is equal to;
equals
everywhere
x = y means x and y represent the same thing or value. 2 = 2
1 + 1 = 2 ≠
is not equal to;
does not equal
everywhere
x ≠ y means that x and y do not represent the same thing or value.
(The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 2 + 2 ≠ 5 <
>
is less than,
is greater than
x < y means x is less than y.
x > y means x is greater than y. 3 < 4
5 > 4 proper subgroup
is a proper subgroup of
H < G means H is a proper subgroup of G. 5Z < Z
A3 < S3 ≪
≫
(very) strict inequality
is much less than,
is much greater than
x ≪ y means x is much less than y.
x ≫ y means x is much greater than y. 0.003 ≪ 1000000 asymptotic comparison
is of smaller order than,
is of greater order than
f ≪ g means the growth of f is asymptotically bounded by g.
(This is I. M. Vinogradov's notation. Another notation is the Big O notation, which looks like f = O(g).) x ≪ ex ≤
≥
is less than or equal to,
is greater than or equal to
x ≤ y means x is less than or equal to y.
x ≥ y means x is greater than or equal to y.
(The forms <= and >= are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5 subgroup
is a subgroup of
H ≤ G means H is a subgroup of G. Z ≤ Z
A3 ≤ S3 reduction
is reducible to
computational complexity theory
A ≤ B means the problemA can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction. If
then
≺is Karp reducible to;
is polynomial-time many-one reducible to
computational complexity theory
L1 ≺ L2 means that the problem L1 is Karp reducible to L2.[1]If L1 ≺ L2 and L2 ∈ , then L1 ∈ P. ∝
is proportional to;
varies as
everywhere
y ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x. Karp reduction[2]
is Karp reducible to;
is polynomial-time many-one reducible to
computational complexity theory
A ∝ B means the problemA can be polynomially reduced to the problem B. If L1 ∝ L2 and L2 ∈ , then L1 ∈ P. +
plus;
add
4 + 6 means the sum of 4 and 6. 2 + 7 = 9 disjoint union
the disjoint union of ... and ...
A1 + A2 means the disjoint union of sets A1 and A2. A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, 9, 10} ⇒
A1 + A2 = {(3,1), (4,1), (5,1), (6,1), (7,2), (8,2), (9,2), (10,2)} −
minus;
take;
subtract
9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5 negative sign
negative;
minus;
the opposite of
−3 means the negative of the number 3. −(−5) = 5 set-theoretic complement
minus;
without
A − B means the set that contains all the elements of A that are not in B.
(∖ can also be used for set-theoretic complement as described below.) {1,2,4} − {1,3,4} = {2} ±
plus or minus
6 ± 3 means both 6 + 3 and 6 − 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3. plus-minus
plus or minus
10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm. ∓
minus or plus
6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5). cos(x± y) = cos(x) cos(y) ∓ sin(x) sin(y). ×
times;
multiplied by
3 × 4 means the multiplication of 3 by 4.
(The symbol * is generally used in programming languages, where ease of typing and use of ASCIItext is preferred.) 7 × 8 = 56 Cartesian product
the Cartesian product of ... and ...;
the direct product of ... and ...
X×Y means the set of all ordered pairswith the first element of each pair selected from X and the second element selected from Y. {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)} cross product
cross
u × v means the cross product of vectorsu and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2) group of units
the group of units of
R× consists of the set of units of the ring R, along with the operation of multiplication.
This may also be written R* as described below, orU(R). *
times;
multiplied by
a * b means the product of a and b.
(Multiplication can also be denoted with × or ⋅, or even simple juxtaposition. * is generally used where ease of typing and use of ASCII text is preferred, such as programming languages.) 4 * 3 means the product of 4 and 3, or 12. convolution
convolution;
convolved with
f * g means the convolution of f and g. . complex conjugate
conjugate
z* means the complex conjugate of z.
( can also be used for the conjugate of z, as described below.) . group of units
the group of units of
R* consists of the set of units of the ring R, along with the operation of multiplication.
This may also be written R× as described above, orU(R). hyperreal numbers
the (set of) hyperreals
*R means the set of hyperreal numbers. Other sets can be used in place of R. *N is the hypernaturalnumbers. Hodge dual
Hodge dual;
Hodge star
*v means the Hodge dual of a vector v. If vis a k-vectorwithin an n-dimensionalorientedinner productspace, then *v is an (n−k)-vector. If are the standard basis vectors of , ·
times;
multiplied by
3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56 dot product
dot
u · v means the dot product of vectorsu and v (1,2,5) · (3,4,−1) = 6 placeholder
(silent)
A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. ⊗
tensor product, tensor product of modules
tensor product of
means the tensor product of V and U.[3]means the tensor product of modules Vand U over the ringR. {1, 2, 3, 4} ⊗ {1, 1, 2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} ÷
⁄
divided by;
over
6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = 0.5
12 ⁄ 4 = 3 quotient group
mod
G / H means the quotient of group Gmodulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a,b+2a}} quotient set
mod
A/~ means the set of all ~ equivalence classes in A. If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then
ℝ/~ = { {x + n : n ∈ ℤ } : x ∈ [0,1) } √
the (principal) square root of
means the nonnegative number whose square is . complex square root
the (complex) square root of
if is represented in polar coordinates with , then . x
overbar;
… bar
(often read as "x bar") is the mean (average value of ). . complex conjugate
conjugate
means the complex conjugate of z.
(z* can also be used for the conjugate of z, as described above.) . finite sequence, tuple
finite sequence, tuple
means the finite sequence/tuple . . algebraic closure
algebraic closure of
is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as because it is the algebraic closure of the rational numbers . topological closure
(topological) closure of
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 rational numbers are dense in the real numbers). |…|
absolute value;
modulus
absolute value of; modulus of
|x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3
|-5| = |5| = 5
| i | = 1
| 3 + 4i | = 5 Euclidean norm or Euclidean length or magnitude
Euclidean norm of
|x| means the (Euclidean) length of vectorx. For x = (3,-4)
determinant
determinant of
|A| means the determinant of the matrix A cardinality
cardinality of;
size of;
order of
|X| means the cardinality of the set X.
(# may be used instead as described below.) |{3, 5, 7, 9}| = 4. …
norm of;
length of
x means the norm of the element x of a normed vector space.[4]x + y ≤ x + y nearest integer function
nearest integer to
x means the nearest integer to x.
(This may also be written [x], ⌊x⌉, nint(x) orRound(x).) 1 = 1, 1.6 = 2, −2.4 = −2, 3.49 = 3 ∣
∤
divides
a|b means a divides b.
a∤b means a does not divide b.
(This symbol can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar |character can be used.) Since 15 = 3×5, it is true that 3|15 and 5|15. conditional probability
given
P(A|B) means the probability of the event a occurring given that b occurs. if X is a uniformly random day of the year P(X is May 25 | X is in May) = 1/31 restriction
restriction of … to …;
restricted to
f|A means the function f restricted to the set A, that is, it is the function with domainA ∩ dom(f) that agrees with f. The function f : R → R defined by f(x) = x2 is not injective, but f|R+ is injective. such that
such that;
so that
everywhere
| means "such that", see ":" (described below). S = {(x,y) | 0 < y < f(x)}
The set of (x,y) such that y is greater than 0 and less than f(x).
is parallel to
x y means x is parallel to y. If l m and m ⊥ n then l ⊥ n. incomparability
is incomparable to
x y means x is incomparable to y. {1,2} {2,3} under set containment. exact divisibility
exactly divides
pa n means pa exactly divides n (i.e. pa divides nbut pa+1 does not). 23 360. #
cardinality of;
size of;
order of
#X means the cardinality of the set X.
(|…| may be used instead as described above.) #{4, 6, 8} = 3 connected sum
connected sum of;
knot sum of;
knot composition of
A#B is the connected sum of the manifolds Aand B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition. A#Sm is homeomorphicto A, for any manifold A, and the sphere Sm. primorial
primorial
n# is product of all prime numbers less than or equal to n. 12# = 2 × 3 × 5 × 7 × 11 = 2310 ℵ
aleph
ℵα represents an infinite cardinality (specifically, theα-th one, where α is an ordinal). |ℕ| = ℵ0, which is called aleph-null. ℶ
beth
ℶα represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ). ?
cardinality of the continuum;
c;
cardinality of the real numbers
The cardinality of is denoted by or by the symbol (a lowercase Frakturletter C). :
such that
such that;
so that
everywhere
: means "such that", and is used in proofs and theset-builder notation (described below). ∃ n ∈ ℕ: n is even. field extension
extends;
over
K : F means the field K extends the field F.
This may also be written as K ≥ F. ℝ : ℚ inner productof matrices
inner product of
A : B means the Frobenius inner product of the matrices A and B.
The general inner product is denoted by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as described below. For spatial vectors, the dot product notation, x·y is common.See also Bra-ket notation. index of a subgroup
index of subgroup
The index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G !
factorial
n! means the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24 logical negation
not
The statement !A is true if and only if A is false.
A slash placed through another operator is the same as "!" placed in front.
(The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation¬Ais preferred.) !(!A) ⇔ A
x ≠ y ⇔ !(x = y) ~
has distribution
X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution row equivalence
is row equivalent to
A~B means that B can be generated by using a series of elementary row operations on A same order of magnitude
roughly similar;
poorly approximates
m ~ n means the quantities m and nhave the sameorder of magnitude, or general size.
(Note that ~ is used for an approximation that is poor, otherwise use ≈ .) 2 ~ 5
8 × 9 ~ 100
but π2 ≈ 10 asymptotically equivalent
is asymptotically equivalent to
f ~ g means . x ~ x+1 equivalence relation
are in the same equivalence class
everywhere
a ~ b means (and equivalently ). 1 ~ 5 mod 4 ≈
approximately equal
is approximately equal to
everywhere
x ≈ y means x is approximately equal to y.
This may also be written ≃, ≅, ~, ♎ (Libra Symbol),or≒. π ≈ 3.14159 isomorphism
is isomorphic to
G ≈ H means that group G is isomorphic (structurally identical) to group H.
(≅ can also be used for isomorphic, as described below.) Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group. ≀
wreath product of … by …
A ≀ H means the wreath product of the group A by the group H.
This may also be written A wr H. is isomorphic to the automorphismgroup of thecomplete bipartite graph on (n,n) vertices. ◅
▻
is a normal subgroup of
N ◅ G means that N is a normal subgroup of groupG. Z(G) ◅ G ideal
is an ideal of
I ◅ R means that I is an ideal of ring R. (2) ◅ Z antijoin
the antijoin of
R ▻ S means the antijoin of the relations Rand S, the tuples in R for which there is not a tuple in S that is equal on their common attribute names. RS = R - R S ⋉
⋊
the semidirect product of
N ⋊φ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ. Also, if G = N ⋊φ H, then G is said to split over N.
(⋊ may also be written the other way round, as ⋉, or as ×.) semijoin
the semijoin of
R ⋉ S is the semijoin of the relations Rand S, the set of all tuples in R for which there is a tuple in Sthat is equal on their common attribute names. R S = a1,..,an(R S) ⋈
the natural join of
R ⋈ S is the natural join of the relations R and S, the set of all combinations of tuples in R and S that are equal on their common attribute names. ∴
therefore;
so;
hence
everywhere
Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal. ∵
because;
since
everywhere
Sometimes used in proofs before reasoning. 3331 is prime ∵ it has no positive integer factors other than itself and one. ■
□
∎
▮
‣
everywhere
Used to mark the end of a proof.
(May also be written Q.E.D.) D'Alembertian
non-Euclidean Laplacian
It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions. ⇒
→
⊃
implies;
if … then
propositional logic, Heyting algebra
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 functionsgiven below.)
(⊃ may mean the same as ⇒,[5]or it may have the meaning for supersetgiven below.) x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since xcould be −2). ⇔
↔
if and only if;
iff
A ⇔ B means A is true if B is true and A is false if Bis false. x + 5 = y+ 2 ⇔ x + 3 = y ¬
˜
not
The statement ¬A is true if and only if A is false.
A slash placed through another operator is the same as "¬" placed in front.
(The symbol ~ has many other uses, so ¬ or the slash notation is preferred. Computer scientists will often use! but this is avoided in mathematical texts.) ¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y) ∧
logical conjunction or meetin a lattice
and;
min;
meet
propositional logic, lattice theory
The statement A ∧ B is true if A and B are both true; else it is false.
For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)). n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number. wedge product
wedge product;
exterior product
u ∧ v means the wedge product of any multivectorsuand v. In three dimensional Euclidean space the wedge product and the cross product of two vectorsare each other's Hodge dual. exponentiation
… (raised) to the power of …
everywhere
a ^ b means a raised to the power of b
(a ^ b is 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 disjunction or joinin a lattice
or;
max;
join
propositional logic, lattice theory
The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.
For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)). n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. ⊕
⊻
xor
propositional logic, Boolean algebra
The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. (¬A) ⊕ A is always true, A ⊕ A is always false. direct sum
direct sum of
The direct sum is a special way of combining several objects into one general object.
(The bun symbol ⊕, or the coproductsymbol ∐, 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}) ∀
for all;
for any;
for each
∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n. ∃
there exists;
there is;
there are
∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even. ∃!
there exists exactly one
∃! x: P(x) means there is exactly one x such thatP(x) is true. ∃! n ∈ ℕ: n + 5 = 2n. =:
:=
≡
:⇔
≜
≝
≐
is defined as;
is equal by definition to
everywhere
x := y, y =: x or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.
(Some writers use ≡ to mean congruence).
P :⇔ Q means P is defined to be logically equivalentto Q. ≅
is congruent to
△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. isomorphic
is isomorphic to
G ≅ H means that group G is isomorphic (structurally identical) to group H.
(≈ can also be used for isomorphic, as described above.) . ≡
... is congruent to ... modulo ...
a ≡ b (mod n) means a − b is divisible by n 5 ≡ 2 (mod 3) { , }
setbrackets
the set of …
{a,b,c} means the set consisting of a, b, and c.[6]ℕ = { 1, 2, 3, …} { : }
{ | }
{ ; }
the set of … such that
{x : P(x)} means the set of all xfor which P(x) is true.[6]{x | P(x)} is the same as {x : P(x)}. {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4} ∅
{ }
the empty set
∅ means the set with no elements.[6]{ } means the same. {n ∈ ℕ : 1 < n2 < 4} = ∅ ∈
∉
is an element of;
is not an element of
everywhere, set theory
a ∈ S means a is an element of the set S;[6]a ∉ Smeans a is not an element of S.[6](1/2)−1 ∈ ℕ
2−1 ∉ ℕ ⊆
⊂
is a subset of
(subset) A ⊆ B means every element of A is also an element of B.[7]
(proper subset) A ⊂ B means A ⊆ B but A ≠ B.
(Some writers use the symbol ⊂ as if it were the same as ⊆.) (A ∩ B) ⊆ A
ℕ ⊂ ℚ
ℚ ⊂ ℝ ⊇
⊃
is a superset of
A ⊇ B means every element of B is also an element of A.
A ⊃ B means A ⊇ B but A ≠ B.
(Some writers use the symbol ⊃ as if it were the same as ⊇.) (A ∪ B) ⊇ B
ℝ ⊃ ℚ ∪
the union of … or …;
union
A ∪ B means the set of those elements which are either in A, or in B, or in both.[7]A ⊆ B ⇔ (A ∪ B) = B ∩
intersected with;
intersect
A ∩ B means the set that contains all those elements that A and B have in common.[7]{x ∈ ℝ : x2 = 1} ∩ ℕ = {1} ∆
symmetric difference
A ∆ B means the set of elements in exactly one of Aor B.
(Not to be confused with delta, Δ, described below.) {1,5,6,8} ∆ {2,5,8} = {1,2,6} ∖
minus;
without
A ∖ B means the set that contains all those elements of A that are not in B.[7]
(− can also be used for set-theoretic complement as described above.) {1,2,3,4} ∖ {3,4,5,6} = {1,2} →
functionarrow
from … to
f: X → Y means the function f maps the set X into the set Y. Let f: ℤ → ℕ∪{0} be defined by f(x) := x2. ↦
functionarrow
maps to
f: a ↦ b means the function f maps the element a to the element b. Let f: x ↦ x+1 (the successor function). ∘
composed with
f∘g is the function, such that (f∘g)(x) = f(g(x)).[8]if f(x) := 2x, and g(x) := x + 3, then (f∘g)(x) = 2(x + 3). o
entrywise product
For two matrices (or vectors) of the same dimensions the Hadamard product is a matrix of the same dimensions with elements given by . This is often used in matrix based programming such as MATLABwhere the operation is done by A.*B
Some common symbols used to denote sets include curly braces { }, element notation |, and the symbol ∈ to indicate membership of an element in a set. Other symbols like ∅ represent the empty set, and ⊆ denotes a subset relationship.
is equal to;
equals
everywhere
x = y means x and y represent the same thing or value. 2 = 2
1 + 1 = 2 ≠
is not equal to;
does not equal
everywhere
x ≠ y means that x and y do not represent the same thing or value.
(The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 2 + 2 ≠ 5 <
>
is less than,
is greater than
x < y means x is less than y.
x > y means x is greater than y. 3 < 4
5 > 4 proper subgroup
is a proper subgroup of
H < G means H is a proper subgroup of G. 5Z < Z
A3 < S3 ≪
≫
(very) strict inequality
is much less than,
is much greater than
x ≪ y means x is much less than y.
x ≫ y means x is much greater than y. 0.003 ≪ 1000000 asymptotic comparison
is of smaller order than,
is of greater order than
f ≪ g means the growth of f is asymptotically bounded by g.
(This is I. M. Vinogradov's notation. Another notation is the Big O notation, which looks like f = O(g).) x ≪ ex ≤
≥
is less than or equal to,
is greater than or equal to
x ≤ y means x is less than or equal to y.
x ≥ y means x is greater than or equal to y.
(The forms <= and >= are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5 subgroup
is a subgroup of
H ≤ G means H is a subgroup of G. Z ≤ Z
A3 ≤ S3 reduction
is reducible to
computational complexity theory
A ≤ B means the problemA can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction. If
then
≺is Karp reducible to;
is polynomial-time many-one reducible to
computational complexity theory
L1 ≺ L2 means that the problem L1 is Karp reducible to L2.[1]If L1 ≺ L2 and L2 ∈ , then L1 ∈ P. ∝
is proportional to;
varies as
everywhere
y ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x. Karp reduction[2]
is Karp reducible to;
is polynomial-time many-one reducible to
computational complexity theory
A ∝ B means the problemA can be polynomially reduced to the problem B. If L1 ∝ L2 and L2 ∈ , then L1 ∈ P. +
plus;
add
4 + 6 means the sum of 4 and 6. 2 + 7 = 9 disjoint union
the disjoint union of ... and ...
A1 + A2 means the disjoint union of sets A1 and A2. A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, 9, 10} ⇒
A1 + A2 = {(3,1), (4,1), (5,1), (6,1), (7,2), (8,2), (9,2), (10,2)} −
minus;
take;
subtract
9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5 negative sign
negative;
minus;
the opposite of
−3 means the negative of the number 3. −(−5) = 5 set-theoretic complement
minus;
without
A − B means the set that contains all the elements of A that are not in B.
(∖ can also be used for set-theoretic complement as described below.) {1,2,4} − {1,3,4} = {2} ±
plus or minus
6 ± 3 means both 6 + 3 and 6 − 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3. plus-minus
plus or minus
10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm. ∓
minus or plus
6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5). cos(x± y) = cos(x) cos(y) ∓ sin(x) sin(y). ×
times;
multiplied by
3 × 4 means the multiplication of 3 by 4.
(The symbol * is generally used in programming languages, where ease of typing and use of ASCIItext is preferred.) 7 × 8 = 56 Cartesian product
the Cartesian product of ... and ...;
the direct product of ... and ...
X×Y means the set of all ordered pairswith the first element of each pair selected from X and the second element selected from Y. {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)} cross product
cross
u × v means the cross product of vectorsu and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2) group of units
the group of units of
R× consists of the set of units of the ring R, along with the operation of multiplication.
This may also be written R* as described below, orU(R). *
times;
multiplied by
a * b means the product of a and b.
(Multiplication can also be denoted with × or ⋅, or even simple juxtaposition. * is generally used where ease of typing and use of ASCII text is preferred, such as programming languages.) 4 * 3 means the product of 4 and 3, or 12. convolution
convolution;
convolved with
f * g means the convolution of f and g. . complex conjugate
conjugate
z* means the complex conjugate of z.
( can also be used for the conjugate of z, as described below.) . group of units
the group of units of
R* consists of the set of units of the ring R, along with the operation of multiplication.
This may also be written R× as described above, orU(R). hyperreal numbers
the (set of) hyperreals
*R means the set of hyperreal numbers. Other sets can be used in place of R. *N is the hypernaturalnumbers. Hodge dual
Hodge dual;
Hodge star
*v means the Hodge dual of a vector v. If vis a k-vectorwithin an n-dimensionalorientedinner productspace, then *v is an (n−k)-vector. If are the standard basis vectors of , ·
times;
multiplied by
3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56 dot product
dot
u · v means the dot product of vectorsu and v (1,2,5) · (3,4,−1) = 6 placeholder
(silent)
A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. ⊗
tensor product, tensor product of modules
tensor product of
means the tensor product of V and U.[3]means the tensor product of modules Vand U over the ringR. {1, 2, 3, 4} ⊗ {1, 1, 2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} ÷
⁄
divided by;
over
6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = 0.5
12 ⁄ 4 = 3 quotient group
mod
G / H means the quotient of group Gmodulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a,b+2a}} quotient set
mod
A/~ means the set of all ~ equivalence classes in A. If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then
ℝ/~ = { {x + n : n ∈ ℤ } : x ∈ [0,1) } √
the (principal) square root of
means the nonnegative number whose square is . complex square root
the (complex) square root of
if is represented in polar coordinates with , then . x
overbar;
… bar
(often read as "x bar") is the mean (average value of ). . complex conjugate
conjugate
means the complex conjugate of z.
(z* can also be used for the conjugate of z, as described above.) . finite sequence, tuple
finite sequence, tuple
means the finite sequence/tuple . . algebraic closure
algebraic closure of
is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as because it is the algebraic closure of the rational numbers . topological closure
(topological) closure of
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 rational numbers are dense in the real numbers). |…|
absolute value;
modulus
absolute value of; modulus of
|x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3
|-5| = |5| = 5
| i | = 1
| 3 + 4i | = 5 Euclidean norm or Euclidean length or magnitude
Euclidean norm of
|x| means the (Euclidean) length of vectorx. For x = (3,-4)
determinant
determinant of
|A| means the determinant of the matrix A cardinality
cardinality of;
size of;
order of
|X| means the cardinality of the set X.
(# may be used instead as described below.) |{3, 5, 7, 9}| = 4. …
norm of;
length of
x means the norm of the element x of a normed vector space.[4]x + y ≤ x + y nearest integer function
nearest integer to
x means the nearest integer to x.
(This may also be written [x], ⌊x⌉, nint(x) orRound(x).) 1 = 1, 1.6 = 2, −2.4 = −2, 3.49 = 3 ∣
∤
divides
a|b means a divides b.
a∤b means a does not divide b.
(This symbol can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar |character can be used.) Since 15 = 3×5, it is true that 3|15 and 5|15. conditional probability
given
P(A|B) means the probability of the event a occurring given that b occurs. if X is a uniformly random day of the year P(X is May 25 | X is in May) = 1/31 restriction
restriction of … to …;
restricted to
f|A means the function f restricted to the set A, that is, it is the function with domainA ∩ dom(f) that agrees with f. The function f : R → R defined by f(x) = x2 is not injective, but f|R+ is injective. such that
such that;
so that
everywhere
| means "such that", see ":" (described below). S = {(x,y) | 0 < y < f(x)}
The set of (x,y) such that y is greater than 0 and less than f(x).
is parallel to
x y means x is parallel to y. If l m and m ⊥ n then l ⊥ n. incomparability
is incomparable to
x y means x is incomparable to y. {1,2} {2,3} under set containment. exact divisibility
exactly divides
pa n means pa exactly divides n (i.e. pa divides nbut pa+1 does not). 23 360. #
cardinality of;
size of;
order of
#X means the cardinality of the set X.
(|…| may be used instead as described above.) #{4, 6, 8} = 3 connected sum
connected sum of;
knot sum of;
knot composition of
A#B is the connected sum of the manifolds Aand B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition. A#Sm is homeomorphicto A, for any manifold A, and the sphere Sm. primorial
primorial
n# is product of all prime numbers less than or equal to n. 12# = 2 × 3 × 5 × 7 × 11 = 2310 ℵ
aleph
ℵα represents an infinite cardinality (specifically, theα-th one, where α is an ordinal). |ℕ| = ℵ0, which is called aleph-null. ℶ
beth
ℶα represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ). ?
cardinality of the continuum;
c;
cardinality of the real numbers
The cardinality of is denoted by or by the symbol (a lowercase Frakturletter C). :
such that
such that;
so that
everywhere
: means "such that", and is used in proofs and theset-builder notation (described below). ∃ n ∈ ℕ: n is even. field extension
extends;
over
K : F means the field K extends the field F.
This may also be written as K ≥ F. ℝ : ℚ inner productof matrices
inner product of
A : B means the Frobenius inner product of the matrices A and B.
The general inner product is denoted by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as described below. For spatial vectors, the dot product notation, x·y is common.See also Bra-ket notation. index of a subgroup
index of subgroup
The index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G !
factorial
n! means the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24 logical negation
not
The statement !A is true if and only if A is false.
A slash placed through another operator is the same as "!" placed in front.
(The symbol ! is primarily from computer science. It is avoided in mathematical texts, where the notation¬Ais preferred.) !(!A) ⇔ A
x ≠ y ⇔ !(x = y) ~
has distribution
X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution row equivalence
is row equivalent to
A~B means that B can be generated by using a series of elementary row operations on A same order of magnitude
roughly similar;
poorly approximates
m ~ n means the quantities m and nhave the sameorder of magnitude, or general size.
(Note that ~ is used for an approximation that is poor, otherwise use ≈ .) 2 ~ 5
8 × 9 ~ 100
but π2 ≈ 10 asymptotically equivalent
is asymptotically equivalent to
f ~ g means . x ~ x+1 equivalence relation
are in the same equivalence class
everywhere
a ~ b means (and equivalently ). 1 ~ 5 mod 4 ≈
approximately equal
is approximately equal to
everywhere
x ≈ y means x is approximately equal to y.
This may also be written ≃, ≅, ~, ♎ (Libra Symbol),or≒. π ≈ 3.14159 isomorphism
is isomorphic to
G ≈ H means that group G is isomorphic (structurally identical) to group H.
(≅ can also be used for isomorphic, as described below.) Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group. ≀
wreath product of … by …
A ≀ H means the wreath product of the group A by the group H.
This may also be written A wr H. is isomorphic to the automorphismgroup of thecomplete bipartite graph on (n,n) vertices. ◅
▻
is a normal subgroup of
N ◅ G means that N is a normal subgroup of groupG. Z(G) ◅ G ideal
is an ideal of
I ◅ R means that I is an ideal of ring R. (2) ◅ Z antijoin
the antijoin of
R ▻ S means the antijoin of the relations Rand S, the tuples in R for which there is not a tuple in S that is equal on their common attribute names. RS = R - R S ⋉
⋊
the semidirect product of
N ⋊φ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ. Also, if G = N ⋊φ H, then G is said to split over N.
(⋊ may also be written the other way round, as ⋉, or as ×.) semijoin
the semijoin of
R ⋉ S is the semijoin of the relations Rand S, the set of all tuples in R for which there is a tuple in Sthat is equal on their common attribute names. R S = a1,..,an(R S) ⋈
the natural join of
R ⋈ S is the natural join of the relations R and S, the set of all combinations of tuples in R and S that are equal on their common attribute names. ∴
therefore;
so;
hence
everywhere
Sometimes used in proofs before logical consequences. All humans are mortal. Socrates is a human. ∴ Socrates is mortal. ∵
because;
since
everywhere
Sometimes used in proofs before reasoning. 3331 is prime ∵ it has no positive integer factors other than itself and one. ■
□
∎
▮
‣
everywhere
Used to mark the end of a proof.
(May also be written Q.E.D.) D'Alembertian
non-Euclidean Laplacian
It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions. ⇒
→
⊃
implies;
if … then
propositional logic, Heyting algebra
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 functionsgiven below.)
(⊃ may mean the same as ⇒,[5]or it may have the meaning for supersetgiven below.) x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since xcould be −2). ⇔
↔
if and only if;
iff
A ⇔ B means A is true if B is true and A is false if Bis false. x + 5 = y+ 2 ⇔ x + 3 = y ¬
˜
not
The statement ¬A is true if and only if A is false.
A slash placed through another operator is the same as "¬" placed in front.
(The symbol ~ has many other uses, so ¬ or the slash notation is preferred. Computer scientists will often use! but this is avoided in mathematical texts.) ¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y) ∧
logical conjunction or meetin a lattice
and;
min;
meet
propositional logic, lattice theory
The statement A ∧ B is true if A and B are both true; else it is false.
For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)). n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number. wedge product
wedge product;
exterior product
u ∧ v means the wedge product of any multivectorsuand v. In three dimensional Euclidean space the wedge product and the cross product of two vectorsare each other's Hodge dual. exponentiation
… (raised) to the power of …
everywhere
a ^ b means a raised to the power of b
(a ^ b is 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 disjunction or joinin a lattice
or;
max;
join
propositional logic, lattice theory
The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.
For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)). n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. ⊕
⊻
xor
propositional logic, Boolean algebra
The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. (¬A) ⊕ A is always true, A ⊕ A is always false. direct sum
direct sum of
The direct sum is a special way of combining several objects into one general object.
(The bun symbol ⊕, or the coproductsymbol ∐, 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}) ∀
for all;
for any;
for each
∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n. ∃
there exists;
there is;
there are
∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even. ∃!
there exists exactly one
∃! x: P(x) means there is exactly one x such thatP(x) is true. ∃! n ∈ ℕ: n + 5 = 2n. =:
:=
≡
:⇔
≜
≝
≐
is defined as;
is equal by definition to
everywhere
x := y, y =: x or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.
(Some writers use ≡ to mean congruence).
P :⇔ Q means P is defined to be logically equivalentto Q. ≅
is congruent to
△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. isomorphic
is isomorphic to
G ≅ H means that group G is isomorphic (structurally identical) to group H.
(≈ can also be used for isomorphic, as described above.) . ≡
... is congruent to ... modulo ...
a ≡ b (mod n) means a − b is divisible by n 5 ≡ 2 (mod 3) { , }
setbrackets
the set of …
{a,b,c} means the set consisting of a, b, and c.[6]ℕ = { 1, 2, 3, …} { : }
{ | }
{ ; }
the set of … such that
{x : P(x)} means the set of all xfor which P(x) is true.[6]{x | P(x)} is the same as {x : P(x)}. {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4} ∅
{ }
the empty set
∅ means the set with no elements.[6]{ } means the same. {n ∈ ℕ : 1 < n2 < 4} = ∅ ∈
∉
is an element of;
is not an element of
everywhere, set theory
a ∈ S means a is an element of the set S;[6]a ∉ Smeans a is not an element of S.[6](1/2)−1 ∈ ℕ
2−1 ∉ ℕ ⊆
⊂
is a subset of
(subset) A ⊆ B means every element of A is also an element of B.[7]
(proper subset) A ⊂ B means A ⊆ B but A ≠ B.
(Some writers use the symbol ⊂ as if it were the same as ⊆.) (A ∩ B) ⊆ A
ℕ ⊂ ℚ
ℚ ⊂ ℝ ⊇
⊃
is a superset of
A ⊇ B means every element of B is also an element of A.
A ⊃ B means A ⊇ B but A ≠ B.
(Some writers use the symbol ⊃ as if it were the same as ⊇.) (A ∪ B) ⊇ B
ℝ ⊃ ℚ ∪
the union of … or …;
union
A ∪ B means the set of those elements which are either in A, or in B, or in both.[7]A ⊆ B ⇔ (A ∪ B) = B ∩
intersected with;
intersect
A ∩ B means the set that contains all those elements that A and B have in common.[7]{x ∈ ℝ : x2 = 1} ∩ ℕ = {1} ∆
symmetric difference
A ∆ B means the set of elements in exactly one of Aor B.
(Not to be confused with delta, Δ, described below.) {1,5,6,8} ∆ {2,5,8} = {1,2,6} ∖
minus;
without
A ∖ B means the set that contains all those elements of A that are not in B.[7]
(− can also be used for set-theoretic complement as described above.) {1,2,3,4} ∖ {3,4,5,6} = {1,2} →
functionarrow
from … to
f: X → Y means the function f maps the set X into the set Y. Let f: ℤ → ℕ∪{0} be defined by f(x) := x2. ↦
functionarrow
maps to
f: a ↦ b means the function f maps the element a to the element b. Let f: x ↦ x+1 (the successor function). ∘
composed with
f∘g is the function, such that (f∘g)(x) = f(g(x)).[8]if f(x) := 2x, and g(x) := x + 3, then (f∘g)(x) = 2(x + 3). o
entrywise product
For two matrices (or vectors) of the same dimensions the Hadamard product is a matrix of the same dimensions with elements given by . This is often used in matrix based programming such as MATLABwhere the operation is done by A.*B ℕ
N;
the (set of) natural numbers
N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.
The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter;analysts, set theoristsand computer scientists prefer the former. To avoid confusion, always check an author's definition of N.
Set theorists often use the notation ω (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation ≤. ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| > 0: a ∈ ℤ} ℤ
Z
Z;
the (set of) integers
ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}.
ℤ+ or ℤ> means {1, 2, 3, ...} . ℤ* or ℤ≥ means {0, 1, 2, 3, ...} .
ℤ = {p, −p : p ∈ ℕ ∪ {0}} ℤnZn;
the (set of) integers modulon
ℤn means {[0], [1], [2], ...[n−1]} with addition and multiplication modulo n.
Note that any letter may be used instead of n, such as p. To avoid confusion with p-adic numbers, useℤ/pℤ or ℤ/(p) instead. ℤ3 = {[0], [1], [2]} p-adic integers
the (set of) p-adic integers
Note that any letter may be used instead of p, such as n or l. ℙ
P;
the projective space;
the projective line;
the projective plane
ℙ means a space with a point at infinity. , probability
the probability of
ℙ(X) means the probability of the event Xoccurring.
This may also be written as P(X), Pr(X), P[X] or Pr[X]. If a fair coin is flipped, ℙ(Heads) = ℙ(Tails) = 0.5. ℚ
Q
Q;
the (set of) rational numbers;
the rationals
ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. 3.14000... ∈ ℚ
π ∉ ℚ ℝ
R
R;
the (set of) real numbers;
the reals
ℝ means the set of real numbers. π ∈ ℝ
√(−1) ∉ ℝ ℂ
C
C;
the (set of) complex numbers
ℂ means {a + b i : a,b ∈ ℝ}. i = √(−1) ∈ ℂ ℍ
H
quaternionsor Hamiltonian quaternions
H;
the (set of) quaternions
ℍ means {a + b i + c j + d k : a,b,c,d ∈ ℝ}. O
big-oh of
Computational complexity theory
The Big O notation describes the limiting behavior of a function, when the argument tends towards a particular value or infinity. If f(x) = 6x4 − 2x3 + 5 and g(x) = x4 , then ∞
infinity
∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. ⌊…⌋
floor;
greatest integer;
entier
⌊x⌋ means the floor of x, i.e. the largest integer less than or equal to x.
(This may also be written [x], floor(x) or int(x).) ⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3 ⌈…⌉
ceiling
⌈x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x.
(This may also be written ceil(x) orceiling(x).) ⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2 ⌊…⌉
nearest integer to
⌊x⌉ means the nearest integer to x.
(This may also be written [x], x, nint(x) orRound(x).) ⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊-3.4⌉ = -3, ⌊4.49⌉ = 4 [ : ]
the degree of
[K : F] means the degree of the extension K: F. [ℚ(√2) : ℚ] = 2
[ℂ : ℝ] = 2
[ℝ : ℚ] = ∞ [ ]
[ , ]
[ , , ]
the equivalence class of
[a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation.
[a]R means the same, but with R as the equivalence relation. Let a ~ b be true iffa ≡ b (mod 5).
Then [2] = {…, −8, −3, 2, 7, …}.
floorfloor;
greatest integer;
entier
[x] means the floor of x, i.e. the largest integer less than or equal to x.
(This may also be written ⌊x⌋, floor(x) or int(x). Not to be confused with the nearest integer function, as described below.) [3] = 3, [3.5] = 3, [3.99] = 3, [−3.7] = −4 nearest integer function
nearest integer to
[x] means the nearest integer to x.
(This may also be written ⌊x⌉, x, nint(x) orRound(x). Not to be confused with the floor function, as described above.) [2] = 2, [2.6] = 3, [-3.4] = -3, [4.49] = 4 Iverson bracket
1 if true, 0 otherwise
[S] maps a true statement S to 1 and a false statement S to 0. [0=5]=0, [7>0]=1, [2 ∈ {2,3,4}]=1, [5 ∈ {2,3,4}]=0 image
image of … under …
everywhere
f[X] means { f(x) : x ∈ X }, the image of the function funder the set X ⊆ dom(f).
(This may also be written as f(X) if there is no risk of confusing the image of f underX with the function application f ofX. Another notation is Im f, the image of f under its domain.) closed interval
closed interval
. 0 and 1/2 are in the interval [0,1]. commutator
the commutator of
[g, h] = g−1h−1gh (or ghg−1h−1), if g, h ∈ G (a group).
[a, b] = ab − ba, if a, b∈ R (a ring or commutative 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
[a, b, c] = a × b · c, the scalar product of a ×b withc. [a, b, c] = [b, c, a] = [c, a, b].
( , )
functionapplication
of
f(x) means the value of the function f at the elementx. If f(x) := x2, then f(3) = 32 = 9. image
image of … under …
everywhere
f(X) means { f(x) : x ∈ X }, the image of the function funder the set X ⊆ dom(f).
(This may also be written as f[X] if there is a risk of confusing the image of f underX with the function application f ofX. Another notation is Im f, the image of f under its domain.) combinations
(from) n choose r
means the number of combinations of relements drawn from a set of n elements.
(This may also be written as nCr.) precedence grouping
parentheses
everywhere
Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. tuple
tuple; n-tuple;
ordered pair/triple/etc;
row vector; sequence
everywhere
An ordered list (or sequence, or horizontal vector, or row vector) of values.
(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. Set theorists and computer scientists often use angle brackets⟨ ⟩ instead of parentheses.)
(a, b) is an ordered pair (or 2-tuple).(a, b, c) is an ordered triple (or 3-tuple).
( ) is the empty tuple(or 0-tuple).
highest common factorhighest common factor;
greatest common divisor; hcf; gcd
number theory
(a, b) means the highest common factor of aand b.
(This may also be written hcf(a, b) orgcd(a, b).) (3, 7) = 1 (they are coprime); (15, 25) = 5. ( , )
] , [
open interval
.
(Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation]a,b[ can be used instead.)
4 is not in the interval (4, 18).(0, +∞) equals the set of positive real numbers.
(( ))multichoose
means n multichoose k. ( , ]
] , ]
half-open interval;
left-open interval
. (−1, 7] and (−∞, −1] [ , )
[ , [
half-open interval;
right-open interval
. [4, 18) and [1, +∞) ⟨⟩
⟨,⟩
inner product of
⟨u,v⟩ means the inner product of u and v, where uand v are members of an inner product space.
Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the inner product or the linear span.
There are many variants of the notation, such as⟨u | v⟩ and (u | v), which are described below. For spatial vectors, the dot product notation, x·y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more "keyboard friendly" forms < and > are sometimes seen. These are avoided in mathematical texts. The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is:
⟨x, y⟩ = 2 × −1 + 3 × 5 = 13 average
average of
let S be a subset of N for example, represents the 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
⟨S⟩ means the span of S ⊆ V. That is, it is the intersection of all subspaces of V which contain S.
⟨u1, u2, …⟩is shorthand for ⟨{u1, u2, …}⟩.
Note that the notation ⟨u, v⟩ may be ambiguous: it could mean the inner productor the linear span.
The span of S may also be written asSp(S).
the subgroup generated by
means the smallest subgroup of G (where S⊆G, a group) containing every element of S.
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 the empty tuple(or 0-tuple).
⟨|⟩inner product of
⟨u | v⟩ means the inner product of u and v, where uand v are members of an inner product space.[9](u| v) means the same.
Another variant of the notation is ⟨u, v⟩ which is described above. For spatial vectors, the dot productnotation, x·y is common. For matrices, the colon notation A : B may be used. As ⟨ and ⟩ can be hard to type, the more "keyboard friendly" forms <and > are sometimes seen. These are avoided in mathematical texts. |⟩
the ket …;
the vector …
|φ⟩ means the vector with label φ, which is in aHilbert space. A qubit's state can be represented as α|0⟩+ β|1⟩, where αand β are complex numbers s.t. |α|2 + |β|2 = 1. ⟨|
the bra …;
the dual of …
⟨φ| means the dual of the vector |φ⟩, a linear functional which maps a ket |ψ⟩ onto the inner product ⟨φ|ψ⟩. ∑
sum over … from … to … of
means a1 + a2 + … + an. = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30 ∏
product over … from … to … of
means a1a2···an. = (1+2)(2+2)(3+2)(4+2) = 3 × 4 × 5 × 6 = 360 Cartesian product
the Cartesian product of;
the direct product of
means the set of all (n+1)-tuples(y0, …, yn). ∐
coproduct over … from … to … of
A general construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. Δ
delta;
change in
Δx means a (non-infinitesimal) change in x.
(If the change becomes infinitesimal, δ and evendare used instead. Not to be confused with the symmetric difference, written ∆, above.) is the gradient of a straight line Laplacian
Laplace operator
The Laplace operator is a second order differential operator in n-dimensional Euclidean space If ƒ is a twice-differentiablereal-valued function, then the Laplacian of ƒ is defined by δ
Dirac delta of
δ(x) Kronecker delta
Kronecker delta of
δij ∂
partial;
d
∂f/∂xi means the partial derivative of f with respect toxi, where f is a function on (x1, …, xn). If f(x,y) := x2y, then ∂f/∂x = 2xy boundary
boundary of
∂M means the boundary of M ∂{x : x ≤ 2} = {x : x = 2} degree of a polynomial
degree of
∂f means the degree of the polynomial f.
(This may also be written deg f.) ∂(x2 − 1) = 2 ∇
∇f (x1, …, xn) is the vector of partial derivatives (∂f /∂x1, …, ∂f / ∂xn). If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z) divergence
del dot;
divergence of
If , then . curl
curl of
If , then . ′
… prime;
derivative of
f ′(x) means the derivative of the function f at the pointx, i.e., the slope of the tangentto f at x.
(The single-quote character ' is sometimes used instead, especially in ASCII text.) If f(x) := x2, then f ′(x) = 2x •
… dot;
time derivative of
means the derivative of x with respect to time. That is . If x(t) := t2, then . ∫
indefinite integral orantiderivative
indefinite integral of
the antiderivative of
∫ f(x) dx means a function whose derivative is f. ∫x2 dx = x3/3 + C definite integral
integral from … to … of … with respect to
∫ab f(x) dx means the signed area between the x-axis and the graph of the functionf between x = a and x =b. ∫ab x2 dx = b3/3 − a3/3; line integral
line/ path/ curve/ integral of… along…
∫C f ds means the integral of falong the curve C, , where r is a parametrization of C.
(If the curve is closed, the symbol ∮ may be used instead, as described below.) ∮
Contour integral;
closed line integral
contour integral of
Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol∰.
The contour integral can also frequently be found with a subscript capital letter C, ∮C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capitalS, ∮S, is used to denote that the integration is over a closed surface.
If C is a Jordan curveabout 0, then . πProjection of
restricts to the attribute set. Pi
pi;
3.1415926;
≈22÷7
Used in various formulas involving circles; π is equivalent to the amount of area a circle would take up in a square of equal width with an area of 4 square units, roughly 3.14/4. It is also the ratio of thecircumferenceto the diameter of a circle. A=πR2=314.16→R=10 σ
Selection of
The selection selects all those tuples in for which holds between the and the attribute. The selection selects all those tuples in for which holds between the attribute and the value .
<:
<·
is covered by
x <• y means that x is covered by y. {1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment. subtype
is a subtype of
T1 <: T2 means that T1 is a subtype of T2. If S <: T and T <: Uthen S <: U (transitivity). †
conjugate transpose;
adjoint;
Hermitian adjoint/conjugate/transpose
A† means the transpose of the complex conjugate ofA.[10]
This may also be written A*T, AT*, A*, AT or AT. If A = (aij) then A† = (aji). T
transpose
AT means A, but with its rows swapped for columns.
This may also be written A', At or Atr. If A = (aij) then AT = (aji). ⊤
the top element
⊤ means the largest element of a lattice. ∀x : x ∨ ⊤ = ⊤ top type
the top type; top
⊤ means the top or universal type; every type in thetype system of interest is a subtype of top. ∀ types T, T <: ⊤ ⊥
is perpendicular to
x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y. If l ⊥ m and m ⊥ n in the plane, then l n. orthogonal complement
orthogonal/ perpendicular complement of;
perp
W⊥ means the orthogonal complement of W(whereW is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W. Within , . coprime
is coprime to
x ⊥ y means x has no factor greater than 1 in common with y. 34 ⊥ 55. independent
is independent of
A ⊥ B means A is an event whose probability is independent of event B. If A ⊥ B, then P(A|B)= P(A). bottom element
the bottom element
⊥ means the smallest element of a lattice. ∀x : x∧ ⊥ = ⊥ bottom type
the bottom type;
bot
⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system. ∀ types T, ⊥ <: T comparability
is comparable to
x ⊥ y means that x is comparable to y. {e, π} ⊥ {1, 2, e, 3, π} under set containment. ⊧
entails
A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. A ⊧ A ∨ ¬A ⊢
infers;
is derived from
propositional logic,predicate logic
x ⊢ y means y is derivable from x. A → B ⊢ ¬B → ¬A. partition
is a partition of
p ⊢ n means that p is a partition of n. (4,3,1,1) ⊢ 9, .
[edit]VariationsIn mathematics written in Arabic, some symbols may be reversed to make right-to-left writing and reading easier. [11]
[edit]See alsoSome Unicode charts of mathematical operators:
Some Unicode cross-references:
View page ratings
Rate this page
Trustworthy
Objective
Complete
Well-written
I am highly knowledgeable about this topic (optional)
Submit ratings
Interaction
Toolbox
Print/export
Languages
X2
what does these symbols mean on this picture
There are more than 700 common hieroglyphics, including three sets of phonetic symbols and hundreds of concept symbols.
There are more than 700 common hieroglyphics, including three sets of phonetic symbols and hundreds of concept symbols.
There are two sets of safety symbols because one of them are for house hold items, while the other one is for workplace items therefore the term used WHMIS.
ALT keyboard symbols or (character map) can be found with every version of Windows. Keyboard symbols are also called character sets.
Mathematics
The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.
Programs
No. Digits are the individual symbols. Consider the statement: "I am 48 years old."48, in this case, is a number.The individual symbols, 4 and 8 in this case, are digits.
the two different sets of symbols is the....... 1:scepter and mirror and 2:swan and dove i think the second one is correct though
The generic name for all symbols in Japanese is "Kana."There are three sets of kana all used together, and each set has a generic name:Hiragana (set of phonetic symbols for syllables, used mainly for Japanese grammar words)Katakana (set of phonetic symbols for syllables, used mainly for foreign words)Kanji (chinese characters)