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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
= |
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. | 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 and non-negative numbers>negative sign | ?3 means the negative of the number 3. | ?(?5) = 5 | |||||||||||
negative | |||||||||||||
arithmetic | |||||||||||||
A ? B means the set that contains all the elements of A that are not in B | {1,2,3,4} ? {3,4,5,6} = {1,2} | ||||||||||||
minus; without | |||||||||||||
set theory
| |||||||||||||
× |
|multiplication
3 × 4 means the multiplication of 3 by 4. | 7 × 8 = 56 | |||||||||||
times | |||||||||||||
arithmetic
| |||||||||||||
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
| |||||||||||||
÷ | division | 6 ÷ 3 or 6/3 means the division of 6 by 3. | 2 ÷ 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 = 2 ? x | 2 = 4 is true, but x2 = 4 ? x = 2 is in general false (since x could be ?2)||||||||||
implies; if .. then | |||||||||||||
propositional logic
| |||||||||||||
? | 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) ? 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 | n | < 4 ? n >2 ? n = 3 when n is a natural number||||||||||
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 | n ? 4 ? n ? 2 ? n ? 3 when | n is a natural number||||||||||
or | |||||||||||||
propositional calculus | , lattice (order)>lattice theory
| ||||||||||||
? | exclusive or | (¬A) | |||||||||||
xor | |||||||||||||
propositional logic, boolean algebra
| |||||||||||||
? |
universal quantification | ? x: P(x) means P(x) is true for all x | ? n ? N: n | 2 ? 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 + 5 = 2n | ||||||||||
there exists | |||||||||||||
predicate logic
| |||||||||||||
:= | definition | x | := y or x ? y means x is defined to be another name for y (but note that ? can also mean other things, such as congruence)cosh x := (1/2)(exp x + exp (?x)); A XOR B :? (A ? B) ? ¬(A ? B) | ||||||||||
is defined as | |||||||||||||
everywhere
| |||||||||||||
{ , } |
set brackets | { | a,b,c} means the set consisting of a, b, and cN = {0,1,2,...} | ||||||||||
the set of ... | |||||||||||||
set theory
| |||||||||||||
{ : } | set theory | {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 : n | 2 < 20} = {0,1,2,3,4}||||||||||
the set of ... such that ... | |||||||||||||
naive set
| |||||||||||||
? | empty set | {} means the set with no elements; ? is the same thing | {n ? N : 1 | < n2 < 4} = {}||||||||||
empty set | |||||||||||||
set theory
| |||||||||||||
? | set membership | 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 an element of; is not an element of | |||||||||||||
everywhere, set theory
| |||||||||||||
? | subset | A ? B means every element of A is also element of B | A ? B ? A; Q ? R | ||||||||||
is a subset of | |||||||||||||
set theory
| |||||||||||||
? | superset | A ? B means every element of B is also element of A | A ? B ? B; R ? Q | ||||||||||
is a superset of | |||||||||||||
set theory
| |||||||||||||
? |
set theoretic union | A ? B means the set that contains all the elements from A and also all those from B, but no others | A ? B ? A ? B = B | ||||||||||
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 : x | 2 = 1} ? N = {1}||||||||||
intersected with; intersect | |||||||||||||
set theory
| |||||||||||||
set theoretic complement | A 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 xIf | 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 | |||||||||||
everywhere
| |||||||||||||
f:X?Y |
function arrow | f | : X ? Y means the function f maps the set X into the set YConsider the function | f: Z ? N defined by f(x) = 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> : a ? Z} = N | ||||||||||
N | |||||||||||||
numbers
| |||||||||||||
Z | integers | Z means {...,?3,?2,?1,0,1,2,3,...} | {a : | a> ? N} = Z | |||||||||
Z | |||||||||||||
numbers
| |||||||||||||
Q | rational numbers | Q means {p/q : p,q ? Z, q ? 0} | 3.14 ? Q; ? ? Q | ||||||||||
Q | |||||||||||||
numbers
| |||||||||||||
R | real numbers | R means {lim | n?? an : ? n ? N: an ? Q, the limit exists}? ? R; ?(?1) ? R | ||||||||||
R | |||||||||||||
numbers
| |||||||||||||
C | complex numbers | C means {a + bi : a,b ? R} | i = ?(?1) ? C | ||||||||||
C | |||||||||||||
numbers
| |||||||||||||
< > |
strict inequality | x | < y means x is less than y; x > y means x is greater than yx | < y ? y > x||||||||||
is less than, is greater than | |||||||||||||
partial orders
| |||||||||||||
? | inequality | x | ? y means x is less than or equal to y; x ? y means x is greater than or equal to yx ? 1 ? x | 2 ? 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 | 2) =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?0 1/x> = ? | ||||||||
infinity | |||||||||||||
numbers
| |||||||||||||
? | pi | ? means the ratio of a circle's circumference to its diameter | A = ?r² is the area of a circle with radius r | ||||||||||
pi | |||||||||||||
Euclidean geometry
| |||||||||||||
! |
factorial | n! is the product 1×2×...×n | 4! = 24 | ||||||||||
factorial | |||||||||||||
combinatorics
| |||||||||||||
| absolute value
| | x | > means the distance in the real line (or the complex plane) between x and zero
| a + bi> = ?(a | 2 + b2)
| |||||||
absolute value of | |||||||||||||
numbers
| |||||||||||||
|
| 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=1n ak means a1 + a2 + ... + an? | k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30||||||||||
sum over ... from ... to ... of | |||||||||||||
arithmetic
| |||||||||||||
? |
product | ? | k=1n ak means a1a2···an? | k=14 (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 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 | ? | ab f(x) dx means the signed area between the x-axis and the graph (functions)of the function (mathematics)>function f between | x = a and x = b? | 0b x2 dx = b3/3; ?x2 dx = x3/3|||||||||
integral from ... to ... of ... with respect to | |||||||||||||
calculus
| |||||||||||||
f ' |
derivative | f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there | If f(x) = x | 2, then f '(x) = 2x and f (x'') = 2||||||||||
derivative of f; f prime | |||||||||||||
calculus
| |||||||||||||
? |
gradient | ?f (x | 1, , xn) is the vector of partial derivatives (df / dx1, , df / dxn)If f (x,y,z) = 3xy + z² then ?f
(3y, 3x, 2z) | ||||||||||
del, nabla, gradient of | |||||||||||||
calculus
| |||||||||||||
? |
partial | With f (x | 1, , xn), ?f/?xi is the derivative of f with respect to xi, with all other variables kept constant.If f(x,y) = x | 2y, 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 | |||||||||||||
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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