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Πίνακας Μαθηματικών Συμβόλων

The following table lists many specialized symbols commonly used in mathematics.

## Basic mathematical symbolsEdit

Σύμβολο Name Εξήγηση Παραδείγματα
Category
=
equality x = y means x and y represent the same thing or value. 1 + 1 = 2
is equal to; equals
everywhere

<>

!=
inequation 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 not equal to; does not equal
everywhere
<

>

strict inequality 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, is greater than, is much less than, is much greater than
order theory

<=

>=
inequality 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 less than or equal to, is greater than or equal to
order theory
proportionality yx means that y = kx for some constant k. if y = 2x, then yx
is proportional to; varies as
everywhere
+
addition 4 + 6 means the sum of 4 and 6. 2 + 7 = 9
plus
arithmetic
disjoint union A1 + A2 means the disjoint union of sets A1 and A2. A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒
A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
the disjoint union of ... and ...
set theory
subtraction 9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5
minus
arithmetic
negative sign −3 means the negative of the number 3. −(−5) = 5
negative; minus
arithmetic
set-theoretic complement A − B means the set that contains all the elements of A that are not in B.

∖ can also be used for set-theoretic complement as described below.
{1,2,4} − {1,3,4}  =  {2}
minus; without
set theory
×
multiplication 3 × 4 means the multiplication of 3 by 4. 7 × 8 = 56
times
arithmetic
Cartesian product X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
the Cartesian product of ... and ...; the direct product of ... and ...
set theory
cross product u × v means the cross product of vectors u and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
cross
vector algebra
·
multiplication 3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56
times
arithmetic
dot product u · v means the dot product of vectors u and v (1,2,5) · (3,4,−1) = 6
dot
vector algebra
÷

division 6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = .5

12 ⁄ 4 = 3
divided by
arithmetic
±
plus-minus 6 ± 3 means both 6 + 3 and 6 - 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
plus or minus
arithmetic
plus-minus 10 ± 2 or eqivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a is ≥ 99 mm and ≤ 101 mm.
plus or minus
measurement
minus-plus 6 ± (3 ∓ 5) means both 6 + (3 - 5) and 6 - (3 + 5). cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
minus or plus
arithmetic
square root x means the positive number whose square is x. √4 = 2
the principal square root of; square root
real numbers
complex square root if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(i φ/2). √(-1) = i
the complex square root of …

square root
complex numbers
|…|
absolute value or modulus |x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3

|–5| = |5|

i | = 1

| 3 + 4i | = 5
absolute value (modulus) of
numbers
Euclidean distance |x – y| means the Euclidean distance between x and y. For x = (1,1), and y = (4,5),
|x – y| = √([1–4]2 + [1–5]2) = 5
Euclidean distance between; Euclidean norm of
Geometry
Determinant |A| means the determinant of the matrix A $\begin{vmatrix} 1&2 \\ 2&4 \\ \end{vmatrix} = 0$
determinant of
Matrix theory
|
divides A single vertical bar is used to denote divisibility.
a|b means a divides b.
Since 15 = 3×5, it is true that 3|15 and 5|15.
divides
Number Theory
!
factorial n ! is the product 1 × 2× ... × n. 4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
T
transpose Swap rows for columns $A_{ij} = (A^T)_{ji}$
transpose
matrix operations
~
probability distribution X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
has distribution
statistics
Row equivalence A~B means that B can be generated by using a series of elementary row operations on A $\begin{bmatrix} 1&2 \\ 2&4 \\ \end{bmatrix} \sim \begin{bmatrix} 1&2 \\ 0&0 \\ \end{bmatrix}$
is row equivalent to
Matrix theory

material implication AB means if A is true then B is also true; if A is false then nothing is said about B.

→ may mean the same as ⇒, or it may have the meaning for functions given below.

⊃ may mean the same as ⇒, or it may have the meaning for superset given below.
x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2).
implies; if … then
propositional logic, Heyting algebra

material equivalence A ⇔ B means A is true if B is true and A is false if B is false. x + 5 = y +2  ⇔  x + 3 = y
if and only if; iff
propositional logic
¬

˜
logical negation The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.

(The symbol ~ has many other uses, so ¬ or the slash notation is preferred.)
¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y)
not
propositional logic
logical conjunction or meet in a lattice The statement AB is true if A and B are both true; else it is false.

For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).
n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.
and; min
propositional logic, lattice theory
logical disjunction or join in a lattice The statement AB is true if A or B (or both) are true; if both are false, the statement is false.

For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).
n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
or; max
propositional logic, lattice theory

exclusive or The statement AB is true when either A or B, but not both, are true. AB means the same. A) ⊕ A is always true, AA is always false.
xor
propositional logic, Boolean algebra
direct sum The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).

Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = VW ⇔ (U = V + W) ∧ (VW = )
direct sum of
Abstract algebra
universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n.
for all; for any; for each
predicate logic
existential quantification ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even.
there exists
predicate logic
∃!
uniqueness quantification ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ ℕ: n + 5 = 2n.
there exists exactly one
predicate logic
:=

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

(Some writers useto mean congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.
cosh x := (1/2)(exp x + exp (−x))

A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
is defined as
everywhere
congruence △ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
is congruent to
geometry
congruence relation a ≡ b (mod n) means a − b is divisible by n 5 ≡ 11 (mod 3)
... is congruent to ... modulo ...
modular arithmetic
{ , }
set brackets {a,b,c} means the set consisting of a, b, and c. ℕ = { 1, 2, 3, …}
the set of …
set theory
{ : }

{ | }
set builder notation {x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. {n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4}
the set of … such that
set theory

{ }
empty set means the set with no elements. { } means the same. {n ∈ ℕ : 1 < n2 < 4} =
the empty set
set theory
set membership a ∈ S means a is an element of the set S; a Πρότυπο:Notin S means a is not an element of S. (1/2)−1 ∈ ℕ

2−1 Πρότυπο:Notin
is an element of; is not an element of
everywhere, set theory

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

(proper subset) A ⊂ B means A ⊆ B but A ≠ B.

(Some writers use the symbol ⊂ as if it were the same as ⊆.)
(A ∩ B) ⊆ A

ℕ ⊂ ℚ

ℚ ⊂ ℝ
is a subset of
set theory

superset A ⊇ B means every element of B is also element of A.

A ⊃ B means A ⊇ B but A ≠ B.

(Some writers use the symbol ⊃ as if it were the same as ⊇.)
(A ∪ B) ⊇ B

ℝ ⊃ ℚ
is a superset of
set theory
set-theoretic union (exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both.
"A or B, but not both."

(inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B.
"A or B or both".
A ⊆ B  ⇔  (A ∪ B) = B (inclusive)
the union of … and …

union
set theory
set-theoretic intersection A ∩ B means the set that contains all those elements that A and B have in common. {x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
intersected with; intersect
set theory
$\Delta$
symmetric difference $A\Delta B$ means the set of elements in exactly one of A or B. {1,5,6,8} $\Delta$ {2,5,8} = {1,2,6}
symmetric difference
set theory
set-theoretic complement A ∖ B means the set that contains all those elements of A that are not in B.

− can also be used for set-theoretic complement as described above.
{1,2,3,4} ∖ {3,4,5,6} = {1,2}
minus; without
set theory
( )
function application f(x) means the value of the function f at the element x. If f(x) := x2, then f(3) = 32 = 9.
of
set theory
precedence grouping Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
parentheses
everywhere
f:XY
function arrow fX → Y means the function f maps the set X into the set Y. Let f: ℤ → ℕ be defined by f(x) := x2.
from … to
set theory,type theory
o
function composition fog is the function, such that (fog)(x) = f(g(x)). if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).
composed with
set theory

N
natural numbers N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention. ℕ = {|a| : a ∈ ℤ, a ≠ 0}
N
numbers

Z
integers ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ+ means {1, 2, 3, ...} = ℕ. ℤ = {p, -p : p ∈ ℕ} ∪ {0}
Z
numbers

Q
rational numbers ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. 3.14000... ∈ ℚ

π ∉ ℚ
Q
numbers

R
real numbers ℝ means the set of real numbers. π ∈ ℝ

√(−1) ∉ ℝ
R
numbers

C
complex numbers ℂ means {a + b i : a,b ∈ ℝ}. i = √(−1) ∈ ℂ
C
numbers
arbitrary constant C can be any number, most likely unknown; usually occurs when calculating antiderivatives. if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C, where F'(x) = f(x)
C
integral calculus
𝕂

K
real or complex numbers K means the statement holds substituting K for R and also for C.
$x^2\in\mathbb{C}\,\forall x\in \mathbb{K}$

because

$x^2\in\mathbb{C}\,\forall x\in \mathbb{R}$

and

$x^2\in\mathbb{C}\,\forall x\in \mathbb{C}$.
K
linear algebra
infinity ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. limx→0 1/|x| = ∞
infinity
numbers
||…||
norm || x || is the norm of the element x of a normed vector space. || x  + y || ≤  || x ||  +  || y ||
norm of

length of
linear algebra
summation

$\sum_{k=1}^{n}{a_k}$ means a1 + a2 + … + an.

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

= 1 + 4 + 9 + 16 = 30
sum over … from … to … of
arithmetic
product

$\prod_{k=1}^na_k$ means a1a2···an.

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

= 3 × 4 × 5 × 6 = 360
product over … from … to … of
arithmetic
Cartesian product

$\prod_{i=0}^{n}{Y_i}$ means the set of all (n+1)-tuples

(y0, …, yn).

$\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3$

the Cartesian product of; the direct product of
set theory
coproduct
coproduct over … from … to … of
category theory

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

The dot notation indicates a time derivative. That is $\dot{x}(t)=\frac{\partial}{\partial t}x(t)$.

If f(x) := x2, then f ′(x) = 2x
… prime

derivative of
calculus
indefinite integral or antiderivative ∫ f(x) dx means a function whose derivative is f. x2 dx = x3/3 + C
indefinite integral of

the antiderivative of
calculus
definite integral 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. 0b x2  dx = b3/3;
integral from … to … of … with respect to
calculus
contour integral or closed line integral Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while this formula involves a closed surface integral, the representation describes only the first or initial integration of the volume, over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∰.

This symbol can also frequently be found with a subscript capital letter C, ∮C, denoting that the closed loop integral is around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮S, is used to denote that the integration is over a closed surface.

contour integral of
calculus
gradient 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, nabla, gradient of
vector calculus
divergence $\nabla \cdot \vec v$ If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$, then $\nabla \cdot \vec v = 3y + 2yz$.
del dot, divergence of
vector calculus
curl $\nabla \times \vec v$ If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$, then $\nabla\times\vec v = -y^2\mathbf{i} - 3x\mathbf{k}$.
curl of
vector calculus
partial differential 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
partial, d
calculus
boundary M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
boundary of
topology
perpendicular xy means x is perpendicular to y; or more generally x is orthogonal to y. If lm and mn then l || n.
is perpendicular to
geometry
bottom element x = ⊥ means x is the smallest element. x : x ∧ ⊥ = ⊥
the bottom element
lattice theory
||
parallel x || y means x is parallel to y. If l || m and mn then ln.
is parallel to
geometry
entailment AB means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. AA ∨ ¬A
entails
model theory
inference xy means y is derived from x. AB ⊢ ¬B → ¬A
infers or is derived from
propositional logic, predicate logic
normal subgroup NG means that N is a normal subgroup of group G. Z(G) ◅ G
is a normal subgroup of
group theory
/
quotient group G/H means the quotient of group G modulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
mod
group theory
quotient set A/~ means the set of all ~ equivalence classes in A. If we define ~ by x~y ⇔ x-y∈Z, then
R/~ = {{x+n : nZ} : x ∈ (0,1]}
mod
set theory
isomorphism GH means that group G is isomorphic to group H Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group.
is isomorphic to
group theory
approximately equal xy means x is approximately equal to y π ≈ 3.14159
is approximately equal to
everywhere
~
same order of magnitude m ~ n, means the quantities m and n have the general size.

(Note that ~ is used for an approximation that is poor, otherwise use ≈ .)
2 ~ 5

8 × 9 ~ 100

but π2 ≈ 10
roughly similar

poorly approximates
Approximation theory

〈,〉

( | )

< , >

·

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

For spatial vectors, the dot product notation, x·y is common.
For matricies, the colon notation may be used.

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

$A:B = \sum_{i,j} A_{ij}B_{ij}$

inner product of
linear algebra
tensor product VU means the tensor product of V and U. {1, 2, 3, 4} ⊗ {1,1,2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
tensor product of
linear algebra
*
convolution f * g means the convolution of f and g. $(f * g )(t) = \int f(\tau) g(t - \tau)\, d\tau$
convolution, convoluted with
functional analysis
$\bar{x}$
mean $\bar{x}$ (often read as "x bar") is the mean (average value of $x_i$). $x = \{1,2,3,4,5\}; \bar{x} = 3$.
overbar, … bar
statistics
$\overline{z}$
complex conjugate $\overline{z}$ is the complex conjugate of z. $\overline{3+4i} = 3-4i$
conjugate
complex numbers
$\triangleq$
delta equal to $\triangleq$ means equal by definition. When $\triangleq$ is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡. $p(x_1,x_2,...,x_n) \triangleq \prod_{i=1}^n p(x_i | x_{\pi_i})$.
equal by definition
everywhere