# ��ѧ����˵��-english

Basic mathematical symbols
Name

Symbol

Explanation

Examples

= �� <> != < > �� <= �� >= ��

is equal to; equals everywhere inequation

x = y means x and y represent the same thing or value.

1+1=2

x �� y means that x and y do not represent the same thing or value. is not equal to; does not equal (The symbols != and <> are primarily from computer science. They are avoided in mathematical texts.) means "not" strict inequality x < y means x is less than y. is less than, is greater than, is x > y means x is greater than y. much less than, is much greater x y means x is much less than y. than x y means x is much greater than y. order theory inequality x �� y means x is less than or equal to y. is less than or x �� y means x is greater than or equal to y. equal to, is greater than or equal to (The symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.) order theory proportionality is proportional y �� x means that y = kx for some constant k. to; varies as everywhere addition plus arithmetic 4 + 6 means the sum of 4 and 6.

1��2

3<4 5>4 0.003 1000000

3 �� 4 and 5 �� 5 5 �� 4 and 5 �� 5

if y = 2x, then y �� x

2+7=9

+

disjoint union the disjoint union of ... and ... set theory subtraction minus arithmetic negative sign 9 4 means the subtraction of 4 from 9. 83=5 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)}

negative; minus 3 means the negative of the number 3. arithmetic set-theoretic complement minus; without set theory multiplication times arithmetic Cartesian 3 �� 4 means the multiplication of 3 by 4. 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.

(5) = 5

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

7 �� 8 = 56

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product the Cartesian product of ... X��Y means the set of all ordered pairs with the first element of each pair selected from X {1,2} �� {3,4} = {(1,3),(1,4),(2,3),(2,4)} and ...; the and the second element selected from Y. direct product of ... and ... set theory cross product cross vector algebra multiplication times 3 4 means the multiplication of 3 by 4. 7 8 = 56 u �� v means the cross product of vectors u and v (1,2,5) �� (3,4,1) = (22, 16, 2)

��

�� ��

arithmetic dot product dot vector algebra division divided by arithmetic plus-minus plus or minus 6 �� 3 means both 6 + 3 and 6 - 3. arithmetic plus-minus plus or minus 10 �� 2 or equivalently 10 �� 20% means the range from 10 2 to 10 + 2. measurement minus-plus minus or plus 6 �� (3 5) means both 6 + (3 - 5) and 6 - (3 + 5). arithmetic square root the principal square root of; ��x means the positive number whose square is x. square root real numbers complex square root 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 absolute value zero. (modulus) of numbers Euclidean distance Euclidean distance |x �C y| means the Euclidean distance between x and y. between; Euclidean norm of Geometry Determinant determinant of |A| means the determinant of the matrix A Matrix theory divides divides Number Theory factorial ��2ҳ For x = (1,1), and y = (4,5), |x �C y| = ��([1�C4]2 + [1�C5]2) = 5 |3| = 3 |�C5| = |5| |i|=1 | 3 + 4i | = 5 if z = r exp(i��) is represented in polar coordinates with -�� < �� �� ��, then ��z = ��r exp (i ��/2). ��4 = 2 cos(x �� y) = cos(x) cos(y) sin(x) sin(y). The equation x = 5 �� ��4, has two solutions, x = 7 and x = 3. If a = 100 �� 1 mm, then a �� 99 mm and a �� 101 mm. 6 �� 3 or 6 3 means the division of 6 by 3. 2 �� 4 = .5 12 4 = 3 u v means the dot product of vectors u and v (1,2,5) (3,4,1) = 6

��

��(-1) = i

|��|

|

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.

! T

factorial combinatorics transpose transpose matrix operations probability distribution has distribution statistics Row equivalence is row equivalent to Matrix theory

n ! is the product 1 �� 2�� ... �� n.

4! = 1 �� 2 �� 3 �� 4 = 24

Swap rows for columns

Aij = (AT)ji

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

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

~

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

�� ��

material implication implies; if �� then A B means if A is true then B is also true; if A is false then nothing is said about B. �� may mean the same as , or it may have the meaning for functions given below. x = 2 x2 = 4 is true, but x2 = 4 x = 2 is in general false (since x could be 2).

propositional may mean the same as , or it may have the meaning for superset given below. logic, Heyting algebra material equivalence if and only if; A B means A is true if B is true and A is false if B is false. iff propositional logic logical negation The statement A is true if and only if A is false. not A slash placed through another operator is the same as "" placed in front. propositional logic (The symbol ~ has many other uses, so or the slash notation is preferred.) logical conjunction or meet in a lattice The statement A �� B is true if A and B are both true; else it is false. and; min propositional For functions A(x) and B(x), A(x) �� B(x) is used to mean min(A(x), B(x)). logic, lattice theory

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

(A) A x �� y (x = y)

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

��

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 n �� 4 �� n �� 2 n �� 3 when n is a false. or; max natural number. propositional For functions A(x) and B(x), A(x) �� B(x) is used to mean max(A(x), B(x)). 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 propositional same. logic, Boolean algebra direct sum direct sum of Abstract algebra universal quantification for all; for any; x: P(x) means P(x) is true for all x. for each predicate logic existential quantification there exists n �� : n2 �� n. The direct sum is a special way of combining several modules into one general module (the symbol �� is used, is only for logic). xor

��

(A) �� A is always true, A �� A is always false.

Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V �� W (U = V + W) �� (V �� W = )

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

n �� : n is even.

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predicate logic uniqueness quantification there exists exactly one predicate logic ! x: P(x) means there is exactly one x such that P(x) is true. ! n �� : n + 5 = 2n.

! := �� : �� {,} {:} {|} {} �� ��

definition x := y or x �� y means x is defined to be another name for y cosh x := (1/2)(exp x + exp (x)) is defined as (Some writers use �� to mean congruence). P : Q means P is defined to be logically equivalent to Q. everywhere congruence is congruent to ��ABC ��DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. geometry A xor B : (A �� B) �� (A �� B)

congruence relation ... is congruent a �� b (mod n) means a b is divisible by n to ... modulo ... modular arithmetic set brackets the set of �� set theory set builder notation the set of �� such that set theory empty set the empty set set theory set membership is an element of; is not an element of everywhere, set theory subset is a subset of (subset) A B means every element of A is also element of B. (proper subset) A B means A B but A �� B. a �� S means a is an element of the set S; a S means a is not an element of S. (1/2)1 �� 21 means the set with no elements. { } means the same. {n �� : 1 < n2 < 4} = {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 �� : n2 < 20} = { 1, 2, 3, 4} {a,b,c} means the set consisting of a, b, and c. = { 1, 2, 3, ��} 5 �� 11 (mod 3)

(A �� B) A (A �� B) B

set theory (Some writers use the symbol as if it were the same as .) superset A B means every element of B is also element of A.

is a superset of A B means A B but A �� B. set theory (Some writers use the symbol as if it were the same as .) 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. the union of �� "A or B, but not both." A B (A �� B) = B (inclusive) and �� (inclusive) A �� B means the set that contains all the elements from A, or all the elements union from B, or all the elements from both A and B. "A or B or both". set theory

��

set-theoretic intersection A �� B means the set that contains all those elements that A and B have in common. intersected with; intersect set theory ��4ҳ {x �� : x2 = 1} �� = {1}

��

symmetric difference symmetric difference set theory set-theoretic complement minus; without set theory function application of 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} A��B means the set of elements in exactly one of A or B. {1,5,6,8} �� {2,5,8} = {1,2,6}

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

If f(x) := x2, then f(3) = 32 = 9.

()

set theory precedence grouping parentheses everywhere function arrow from �� to set theory,type theory function composition composed with set theory natural numbers N N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention. = {|a| : a �� , a �� 0} f: X �� Y means the function f maps the set X into the set Y. Let f: �� be defined by f(x) := x2.

Perform the operations inside the parentheses first.

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

f:X�� Y
o

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).

N Z Q R C

numbers integers Z numbers rational numbers Q numbers real numbers �С� R numbers complex numbers C numbers arbitrary constant C integral calculus real or complex numbers K means the statement holds substituting K for R and also for C. and linear algebra . ��5ҳ means the set of real numbers. ��(1) means {p/q : p �� , q �� }. means {..., 3, 2, 1, 0, 1, 2, 3, ...} and + means {1, 2, 3, ...} = . = {p, -p : p �� } �� {0}

3.14000... �� ��

means {a + b i : a,b �� }.

i = ��(1) ��

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

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

because

K

K

�� ||��|| ��

infinity infinity numbers norm norm of length of linear algebra summation sum over �� from �� to �� of arithmetic product product over �� from �� to �� of arithmetic

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

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

|| x + y || �� || x || + || y ||

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

= (1+2)(2+2)(3+2)(4+2) means a1a2an. = 3 �� 4 �� 5 �� 6 = 360

��

Cartesian product the Cartesian product of; the direct product of set theory coproduct means the set of all (n+1)-tuples (y0, ��, yn).

��

coproduct over �� from �� to �� of category theory derivative �� prime derivative of calculus indefinite integral or antiderivative indefinite integral of �� f(x) dx means a function whose derivative is f. ��x2 dx = x3/3 + C f ��(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. If f(x) := x2, then f ��(x) = 2x The dot notation indicates a time derivative. That is .

��

the antiderivative of calculus definite integral integral from �� �� b f(x) dx means the signed area between the x-axis and the graph of the function f to �� of �� with a between x = a and x = b. respect to calculus Similar to the integral, but used to denote a single integration over a closed curve or loop. contour integral It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only or closed line the first integration of the volume over the enclosing surface. Instances where the latter integral requires simultaneous double integration, the symbol would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol . contour integral The contour integral can also frequently be found with a subscript capital letter C, ��C, of denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ��S, is calculus used to denote that the integration is over a closed surface. gradient del, nabla, gradient of vector calculus divergence del dot, divergence of If , then ��6ҳ f (x1, ��, xn) is the vector of partial derivatives (f / x1, ��, f / xn). If f (x,y,z) := 3xy + z, then f = (3y, 3x, 2z) ��0b x2 dx = b3/3;

��

vector calculus curl curl of vector calculus partial differential partial, d With f (x1, ��, xn), f/xi is the derivative of f with respect to xi, with all other variables kept constant. If

.

, then .

If f(x,y) := x2y, then f/x = 2xy

calculus boundary boundary of topology perpendicular is perpendicular x �� y means x is perpendicular to y; or more generally x is orthogonal to y. to If l �� m and m �� n then l || n. M means the boundary of M {x : ||x|| �� 2} = {x : ||x|| = 2}

��

geometry bottom element the bottom element lattice theory parallel is parallel to geometry entailment entails model theory inference infers or is derived from propositional logic, predicate logic normal subgroup is a normal subgroup of group theory quotient group mod 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}} If we define ~ by x ~ y x y �� , then /~ = {{x + n : n �� } : x �� (0,1]} N G means that N is a normal subgroup of group G. Z(G) G 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 x || y means x is parallel to y. If l || m and m �� n then l �� n. x = �� means x is the smallest element. x : x �� �� = ��

||

x y means y is derived from x.

A �� B B �� A

/

group theory quotient set mod set theory approximately equal is x �� y means x is approximately equal to y. approximately equal to everywhere isomorphism is isomorphic to G �� H means that group G is isomorphic to group H. group theory same order of magnitude roughly similar A/~ means the set of all ~ equivalence classes in A.

�� �� 3.14159

��

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

2~5 m ~ n means the quantities m and n have the same order of magnitude, or general size. (Note that ~ is used for an approximation that is poor, otherwise use �� .) 8 �� 9 ~ 100 but ��2 �� 10

~

poorly approximates Approximation theory

��,��
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inner product

(|) <,>
linear algebra inner product of ��x,y�� means the inner product of x and y as defined in an inner product space. For spatial vectors, the dot product notation, xy is common. For matricies, the colon notation may be used. The standard inner product between two vectors x = (2, 3) and y = (1, 5) is: ��x, y�� = 2 �� 1 + 3 �� 5 = 13 A:B =

��
i,j

AijBij

:
tensor product

*

tensor product V U means the tensor product of V and U. of linear algebra convolution convolution, convoluted with f * g means the convolution of f and g. functional analysis mean overbar, �� bar statistics complex conjugate conjugate complex numbers delta equal to equal by definition everywhere means equal by definition. When is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ��. is the complex conjugate of z. (often read as "x bar") is the mean (average value of xi).

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

x

.

.

Mathematical alphanumeric symbols Table of logic symbols Mathematical notation ISO 31-11 Roman letters used in mathematics Greek letters used in mathematics Notation in probability Physical constants [[millimeters

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