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# GetWiki:Symbols Table

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GetWiki:Symbols Table

In Mathematics, a set of symbols is frequently used in mathematical expressions. As mathematicians are familiar with these symbols, they are not explained each time they are used. So, for mathematical novices, the following table lists many common symbols together with their name, pronunciation and related field of Mathematics. Additionally, the third column contains an informal definition, and the fourth column gives a short example.

Be aware that, in some cases, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.

Note: The large table below has a mysterious origin, but it has often served as a complex testbed for GetWiki’s new table rendering code.

## Basic Mathematical Symbols

Symbol
Name
Explanation
Example
Category
=
equality xÂ = y means x and y represent the same thing or value. 1Â + 1Â = 2
is equal to; equals
everywhere
?
Inequation x ? y means that x and y do not represent the same thing or value. 1 ? 2
is not equal to; does not equal
everywhere
+
addition 4 + 6 means the sum of 4 and 6. 2 + 7 = 9
plus
arithmetic
?
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
arithmetic
set theoretic complement AÂ ?Â B means the set that contains all the elements of A that are not in B {1,2,3,4}Â ?Â {3,4,5,6}Â Â =Â Â {1,2}
minus; without
set_theory
Ã—
multiplication 3 Ã— 4 means the multiplication of 3 by 4. 7 Ã— 8 = 56
times
arithmetic
cartesian product XÃ—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} Ã— {3,4} = {(1,3),(1,4),(2,3),(2,4)}
the cartesian product of â€¦ and â€¦; the direct product of â€¦ and â€¦
set_theory
Ã·

/
division 6 Ã· 3 or 6/3 means the division of 6 by 3. 2 Ã· 4 = .5

12/4 = 3
divided by
arithmetic
?

?

?
material implication 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 = 2Â Â ?Â  x2 = 4 is true, but x2 = 4 Â Â ?Â  x = 2 is in general false (since x could be ?2)
implies; if .. then
propositional_logic
?

?
material equivalence AÂ ? B means A is true if B is true and A is false if B is false xÂ + 5Â = yÂ +2Â Â ?Â  xÂ + 3Â = y
if and only if; iff
propositional_logic
Â¬
logical negation the statement Â¬A is true if and only if A is false

a slash placed through another operator is the same as “Â¬” placed in front
Â¬(Â¬A)Â ? A
xÂ ?Â yÂ Â ?Â  Â¬(xÂ =Â  y)
not
propositional_logic
?
logical conjunction or meet in a lattice the statement A ? B is true if A and B are both true; else it is false nÂ < 4Â Â ?Â  nÂ >2Â Â ?Â  nÂ = 3 when n is a natural number
and
propositional_logic, lattice_theory
?
logical disjunction or join in a lattice the statement A ? B is true if A or B (or both) are true; if both are false, the statement is false nÂ ? 4Â Â ?Â  nÂ ? 2Â Â ? nÂ ? 3 when n is a natural number
or
propositional_logic, lattice_theory

?

?
exclusive or
A &o(lus; B
is true when either A or B are true, but not when both are true
(Â¬A)
&o(lus;
A is always true, A
&o(lus;
A is always false
xor
propositional_logic, boolean algebra
?
universal quantification ?Â x: P(x) means P(x) is true for all x ?Â nÂ ? N: n2Â ? n
for all; for any; for each
predicate logic
?
existential quantification ?Â x: P(x) means there is at least one x such that P(x) is true ?Â nÂ ? N: nÂ + 5Â = 2n
there exists
predicate logic
:=

?

:?
definition xÂ := y or xÂ ? y means x is defined to be another name for y (but note that ? can also mean other things, such as congruence)

PÂ :? Q means P is defined to be logically equivalent to Q
coshÂ xÂ := (1/2)(expÂ xÂ + expÂ (?x)); A XOR BÂ :? (AÂ ? B)Â ? Â¬(AÂ ? B)
is defined as
everywhere
{ , }
set brackets {a,b,c} means the set consisting of a, b, and c NÂ = {0,1,2,...}
the set of ...
set_theory
{ : }

{ | }
set_theory {xÂ : P(x)} means the set of all x for which P(x) is true. {xÂ | P(x)} is the same as {xÂ : P(x)}. {nÂ ? NÂ : n2Â <Â 20}Â = {0,1,2,3,4}
the set of ... such that ...
naive set

?

{}
empty set {} means the set with no elements; ? is the same thing {nÂ ? NÂ : 1Â < n2Â < 4}Â = {}
empty set
set_theory
?

?
set membership aÂ ? S means a is an element of the set S; aÂ ? S means a is not an element of S (1/2)?1Â ? N; 2?1Â ? N
is an element of; is not an element of
everywhere, set_theory
?

?
subset AÂ ? B means every element of A is also element of B
AÂ ? B means AÂ ? B but AÂ ? B
AÂ ? B ? A; QÂ ? R
is a subset of
set_theory
?

?
superset AÂ ? B means every element of B is also element of A
AÂ ? B means AÂ ? B but AÂ ? B
AÂ ? B ? B; RÂ ? Q
is a superset of
set_theory
?
set theoretic union AÂ ? B means the set that contains all the elements from A and also all those from B, but no others AÂ ? BÂ Â ?Â  AÂ ? BÂ = B
the union of ... and ...; union
set_theory
?
set theoretic intersection AÂ ? B means the set that contains all those elements that A and B have in common {xÂ ? RÂ : x2Â = 1}Â ? NÂ = {1}
intersected with; intersect
set_theory
set theoretic complement AÂ  B means the set that contains all those elements of A that are not in B {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)Â := x2, then f(3)Â = 32Â = 9
of
set_theory
precedence grouping perform the operations inside the parentheses first (8/4)/2Â = 2/2Â = 1, but 8/(4/2)Â = 8/2Â = 4
everywhere
f:X?Y
function arrow f:Â XÂ ? Y means the function f maps the set X into the set Y Consider the function f:Â ZÂ ? N defined by f(x)Â = x2
from ... to
functions

N

?
natural numbers N means {0,1,2,3,...}, but see the article on natural numbers for a different convention. { a>Â : aÂ ? Z}Â = N
N
numbers

Z

?
integers Z means {...,?3,?2,?1,0,1,2,3,...} {aÂ : a>Â ? N}Â = Z
Z
numbers

Q

?
rational numbers Q means {p/qÂ : p,qÂ ? Z, qÂ ? 0} 3.14Â ? Q; ?Â ? Q
Q
numbers

R

?
real numbers R means {limn??Â anÂ : ?Â nÂ ? N: anÂ ? Q, the limit exists} ?Â ? R; ?(?1)Â ? R
R
numbers

C

?
complex numbers C means {aÂ + biÂ : a,bÂ ? R} iÂ = ?(?1)Â ? C
C
numbers
<

>
strict_inequality xÂ < y means x is less than y; xÂ > y means x is greater than y xÂ < yÂ Â ?Â  yÂ > x
is less than, is greater than
partial orders
?

?
inequality xÂ ? y means x is less than or equal to y; xÂ ? y means x is greater than or equal to y xÂ ? 1Â Â ?Â  x2Â ? x
is less than or equal to, is greater than or equal to
partial orders
?
square root ?x means the positive number whose square is x ?(x2)Â = x>
the principal square root of; square root
real numbers
?
infinity ? is an element of the extended_number_line that is greater than all real numbers; it often occurs in limits limx?0Â 1/ x>Â = ?
infinity
numbers
?
pi ? means the ratio of a circle’s circumference to its diameter AÂ = ?rÂ² is the area of a circle with radius r
pi
Euclidean geometry
!
factorial n! is the product 1Ã—2Ã—...Ã—n 4! = 24
factorial
combinatorics
absolute value x> means the distance in the real line (or the complex plane) between x and zero aÂ + bi>Â = ?(a2Â + b2)
absolute value of
numbers
Â  norm x is the norm of the element x of a normed vector space x+y ? x + y
norm of; length of
functional analysis
?
summation ?k=1nÂ ak means a1Â + a2Â + ...Â + an ?k=14Â k2Â = 12Â + 22Â + 32Â + 42Â = 1Â + 4Â + 9Â + 16Â = 30
sum over ... from ... to ... of
arithmetic
?
product ?k=1nÂ ak means a1a2Â·Â·Â·an ?k=14Â (kÂ + 2)Â = (1Â  + 2)(2Â + 2)(3Â + 2)(4Â + 2)Â = 3Â Ã— 4Â Ã— 5Â Ã— 6Â = 360
product over ... from ... to ... of
arithmetic
cartesian product ?i=0nYi means the set of all (n+1)-tuples (y0,...,yn). ?n=13R = Rn
the cartesian product of; the direct product of
set_theory
?
integration ?abÂ f(x)Â dx means the signed area between the x-axis and the graph of the function f between xÂ = a and xÂ = b ?0bÂ x2 Â dxÂ = b3/3; ?x2Â dxÂ = x3/3
integral from ... to ... of ... with respect to
calculus
fÂ ’
derivative fÂ ’(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there If f(x) = x2, then fÂ ’(x) = 2x and fÂ (x’’) = 2
derivative of f; f prime
calculus
?
gradient ?f (x1, â€¦, xn) is the vector of partial derivatives (df / dx1, â€¦, df / dxn) If f (x,y,z) = 3xy + zÂ² then ?f = (3y, 3x, 2z)
calculus
?
partial With f (x1, â€¦, 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 derivative of
calculus
?
perpendicular x ? y means x is perpendicular to y; or more generally x is orthogonal to y.
is perpendicular to
orthogonality
bottom element x = ? means x is the smallest element.
the bottom element
lattice_theory
?
entailment
a models b
means the sentence a entails the sentence b. Formal definition:
a models b
if and only if, in every model in which a is true, b is also true.
entails
propositional_logic, predicate logic
?
inference x
vdash
y means y is derived from x.
infers or is derived from
propositional_logic, predicate logic

NOTE: If some of these symbols are used in an article intended for beginners, it may be a good idea to include a statement like the below, included with the {{msg:symbols}} code. This will help the article reach a broader audience:

Some content adapted from the Wikinfo article “Table of mathematical symbols” under the GNU Free Documentation License.
[ last updated: 4:48pm EDT - Sun, Jun 02 2024 ]
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