Table of mathematical symbols - Ies Jovellanos
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Table of mathematical symbols
From Wikipedia, the free encyclopedia For the HTML codes of mathematical symbols see mathematical HTML. Note: This article contains special characters.
The following table lists many specialized symbols commonly used in mathematics.
Basic mathematical symbols
Symbol
=
Name
Read as
Explanation
Category
equality
is equal to; equals
x = y means x and y represent the same thing or value.
everywhere
1 + 1 = 2
Examples
!=
inequation
x y means that x and y
do not represent the same thing or value.
is not equal to; does not equal
(The symbols != and 1 2
are primarily from
computer science. They
are avoided in
everywhere mathematical texts.)
<
strict inequality
x < y means x is less
than y.
>
is less than, is greater than, is much less than,
x > y means x is greater than y.
3 < 4 5 > 4.
is much greater than
x y means x is much less than y.
0.003 1000000
x y means x is much
greater than y. order theory
= +
-
?
equal to, is y.
greater than or
equal to
(The symbols = are primarily from computer science. They
3 4 and 5 5 5 4 and 5 5
order theory are avoided in
mathematical texts.)
proportionality
is proportional to; varies as
y x means that y = kx for some constant k.
if y = 2x, then y x
everywhere
addition
plus
4 + 6 means the sum of 4 and 6.
2 + 7 = 9
arithmetic
disjoint union
the disjoint A1 + A2 means the
A1 = {1, 2, 3, 4} A2 = {2, 4, 5, 7}
union of ... and ...
disjoint union of sets A1 A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2),
and A2.
(4,2), (5,2), (7,2)}
set theory
subtraction
minus
9 - 4 means the subtraction of 4 from 9.
8 - 3 = 5
arithmetic
negative sign
negative ; minus
-3 means the negative of the number 3.
arithmetic
-(-5) = 5
set-theoretic complement
A - B means the set that contains all the
minus; without elements of A that are
set theory not in B.
{1,2,4} - {1,3,4} = {2}
multiplication 3 ? 4 means the
times
multiplication of 3 by
arithmetic 4.
7 ? 8 = 56
Cartesian
product
X?Y means the set of
the Cartesian all ordered pairs with
product of ... and ...; the direct
the first element of each pair selected from X
{1,2} ? {3,4} = {(1,3),(1,4),(2,3),(2,4)}
product of ... and the second element
and ...
selected from Y.
set theory
cross product u ? v means the cross
cross
product of vectors u
vector algebra and v
(1,2,5) ? (3,4,-1) = (-22, 16, - 2)
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? ? / ?
|...|
multiplication 3 ? 4 means the
times
multiplication of 3 by
arithmetic 4.
dot product
u ? v means the dot
dot
product of vectors u
vector algebra and v
7 ? 8 = 56 (1,2,5) ? (3,4,-1) = 6
division
divided by
6 ? 3 or 6 / 3 means the 2 ? 4 = .5
division of 6 by 3.
12 / 4 = 3
arithmetic
plus-minus
plus or minus
6 ? 3 means both 6 + 3 The equation x = 5 ? 4, has two
and 6 - 3.
solutions, x = 7 and x = 3.
arithmetic
plus-minus plus or minus measurement
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.
minus-plus
6 ? (3 5) means both
minus or plus 6 + (3 - 5) and 6 - (3 + cos(x ? y) = cos(x) cos(y)
arithmetic 5).
sin(x) sin(y).
square root
the principal x means the positive square root of; number whose square is 4 = 2
square root x.
real numbers
complex square root
if z = r exp(i) is the complex represented in polar square root of ... coordinates with - < (-1) = i square root , then z = r exp
(i /2). complex numbers
absolute value or
modulus
|x| means the distance
along the real line (or
absolute value across the complex
(modulus) of plane) between x and
zero. numbers
|3| = 3 |?5| = |5| | i | = 1 | 3 + 4i | = 5
Euclidean distance
Euclidean distance |x ? y| means the
For x = (1,1), and y = (4,5),
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between; Euclidean distance Euclidean norm between x and y.
of
|x ? y| = ([1?4]2 + [1?5]2) = 5
Geometry
Determinant
|A| means the
determinant of determinant of the
Matrix theory matrix A
divides
A single vertical bar is
|
divides
used to denote divisibility.
Since 15 = 3?5, it is true that 3|15 and 5|15.
Number Theory a|b means a divides b.
factorial
!
factorial
n ! is the product 1 ? 2? ... ? n.
4! = 1 ? 2 ? 3 ? 4 = 24
combinatorics
transpose
T
transpose Swap rows for columns Aij = (AT)ji
matrix
operations
probability distribution
X ~ D, means the random variable X has X ~ N(0,1), the standard normal
has distribution the probability
distribution
~
statistics distribution D. Row equivalence A~B means that B can
is row equivalent be generated by using a
to
series of elementary
Matrix theory row operations on A
material
implication
A B means if A is
true then B is also true; if A is false then nothing is said about B.
implies; if ... may mean the same
then
as , or it may have the x = 2 x2 = 4 is true, but x2 = 4 x =
meaning for functions 2 is in general false (since x could be -2).
given below.
propositional may mean the same
logic, Heyting algebra
as , or it may have the
meaning for superset
given below.
material
equivalence
A B means A is true
if and only if; iff
if B is true and A is false if B is false.
x + 5 = y +2 x + 3 = y
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? ~
propositional logic
The statement ?A is logical negation true if and only if A is
false.
A slash placed through
not
another operator is the ?(?A) A
same as "?" placed in front.
x y ?(x = y)
propositional (The symbol ~ has logic many other uses, so ? or the slash notation is
preferred.)
logical
The statement A B is
conjunction or true if A and B are both meet in a lattice true; else it is false.
and; min
n < 4 n >2 n = 3 when n is a For functions A(x) and natural number.
propositional B(x), A(x) B(x) is
logic, lattice used to mean min(A(x), theory B(x)).
logical disjunction or join in a lattice
or; max
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
propositional B(x), A(x) B(x) is
logic, lattice
theory
used to mean max(A (x), B(x)).
n 4 n 2 n 3 when n is a natural number.
exclusive or
The statement A B is
xor
true when either A or
propositional B, but not both, are logic, Boolean true. A B means the
algebra same.
(?A) A is always true, A A is always false.
The direct sum is a
direct sum
special way of
Most commonly, for vector spaces U, V,
combining several one and W, the following consequence is
direct sum of
modules into one
used:
general module (the symbol is used,
is
U = V W (U = V + W) (V W =
Abstract algebra only for logic).
)
universal quantification
for all; for any;
x: P(x) means P(x) is
n
: n2 n.
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for each true for all x. predicate logic
existential
quantification x: P(x) means there is there exists at least one x such that n : n is even.
P(x) is true.
predicate logic
uniqueness
!
quantification ! x: P(x) means there there exists is exactly one x such ! n : n + 5 = 2n.
exactly one that P(x) is true.
predicate logic
:= definition
x := y or x y means x
is defined to be another name for y
is defined as (Some writers use to cosh x := (1/2)(exp x + exp (-x))
mean congruence).
A xor B : (A B) ?(A B)
:
{ , } { : } { | }
P : Q means P is everywhere defined to be logically
equivalent to Q.
congruence
ABC DEF means
triangle ABC is is congruent to congruent to (has the
same measurements as) geometry triangle DEF.
congruence relation
... is congruent a b (mod n) means a to ... modulo ... - b is divisible by n
modular arithmetic
5 11 (mod 3)
set brackets
{a,b,c} means the set
the set of ... consisting of a, b, and
set theory c.
= { 1, 2, 3, ...}
set builder
notation
{x : P(x)} means the set
the set of ... such that
of all x for which P(x) is true. {x | P(x)} is the
{n
same as {x : P(x)}.
set theory
: n2 < 20} = { 1, 2, 3, 4}
empty set the empty set
means the set with
no elements. { } means {n the same.
: 1 < n2 < 4} =
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{ }
set theory
set membership
is an element of; is not an element
a S means a is an element of the set S; a
(1/2)-1
of
S means a is not an
everywhere, set element of S.
2-1
theory
subset
(subset) A B means
every element of A is also element of B.
(A B) A
(proper subset) A B is a subset of means A B but A B.
(Some writers use the
set theory symbol as if it were
the same as .)
superset
A B means every
element of B is also element of A.
A B means A B but (A B) B is a superset of A B.
(Some writers use the set theory symbol as if it were
the same as .)
(exclusive) A B
set-theoretic union
means the set that contains all the
elements from A, or all
the elements from B,
but not both.
the union of ... "A or B, but not both." and
A B (A B) = B (inclusive)
(inclusive) A B means
union
the set that contains all the elements from A, or
all the elements from B,
or all the elements from
set theory both A and B. "A or B or both".
set-theoretic
intersection
A B means the set that contains all those
{x
: x2 = 1}
= {1}
intersected with; elements that A and B
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intersect have in common. set theory
symmetric
difference
AB means the set of
symmetric elements in exactly one {1,5,6,8} {2,5,8} = {1,2,6}
difference of A or B.
set theory
set-theoretic complement
A B means the set that contains all those
minus; without elements of A that are
set theory not in B.
{1,2,3,4}
{3,4,5,6} = {1,2}
function
application
f(x) means the value of
of
the function f at the element x.
If f(x) := x2, then f(3) = 32 = 9.
( )
set theory precedence
grouping
Perform the operations
parentheses
inside the parentheses (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. first.
everywhere
function arrow
f:XY
from ... to
f: X Y means the function f maps the set Let f: be defined by f(x) := x2.
set theory,type X into the set Y.
theory
function
o
composition
fog is the function, such if f(x) := 2x, and g(x) := x + 3, then (fog)
composed with that (fog)(x) = f(g(x)). (x) = 2(x + 3).
set theory
natural numbers N means { 1, 2, 3, ...},
N
but see the article on natural numbers for a
= {|a| : a , a 0}
N
different convention. numbers
integers
means {..., -3, -2,
Z
-1, 0, 1, 2, 3, ...} and + means {1, 2, 3,
= {p, -p : p } {0}
Z
numbers ...} = .
rational numbers
means {p/q : p , 3.14000...
Q
q }.
Q
numbers
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