org/wiki/Gradienthttp://en.wikipedia.org/wiki/Delhttp://en.wikipedia.org/wiki/Nabla_symbolhttp://en.wikipedia.org/wiki/Delta_(letter)http://en.wikipedia.org/wiki/Dirac_delta_functionhttp://en.wikipedia.org/wiki/Hyperfunctionhttp://en.wikipedia.org/wiki/Kronecker_deltahttp://en.wikipedia.org/wiki/Hyperfunctionhttp://en.wikipedia.org/wiki/Rounded_dhttp://en.wikipedia.org/wiki/Partial_derivativehttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Boundary_(topology)http://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Degree_of_a_polynomialhttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Nabla_symbolhttp://en.wikipedia.org/wiki/Gradienthttp://en.wikipedia.org/wiki/Delhttp://en.wikipedia.org/wiki/Nabla_symbol


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gradient of

vector calculusdivergence

del dot;divergence of

vector calculus

If , then.

curl

curl of

vector calculus

If , then

.

derivative

prime;derivative of

calculus

f(x) means the derivative of the functionfat thepointx, i.e., theslopeof the tangent tofatx.

(The single-quote character' is sometimes usedinstead, especially in ASCII text.)

Iff(x) :=x2, thenf(x) = 2x

derivative

dot;time derivative of

calculus

means the derivative ofx with respect to time.

That is .Ifx(t) := t2, then .
http://en.wikipedia.org/wiki/Gradienthttp://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Divergencehttp://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Curl_(mathematics)http://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Prime_(symbol)http://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Tangenthttp://en.wikipedia.org/wiki/Newton's_notationhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Gradienthttp://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Divergencehttp://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Curl_(mathematics)http://en.wikipedia.org/wiki/Vector_calculushttp://en.wikipedia.org/wiki/Prime_(symbol)http://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Tangenthttp://en.wikipedia.org/wiki/Newton's_notationhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Calculus


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indefinite integral orantiderivative

indefinite integral ofthe antiderivative of

calculus

f(x) dx means a function whose derivative isf. x2

dx =x3

/3 + C

definite integral

integral from to of with respect to

calculus

abf(x) dx means the signed area between thex-axis and the graph of thefunctionfbetweenx = aandx = b.

abx2 dx = b3/3 a3/3;

line integral

line/ path/ curve/ integralof along

calculus

Cfds means the integral offalong the curve C,, where r is a

parametrization ofC.

(If the curve is closed, the symbol may be usedinstead, as described below.)

Contour integral;closed line integral

contour integral of

calculus

Similar to the integral, but used to denote a singleintegration over a closed curve or loop. It issometimes used in physics texts involvingequations regarding Gauss's Law, and while theseformulas involve a closed surface integral, therepresentations describe only the first integrationof the volume over the enclosing surface.Instances where the latter requires simultaneousdouble integration, the symbol would be moreappropriate. A third related symbol is the closed

IfCis a Jordan curve about 0, then .
http://en.wikipedia.org/wiki/Integral_symbolhttp://en.wikipedia.org/wiki/Indefinite_integralhttp://en.wikipedia.org/wiki/Antiderivativehttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Definite_integralhttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Graph_(functions)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Line_integralhttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Line_integralhttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Gauss's_Lawhttp://en.wikipedia.org/wiki/Surface_integralhttp://en.wikipedia.org/wiki/Jordan_curvehttp://en.wikipedia.org/wiki/Integral_symbolhttp://en.wikipedia.org/wiki/Indefinite_integralhttp://en.wikipedia.org/wiki/Antiderivativehttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Definite_integralhttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Graph_(functions)http://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Line_integralhttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Line_integralhttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Gauss's_Lawhttp://en.wikipedia.org/wiki/Surface_integralhttp://en.wikipedia.org/wiki/Jordan_curve


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volume integral, denoted by the symbol .

The contour integral can also frequently be found

with a subscript capital letterC,

C, denotingthat a closed loop integral is, in fact, around acontourC, or sometimes dually appropriately, acircle C. In representations of Gauss's Law, asubscript capital S, S, is used to denote that theintegration is over a closed surface.

projection

Projection of

relational algebra

restricts to theattribute set.

Pi

pi;3.1415926;

227

mathematical constant

Used in various formulasinvolving circles; isequivalent to the amount of area a circle wouldtake up in a square of equal width with an area of4 square units, roughly 3.14/4. It is also the ratioof the circumference to the diameter of a circle.

A=R2=314.16R=10

selection

Selection of

relational algebra

The selection selects all those tuplesinfor which holds between the and the

attribute. The selection selects all thosetuples in for which holds between theattribut

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