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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.
Basic Mathematical Symbols
bgcolor=#a0e0a0 ! rowspan="3" align=centerSymbol
!align=leftName
! rowspan="3" Explanation
! rowspan="3" Example
bgcolor=#a0e0a0
!align=centerShould be read as
bgcolor=#a0e0a0
!align=rightCategory
Â = y means x and y represent the same thing or value. addition subtraction
set theoretic complement multiplication cartesian product
/
12/4 = 3
?
?
? 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; ^{2} = 4 is true, but x^{2} = 4 Â Â ?Â x = 2 is in general false (since x could be ?2)
?
a slash placed through another operator is the same as "Â¬" placed in front
xÂ ?Â yÂ Â ?Â Â¬(xÂ =Â y) < 4Â Â ?Â nÂ >2Â Â ?Â nÂ = 3 when n is a natural numbern is a natural number
?
A &o(lus; B
is true when either A or B are true, but not when both are true
&o(lus;
A is always true, A &o(lus;
A is always false
^{2}Â ? n?
:?Â := 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 Qa,b,c} means the set consisting of a, b, and c
{  }^{2}Â <Â 20}Â = {0,1,2,3,4}
{}< n^{2}Â < 4}Â = {}
?^{?1}Â ? N; 2^{?1}Â ? N
?
AÂ ? B means AÂ ? B but AÂ ? B
?
AÂ ? B means AÂ ? B but AÂ ? B^{2}Â = 1}Â ? NÂ = {1} (x) means the value of the function f at the element x
f(x)Â := x^{2}, then f(3)Â = 3^{2}Â = 9 :Â XÂ ? Y means the function f maps the set X into the set Yf:Â ZÂ ? N defined by f(x)Â = x^{2}
?
?
?
?
_{n??}Â a_{n}Â : ?Â nÂ ? N:
a_{n}Â ? Q, the limit
exists}
?
< y means x is less than y; xÂ > y means x is greater than y< yÂ Â ?Â yÂ > x?Â ? y means x is less than or equal to y; xÂ ? y means x is greater than or equal to y^{2}Â ? x^{2})Â = _{x?0}Â 1/> means the distance in the real line (or the complex plane) between x and zero^{2}Â + b^{2})
A transparent image for text is: Image:Del.gif (). _{1}, â€¦, x_{n}), ?f/?x_{i} is the derivative of f with respect to x_{i}, with all other variables kept constant. ^{2}y, then ?f/?x = 2xy 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.
y means y is derived from x.
= 
equality  x  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
 
+ 
4 + 6 means the sum of 4 and 6.  2 + 7 = 9  
plus  
arithmetic
 
? 
9 ? 4 means the subtraction of 4 from 9.  8 ? 3 = 5  
minus  
arithmetic  
negative and nonnegative numbers>negative sign  ?3 means the negative of the number 3.  ?(?5) = 5  
negative  
arithmetic  
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— 
3 Ã— 4 means the multiplication of 3 by 4.  7 Ã— 8 = 56  
times  
arithmetic
 
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  
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.  x = 2Â Â ?Â x  
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  Â¬(Â¬A)Â ? A  
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  
and  
propositional calculus  , lattice (order)>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  
or  
propositional calculus  , lattice (order)>lattice theory
 
?  exclusive or  (Â¬A)  
xor  
propositional logic, boolean algebra
 
? 
universal quantification  ?Â x: P(x) means P(x) is true for all x  ?Â nÂ ? N: 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  x  coshÂ xÂ := (1/2)(expÂ xÂ + expÂ (?x)); A XOR BÂ :? (AÂ ? B)Â ? Â¬(AÂ ? B)  
is defined as  
everywhere
 
{ , } 
set brackets  {  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 : n  
the set of ... such that ...  
naive set
 
? 
empty set  {} means the set with no elements; ? is the same thing  {nÂ ? NÂ : 1Â  
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)  
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 ? A; QÂ ? R  
is a subset of  
set theory
 
? 
superset  AÂ ? B means every element of B is also element of A  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 : x  
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  If  
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  Consider the function  
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 {lim  ?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  xÂ  
is less than, is greater than  
partial orders
 
? 
inequality  x  xÂ ? 1Â Â ?Â 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  ?(x  x>  
the principal square root of; square root  
real numbers
 
? 
infinity  ? is an element of the extended real number line  that is greater than all real numbers; it often occurs in limit (mathematics)>limits  lim  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  aÂ + bi>Â = ?(a  
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  ?  ?  
sum over ... from ... to ... of  
arithmetic
 
? 
product  ?  ?  
product over ... from ... to ... of  
arithmetic
 
cartesian product  ?  ?  
the cartesian product of; the direct product of  
set theory
 
? 
integration  ?  of the function (mathematics)>function f between  ?  
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) = x  
derivative of f; f prime  
calculus
 
? 
gradient  ?f (x  If f (x,y,z) = 3xy + zÂ² then ?f
(3y, 3x, 2z)  
del, nabla, gradient of  
calculus
 
? 
partial  With f (x  If f(x,y) = x  
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  
entails  
propositional logic, predicate logic
 
inference  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
This article uses mathematical symbols.
External Links
 Official Code Chart
 Jeff Miller: ''Earliest Uses of Various Mathematical Symbols"
 TCAEP  Institute of Physics
Some content adapted from the Wikinfo article "Table of mathematical symbols" under the GNU Free Documentation License.
[ last updated: 6:18pm EDT  Sat, Aug 02 2008 ]
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