This is an experimental version of Table of mathematical symbols. (Structure is based on the article Mathematische Symbole on the German Wikipedia.)
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Symbol ( TeX) |
Name | Explanation | Examples |
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Category | ||||
AT
Atr |
transpose
|
AT means A, but with its rows swapped for columns. | If A = (aij) then AT = (aji). | |
|A|
det(A) |
determinant of
|
|A| means the determinant of the matrix A | ||
W⊥
|
orthogonal/perpendicular complement of; perp
|
If W is a subspace of the inner product space V, then W⊥ is the set of all vectors in V orthogonal to every vector in W. | Within , . | |
V ⊕ W
|
direct sum of
|
The direct sum is a special way of combining several modules into one general module. | Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0}) | |
〈,〉
( | ) < , > · : |
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, x·y is common. |
The
standard inner product between two vectors x = (2, 3) and y = (−1, 5) is: 〈x, y〉 = 2 × −1 + 3 × 5 = 13
| |
〈 , 〉
< , > Sp |
(linear) span of; linear hull of
|
If u,v,w ∈ V then 〈u, v, w〉 means the span of u, v and w, as does Sp(u, v, w). That is, it is the intersection of all subspaces of V which contain u, v and w.
(Note that the notation 〈u, v〉 may be ambiguous: it could mean the inner product or the span.) |
. | |
⊗
|
tensor product of
|
means the tensor product of V and U. means the tensor product of modules V and U over the ring R. | {1, 2, 3, 4} ⊗ {1, 1, 2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} |
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|…|
|
absolute value (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 |
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·
|
dot
|
u · v means the dot product of vectors u and v | (1,2,5) · (3,4,−1) = 6 | |
×
|
cross
|
u × v means the cross product of vectors u and v | (1,2,5) × (3,4,−1) = (−22, 16, − 2) | |
∧
|
wedge product; exterior product
|
u ∧ v means the wedge product of
vectors u and v. This generalizes the cross product to higher dimensions. (For vectors in R3, × can also be used.) |
Symbol (HTML) |
Symbol ( TeX) |
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Category | ||||
|…|
# ♯ |
cardinality of; size of
|
|X| means the cardinality of the set X. | |{3, 5, 7, 9}| = 4. |
Symbol (HTML) |
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×
|
the Cartesian product of ... and ...; the direct product of ... and ...
|
X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. | {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)} | |
∏
|
the Cartesian product of; the direct product of
|
means the set of all
(n+1)-tuples
|
||
−
∖ |
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.) |
{1,2,4} − {1,3,4} = {2} | |
∪
|
the union of … or …; union
|
A ∪ B means the set of those elements which are either in A, or in B, or in both. | 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. | {x ∈ ℝ : x2 = 1} ∩ ℕ = {1} | |
∆
|
symmetric difference
|
A ∆ B means the set of elements in exactly one of A or B. | {1,5,6,8} ∆ {2,5,8} = {1,2,6} |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
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Category | ||||
∈
∉ |
set membership
is an element of; is not an element of
everywhere,
set theory
|
a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S. | (1/2)−1 ∈ ℕ 2−1 ∉ ℕ | |
⊆
⊂ |
is a subset of
|
(subset) A ⊆ B means every element of A is also an element of B. (proper subset) A ⊂ B means A ⊆ B but A ≠ B. (Some writers use the symbol ⊂ as if it were the same as ⊆.) |
(A ∩ B) ⊆ A ℕ ⊂ ℚ ℚ ⊂ ℝ | |
⊇
⊃ |
is a superset of
|
A ⊇ B means every element of B is also element of A. A ⊃ B means A ⊇ B but A ≠ B. (Some writers use the symbol ⊃ as if it were the same as ⊇.) |
(A ∪ B) ⊇ B ℝ ⊃ ℚ |
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ℕ
N |
N
|
N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention. | ℕ = {|a| : a ∈ ℤ, a ≠ 0} | |
ℤ
Z |
Z
|
ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ+ means {1, 2, 3, ...} = ℕ. | ℤ = {p, −p : p ∈ ℕ ∪ {0} | |
ℚ
Q |
Q
|
ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. | 3.14000... ∈ ℚ π ∉ ℚ | |
ℝ
R |
R
|
ℝ means the set of real numbers. | π ∈ ℝ √(−1) ∉ ℝ | |
ℂ
C |
C
|
ℂ means {a + b i : a,b ∈ ℝ}. | i = √(−1) ∈ ℂ | |
𝕂
K |
K
|
K means the statement holds substituting K for R and also for C. |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
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Category | ||||
|
|
divides
|
a|b means a divides b. | Since 15 = 3×5, it is true that 3|15 and 5|15.
| |
||
|
exact
divisibility
exactly divides
|
pa || n means pa exactly divides n (i.e. pa divides n but pa+1 does not). | 23 || 360. | |
⊥
|
is coprime to
|
x ⊥ y means x has no factor in common with y. | 34 ⊥ 55. |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
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Category | ||||
⌊…⌋
|
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) or ceiling(x).) |
⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2 |
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∴
|
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. | |
⇒
→ ⊃ |
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.) (⊃ may mean the same as ⇒, or it may have the meaning for superset given below.) |
x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since x could be −2). | |
⇔
↔ |
if and only if;
iff
|
A ⇔ B means A is true if B is true and A is false if B is false. | x + 5 = y +2 ⇔ x + 3 = y | |
¬
˜ |
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.) |
¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y) | |
∧
|
logical conjunction or meet in a
lattice
and; min; meet
|
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. | |
∨
|
logical disjunction or join in a
lattice
or; max; join
|
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
|
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. | |
∀
|
for all; for any; for each
|
∀ x: P(x) means P(x) is true for all x. | ∀ n ∈ ℕ: n2 ≥ n. | |
∃
|
there exists
|
∃ 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 that P(x) is true. | ∃! n ∈ ℕ: n + 5 = 2n. | |
⊧
|
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
|
x ⊢ y means y is derivable from x. | A → B ⊢ ¬B → ¬A. |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
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Category | ||||
=
|
is equal to; equals
everywhere
|
x = y means x and y represent the same thing or value. | 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 symbols != and <> are primarily from computer science. They are avoided in mathematical texts.) |
1 ≠ 2 | |
<
> ≪ ≫ |
is less than, is greater than, is much less than, is much greater than
|
x < y means x is less than y. x > y means x is greater than y. x ≪ y means x is much less than y. x ≫ y means x is much greater than y. |
3 < 4 5 > 4 0.003 ≪ 1000000 | |
≤
<= ≥ >= |
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 symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.) |
3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5 | |
∝
|
is proportional to; varies as
everywhere
|
y ∝ x means that y = kx for some constant k. | if y = 2x, then y ∝ x | |
+
|
4 + 6 means the sum of 4 and 6. | 2 + 7 = 9 | ||
the disjoint union of ... and ...
|
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)} | ||
−
|
9 − 4 means the subtraction of 4 from 9. | 8 − 3 = 5 | ||
negative; minus; the opposite of
|
−3 means the negative of the number 3. | −(−5) = 5 | ||
×
|
times
|
3 × 4 means the multiplication of 3 by 4. | 7 × 8 = 56 | |
·
|
times
|
3 · 4 means the multiplication of 3 by 4. | 7 · 8 = 56 | |
÷
⁄ |
divided by
|
6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. | 2 ÷ 4 = .5 12 ⁄ 4 = 3 | |
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}} | ||
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]} | ||
±
|
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 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). | |
√
|
the principal square root of; square root
|
means the positive number whose square is . | ||
the complex square root of …; square root
|
if is represented in polar coordinates with , then . | |||
|…|
|
Euclidean distance between; Euclidean norm of
|
|x – y| means the Euclidean distance between x and y. | For x = (1,1), and y = (4,5), |x – y| = √([1–4]2 + [1–5]2) = 5 | |
|
|
given
|
P(A|B) means the probability of the event a occurring given that b occurs. | If P(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4 | |
!
|
factorial
|
n! is the product 1 × 2 × ... × n. | 4! = 1 × 2 × 3 × 4 = 24 | |
~
|
has distribution
|
X ~ D, means the random variable X has the probability distribution D. | X ~ N(0,1), the standard normal distribution | |
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 n have the same
order 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 | ||
is asymptotically equivalent to
|
f ~ g means . | x ~ x+1 | ||
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. | π ≈ 3.14159 | |
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. | ||
◅
|
is a normal subgroup of
|
N ◅ G means that N is a normal subgroup of group G. | Z(G) ◅ G | |
is an ideal of
|
I ◅ R means that I is an ideal of ring R. | (2) ◅ Z | ||
:=
≡ :⇔ |
is defined as; equal by definition
everywhere
|
x := y 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 equivalent to Q. |
||
≅
|
is congruent to
|
△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. | ||
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 ≡ 11 (mod 3) | |
{ , }
|
set brackets
the set of …
|
{a,b,c} means the set consisting of a, b, and c. | ℕ = { 1, 2, 3, …} | |
{ : }
{ | } |
the set of … such that
|
{x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. | {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4} | |
∅
{ } |
the empty set
|
∅ means the set with no elements. { } means the same. | {n ∈ ℕ : 1 < n2 < 4} = ∅ | |
→
|
function arrow
from … to
|
f: X → Y means the function f maps the set X into the set Y. | Let f: ℤ → ℕ be defined by f(x) := x2. | |
↦
|
function arrow
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). | |
o
|
composed with
|
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). | |
∞
|
infinity
|
∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. | ||
[ ]
[ , ] [ , , ] |
the equivalence class of
|
a] is the equivalence class of a, i.e. {x : x ~ a}, where ~ is an
equivalence relation. aR is the same, but with R as the equivalence relation. |
Let a ~ b be true
iff a ≡ b (
mod 5).
Then [2] = {…, −8, −3, 2, 7, …}. | |
closed interval
|
. | [0,1] | ||
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 = xx, y] (group theory). AB, C] = AB, C] + [A, CB (ring theory). | ||
the triple scalar product of
|
a, b, c] = a × b · c, the scalar product of a × b with c. | a, b, c] = [b, c, a] = [c, a, b]. | ||
( )
( , ) |
function application
of
|
f(x) means the value of the function f at the element x. | If f(x) := x2, then f(3) = 32 = 9. | |
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; n-tuple; ordered pair/triple/etc; row vector
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.) |
(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; hcf
number theory
|
(a, b) means the highest common factor of a and b. (This may also be written hcf(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.) |
(4,18) | |
( , ]
] , ] |
half-open interval; left-open interval
|
. | (−1, 7] and (−∞, −1] | |
[ , )
[ , [ |
half-open interval; right-open interval
|
. | [4, 18) and [1, +∞) | |
∑
|
sum over … from … to … of
|
means a1 + a2 + … + an. | = 12 + 22 + 32 + 42
| |
∏
|
product over … from … to … of
|
means a1a2···an. | = (1+2)(2+2)(3+2)(4+2)
| |
∐
|
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. | ||
′
• |
… prime
derivative of |
f ′(x) is the derivative of the function f at the point x, i.e., the
slope of the
tangent to f at x. The dot notation indicates a time derivative. That is . |
If f(x) := x2, then f ′(x) = 2x | |
∫
|
indefinite integral of
the antiderivative of |
∫ f(x) dx means a function whose derivative is f. | ∫x2 dx = x3/3 + C | |
integral from … to … of … with respect to
|
∫ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. | ∫ab x2 dx = b3/3 − a3/3; | ||
∮
|
contour integral or 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 capital S, ∮S, is used to denote that the integration is over a closed surface. |
If C is a Jordan curve about 0, then . | |
∇
|
∇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) | ||
del dot, divergence of
|
If , then . | |||
curl of
|
If , then . | |||
∂
|
partial, d
|
With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant. | If f(x,y) := x2y, then ∂f/∂x = 2xy | |
boundary of
|
∂M means the boundary of M | ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2} | ||
degree of
|
∂f means the degree of the polynomial f. (This may also be written deg f.) |
∂(x2 − 1) = 2 | ||
δ
|
Dirac delta of
|
δ(x) | ||
Kronecker delta of
|
δij | |||
<:
<· |
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. | |
is a subtype of
|
T1 <: T2 means that T1 is a subtype of T2. | If S <: T and T <: U then S <: U ( transitivity). | ||
⊤
|
the top element
|
⊤ means the largest element of a lattice. | ∀x : x ∨ ⊤ = ⊤ | |
the top type; top
|
⊤ means the top or universal type; every type in the type 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. | |
the bottom element
|
⊥ means the smallest element of a lattice. | ∀x : x ∧ ⊥ = ⊥ | ||
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 | ||
is comparable to
|
x ⊥ y means that x is comparable to y. | {e, π} ⊥ {1, 2, e, 3, π} under set containment. | ||
||…||
|
norm of; length of
|
|| x || is the norm of the element x of a normed vector space. | || x + y || ≤ || x || + || y || | |
||
|
is parallel to
|
x || y means x is parallel to y. | If l || m and m ⊥ n then l ⊥ n. In physics this is also used to express . | |
is incomparable to
|
x || y means x is incomparable to y. | {1,2} || {2,3} under set containment. | ||
*
|
convolution, convolved with
|
f * g means the convolution of f and g. | . | |
conjugate
|
z* is the complex conjugate of z. ( can also be used for the conjugate of z, as described below.) |
. | ||
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× or U(R).) |
. | ||
x̄
|
overbar, … bar
|
(often read as "x bar") is the mean (average value of ). | . | |
conjugate
|
is the complex conjugate of z. (z* can also be used for the conjugate of z, as described above.) |
. |
This is an experimental version of Table of mathematical symbols. (Structure is based on the article Mathematische Symbole on the German Wikipedia.)
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
AT
Atr |
transpose
|
AT means A, but with its rows swapped for columns. | If A = (aij) then AT = (aji). | |
|A|
det(A) |
determinant of
|
|A| means the determinant of the matrix A | ||
W⊥
|
orthogonal/perpendicular complement of; perp
|
If W is a subspace of the inner product space V, then W⊥ is the set of all vectors in V orthogonal to every vector in W. | Within , . | |
V ⊕ W
|
direct sum of
|
The direct sum is a special way of combining several modules into one general module. | Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0}) | |
〈,〉
( | ) < , > · : |
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, x·y is common. |
The
standard inner product between two vectors x = (2, 3) and y = (−1, 5) is: 〈x, y〉 = 2 × −1 + 3 × 5 = 13
| |
〈 , 〉
< , > Sp |
(linear) span of; linear hull of
|
If u,v,w ∈ V then 〈u, v, w〉 means the span of u, v and w, as does Sp(u, v, w). That is, it is the intersection of all subspaces of V which contain u, v and w.
(Note that the notation 〈u, v〉 may be ambiguous: it could mean the inner product or the span.) |
. | |
⊗
|
tensor product of
|
means the tensor product of V and U. means the tensor product of modules V and U over the ring R. | {1, 2, 3, 4} ⊗ {1, 1, 2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
|…|
|
absolute value (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 |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
·
|
dot
|
u · v means the dot product of vectors u and v | (1,2,5) · (3,4,−1) = 6 | |
×
|
cross
|
u × v means the cross product of vectors u and v | (1,2,5) × (3,4,−1) = (−22, 16, − 2) | |
∧
|
wedge product; exterior product
|
u ∧ v means the wedge product of
vectors u and v. This generalizes the cross product to higher dimensions. (For vectors in R3, × can also be used.) |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
|…|
# ♯ |
cardinality of; size of
|
|X| means the cardinality of the set X. | |{3, 5, 7, 9}| = 4. |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
×
|
the Cartesian product of ... and ...; the direct product of ... and ...
|
X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. | {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)} | |
∏
|
the Cartesian product of; the direct product of
|
means the set of all
(n+1)-tuples
|
||
−
∖ |
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.) |
{1,2,4} − {1,3,4} = {2} | |
∪
|
the union of … or …; union
|
A ∪ B means the set of those elements which are either in A, or in B, or in both. | 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. | {x ∈ ℝ : x2 = 1} ∩ ℕ = {1} | |
∆
|
symmetric difference
|
A ∆ B means the set of elements in exactly one of A or B. | {1,5,6,8} ∆ {2,5,8} = {1,2,6} |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
∈
∉ |
set membership
is an element of; is not an element of
everywhere,
set theory
|
a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S. | (1/2)−1 ∈ ℕ 2−1 ∉ ℕ | |
⊆
⊂ |
is a subset of
|
(subset) A ⊆ B means every element of A is also an element of B. (proper subset) A ⊂ B means A ⊆ B but A ≠ B. (Some writers use the symbol ⊂ as if it were the same as ⊆.) |
(A ∩ B) ⊆ A ℕ ⊂ ℚ ℚ ⊂ ℝ | |
⊇
⊃ |
is a superset of
|
A ⊇ B means every element of B is also element of A. A ⊃ B means A ⊇ B but A ≠ B. (Some writers use the symbol ⊃ as if it were the same as ⊇.) |
(A ∪ B) ⊇ B ℝ ⊃ ℚ |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
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Read as | ||||
Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
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Read as | ||||
Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
ℕ
N |
N
|
N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention. | ℕ = {|a| : a ∈ ℤ, a ≠ 0} | |
ℤ
Z |
Z
|
ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ+ means {1, 2, 3, ...} = ℕ. | ℤ = {p, −p : p ∈ ℕ ∪ {0} | |
ℚ
Q |
Q
|
ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. | 3.14000... ∈ ℚ π ∉ ℚ | |
ℝ
R |
R
|
ℝ means the set of real numbers. | π ∈ ℝ √(−1) ∉ ℝ | |
ℂ
C |
C
|
ℂ means {a + b i : a,b ∈ ℝ}. | i = √(−1) ∈ ℂ | |
𝕂
K |
K
|
K means the statement holds substituting K for R and also for C. |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
|
|
divides
|
a|b means a divides b. | Since 15 = 3×5, it is true that 3|15 and 5|15.
| |
||
|
exact
divisibility
exactly divides
|
pa || n means pa exactly divides n (i.e. pa divides n but pa+1 does not). | 23 || 360. | |
⊥
|
is coprime to
|
x ⊥ y means x has no factor in common with y. | 34 ⊥ 55. |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
⌊…⌋
|
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) or ceiling(x).) |
⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2 |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
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Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
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Category |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
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Category | ||||
∴
|
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. | |
⇒
→ ⊃ |
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.) (⊃ may mean the same as ⇒, or it may have the meaning for superset given below.) |
x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since x could be −2). | |
⇔
↔ |
if and only if;
iff
|
A ⇔ B means A is true if B is true and A is false if B is false. | x + 5 = y +2 ⇔ x + 3 = y | |
¬
˜ |
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.) |
¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y) | |
∧
|
logical conjunction or meet in a
lattice
and; min; meet
|
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. | |
∨
|
logical disjunction or join in a
lattice
or; max; join
|
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
|
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. | |
∀
|
for all; for any; for each
|
∀ x: P(x) means P(x) is true for all x. | ∀ n ∈ ℕ: n2 ≥ n. | |
∃
|
there exists
|
∃ 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 that P(x) is true. | ∃! n ∈ ℕ: n + 5 = 2n. | |
⊧
|
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
|
x ⊢ y means y is derivable from x. | A → B ⊢ ¬B → ¬A. |
Symbol (HTML) |
Symbol ( TeX) |
Name | Explanation | Examples |
---|---|---|---|---|
Read as | ||||
Category | ||||
=
|
is equal to; equals
everywhere
|
x = y means x and y represent the same thing or value. | 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 symbols != and <> are primarily from computer science. They are avoided in mathematical texts.) |
1 ≠ 2 | |
<
> ≪ ≫ |
is less than, is greater than, is much less than, is much greater than
|
x < y means x is less than y. x > y means x is greater than y. x ≪ y means x is much less than y. x ≫ y means x is much greater than y. |
3 < 4 5 > 4 0.003 ≪ 1000000 | |
≤
<= ≥ >= |
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 symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.) |
3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5 | |
∝
|
is proportional to; varies as
everywhere
|
y ∝ x means that y = kx for some constant k. | if y = 2x, then y ∝ x | |
+
|
4 + 6 means the sum of 4 and 6. | 2 + 7 = 9 | ||
the disjoint union of ... and ...
|
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)} | ||
−
|
9 − 4 means the subtraction of 4 from 9. | 8 − 3 = 5 | ||
negative; minus; the opposite of
|
−3 means the negative of the number 3. | −(−5) = 5 | ||
×
|
times
|
3 × 4 means the multiplication of 3 by 4. | 7 × 8 = 56 | |
·
|
times
|
3 · 4 means the multiplication of 3 by 4. | 7 · 8 = 56 | |
÷
⁄ |
divided by
|
6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. | 2 ÷ 4 = .5 12 ⁄ 4 = 3 | |
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}} | ||
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]} | ||
±
|
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 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). | |
√
|
the principal square root of; square root
|
means the positive number whose square is . | ||
the complex square root of …; square root
|
if is represented in polar coordinates with , then . | |||
|…|
|
Euclidean distance between; Euclidean norm of
|
|x – y| means the Euclidean distance between x and y. | For x = (1,1), and y = (4,5), |x – y| = √([1–4]2 + [1–5]2) = 5 | |
|
|
given
|
P(A|B) means the probability of the event a occurring given that b occurs. | If P(A)=0.4 and P(B)=0.5, P(A|B)=((0.4)(0.5))/(0.5)=0.4 | |
!
|
factorial
|
n! is the product 1 × 2 × ... × n. | 4! = 1 × 2 × 3 × 4 = 24 | |
~
|
has distribution
|
X ~ D, means the random variable X has the probability distribution D. | X ~ N(0,1), the standard normal distribution | |
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 n have the same
order 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 | ||
is asymptotically equivalent to
|
f ~ g means . | x ~ x+1 | ||
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. | π ≈ 3.14159 | |
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. | ||
◅
|
is a normal subgroup of
|
N ◅ G means that N is a normal subgroup of group G. | Z(G) ◅ G | |
is an ideal of
|
I ◅ R means that I is an ideal of ring R. | (2) ◅ Z | ||
:=
≡ :⇔ |
is defined as; equal by definition
everywhere
|
x := y 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 equivalent to Q. |
||
≅
|
is congruent to
|
△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. | ||
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 ≡ 11 (mod 3) | |
{ , }
|
set brackets
the set of …
|
{a,b,c} means the set consisting of a, b, and c. | ℕ = { 1, 2, 3, …} | |
{ : }
{ | } |
the set of … such that
|
{x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. | {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4} | |
∅
{ } |
the empty set
|
∅ means the set with no elements. { } means the same. | {n ∈ ℕ : 1 < n2 < 4} = ∅ | |
→
|
function arrow
from … to
|
f: X → Y means the function f maps the set X into the set Y. | Let f: ℤ → ℕ be defined by f(x) := x2. | |
↦
|
function arrow
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). | |
o
|
composed with
|
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). | |
∞
|
infinity
|
∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. | ||
[ ]
[ , ] [ , , ] |
the equivalence class of
|
a] is the equivalence class of a, i.e. {x : x ~ a}, where ~ is an
equivalence relation. aR is the same, but with R as the equivalence relation. |
Let a ~ b be true
iff a ≡ b (
mod 5).
Then [2] = {…, −8, −3, 2, 7, …}. | |
closed interval
|
. | [0,1] | ||
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 = xx, y] (group theory). AB, C] = AB, C] + [A, CB (ring theory). | ||
the triple scalar product of
|
a, b, c] = a × b · c, the scalar product of a × b with c. | a, b, c] = [b, c, a] = [c, a, b]. | ||
( )
( , ) |
function application
of
|
f(x) means the value of the function f at the element x. | If f(x) := x2, then f(3) = 32 = 9. | |
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; n-tuple; ordered pair/triple/etc; row vector
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.) |
(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; hcf
number theory
|
(a, b) means the highest common factor of a and b. (This may also be written hcf(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.) |
(4,18) | |
( , ]
] , ] |
half-open interval; left-open interval
|
. | (−1, 7] and (−∞, −1] | |
[ , )
[ , [ |
half-open interval; right-open interval
|
. | [4, 18) and [1, +∞) | |
∑
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sum over … from … to … of
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means a1 + a2 + … + an. | = 12 + 22 + 32 + 42
| |
∏
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product over … from … to … of
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means a1a2···an. | = (1+2)(2+2)(3+2)(4+2)
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∐
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coproduct over … from … to … of
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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. | ||
′
• |
… prime
derivative of |
f ′(x) is the derivative of the function f at the point x, i.e., the
slope of the
tangent to f at x. The dot notation indicates a time derivative. That is . |
If f(x) := x2, then f ′(x) = 2x | |
∫
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indefinite integral of
the antiderivative of |
∫ f(x) dx means a function whose derivative is f. | ∫x2 dx = x3/3 + C | |
integral from … to … of … with respect to
|
∫ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. | ∫ab x2 dx = b3/3 − a3/3; | ||
∮
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contour integral or closed
line integral
contour integral of
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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 capital S, ∮S, is used to denote that the integration is over a closed surface. |
If C is a Jordan curve about 0, then . | |
∇
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∇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) | ||
del dot, divergence of
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If , then . | |||
curl of
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If , then . | |||
∂
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partial, d
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With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant. | If f(x,y) := x2y, then ∂f/∂x = 2xy | |
boundary of
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∂M means the boundary of M | ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2} | ||
degree of
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∂f means the degree of the polynomial f. (This may also be written deg f.) |
∂(x2 − 1) = 2 | ||
δ
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Dirac delta of
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δ(x) | ||
Kronecker delta of
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δij | |||
<:
<· |
is covered by
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x <• y means that x is covered by y. | {1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment. | |
is a subtype of
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T1 <: T2 means that T1 is a subtype of T2. | If S <: T and T <: U then S <: U ( transitivity). | ||
⊤
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the top element
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⊤ means the largest element of a lattice. | ∀x : x ∨ ⊤ = ⊤ | |
the top type; top
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⊤ means the top or universal type; every type in the type system of interest is a subtype of top. | ∀ types T, T <: ⊤ | ||
⊥
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is perpendicular to
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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. | |
the bottom element
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⊥ means the smallest element of a lattice. | ∀x : x ∧ ⊥ = ⊥ | ||
the bottom type; bot
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⊥ 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 | ||
is comparable to
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x ⊥ y means that x is comparable to y. | {e, π} ⊥ {1, 2, e, 3, π} under set containment. | ||
||…||
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norm of; length of
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|| x || is the norm of the element x of a normed vector space. | || x + y || ≤ || x || + || y || | |
||
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is parallel to
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x || y means x is parallel to y. | If l || m and m ⊥ n then l ⊥ n. In physics this is also used to express . | |
is incomparable to
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x || y means x is incomparable to y. | {1,2} || {2,3} under set containment. | ||
*
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convolution, convolved with
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f * g means the convolution of f and g. | . | |
conjugate
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z* is the complex conjugate of z. ( can also be used for the conjugate of z, as described below.) |
. | ||
the group of units of
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R* consists of the set of units of the ring R, along with the operation of multiplication. (This may also be written R× or U(R).) |
. | ||
x̄
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overbar, … bar
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(often read as "x bar") is the mean (average value of ). | . | |
conjugate
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is the complex conjugate of z. (z* can also be used for the conjugate of z, as described above.) |
. |