Table of mathematical symbols
this wiki
Assessment |
Biopsychology |
Comparative |
Cognitive |
Developmental |
Language |
Individual differences |
Personality |
Philosophy |
Social |
Methods |
Statistics |
Clinical |
Educational |
Industrial |
Professional items |
World psychology |
Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory
The following table lists many specialized symbols commonly used in mathematics. For the HTML codes of mathematical symbols see mathematical HTML.
- Note: This article contains special characters.
Contents
[show]Basic mathematical symbolsEdit
Symbol
| Name | Explanation | Examples |
---|---|---|---|
Should be read as | |||
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. | 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 ≪1,000,000 |
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. | 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 | y ∝ x means that y = kx for some constant k. | if y = 2x, then y ∝ x |
is proportional to | |||
everywhere | |||
+
| addition | 4 + 6 means the sum of 4 and 6. | 2 + 7 = 9 |
plus | |||
arithmetic | |||
disjoint union | A_{1} + A_{2} means the disjoint union of sets A_{1} and A_{2}. | A_{1}={1,2,3,4} ∧ A_{2}={2,4,5,7} ⇒ A_{1} + A_{2} = {(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. | {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 | |||
÷
/ | division | 6 ÷ 3 or 6/3 means the division of 6 by 3. | 2 ÷ 4 = .5 12/4 = 3 |
divided by | |||
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(iφ/2). | √(-1) = i | |
the complex square root of; square root | |||
complex numbers | |||
| |
| absolute value | |x| means the distance in the real line (or the complex plane) between x and zero. | |3| = 3, |-5| = |5| |i| = 1, |3+4i| = 5 |
absolute value of | |||
numbers | |||
!
| factorial | n! is the product 1×2×...×n. | 4! = 1 × 2 × 3 × 4 = 24 |
factorial | |||
combinatorics | |||
~
| 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 | |||
⇒
→ ⊃ | 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 = 2 ⇒ x^{2} = 4 is true, but x^{2} = 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 slash placed through another operator is the same as "¬" placed in front. | ¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = 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. | n < 4 ∧ n >2 ⇔ n = 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. | n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. |
or | |||
propositional logic, lattice theory | |||
⊕ ⊻ | exclusive or | The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. | (¬A) ⊕ A is always true, A ⊕ A is always false. |
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 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 (but note that ≡ can also mean other things, such as 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 | |||
{ , }
| set brackets | {a,b,c} means the set consisting of a, b, and c. | N = {0,1,2,...} |
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 : n^{2} < 20} = {0,1,2,3,4} |
the set of ... such that ... | |||
set theory | |||
empty set | Template:0/ means the set with no elements. {} means the same. | {n ∈ N : 1 < n^{2} < 4} = Template:0/ | |
the empty set | |||
set theory | |||
set membership | a ∈ S means a is an element of the set S; a Template:Notin S means a is not an element of S. | (1/2)^{−1} ∈ N 2^{−1} Template:Notin N | |
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. | 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 means A ⊇ B but A ≠ B. | 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 : 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 x. | If f(x) := x^{2}, then f(3) = 3^{2} = 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 Y. | Let f: Z → N be defined by f(x) := x^{2}. |
from ... to | |||
set 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 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 ℤ | integers | Z means {...,−3,−2,−1,0,1,2,3,...}. | {a : |a| ∈ N} = Z |
Z | |||
numbers | |||
Q ℚ | rational numbers | Q means {p/q : p,q ∈ Z, q ≠ 0}. | 3.14 ∈ Q π ∉ Q |
Q | |||
numbers | |||
R ℝ | real numbers | R means the set of real numbers. | π ∈ R √(−1) ∉ R |
R | |||
numbers | |||
C ℂ | complex numbers | C means {a + bi : a,b ∈ R}. | i = √(−1) ∈ C |
C | |||
numbers | |||
∞
| infinity | ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. | lim_{x→0} 1/|x| = ∞ |
infinity | |||
numbers | |||
pi | π is the ratio of a circle's circumference to its diameter. Its value is 3.1415.... | A = πr² is the area of a circle with radius r | |
pi | |||
Euclidean geometry | |||
|| ||
| 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 | ∑_{k=1}^{n} a_{k} means a_{1} + a_{2} + ... + a_{n}. | ∑_{k=1}^{4} k^{2} = 1^{2} + 2^{2} + 3^{2} + 4^{2} = 1 + 4 + 9 + 16 = 30 |
sum over ... from ... to ... of | |||
arithmetic | |||
∏
| product | ∏_{k=1}^{n} a_{k} means a_{1}a_{2}···a_{n}. | ∏_{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 | ∏_{i=0}^{n}Y_{i} means the set of all (n+1)-tuples (y_{0},...,y_{n}). | ∏_{n=1}^{3}R = R^{n} | |
the Cartesian product of; the direct product of | |||
set 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. | If f(x) := x^{2}, then f '(x) = 2x |
… prime; derivative of … | |||
calculus | |||
∫
| indefinite integral or antiderivative | ∫ f(x) dx means a function whose derivative is f. | ∫x^{2} dx = x^{3}/3 + C |
indefinite integral of …; the antiderivative of … | |||
calculus | |||
definite integral | ∫_{a}^{b} f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. | ∫_{0}^{b} x^{2 } dx = b^{3}/3; | |
integral from ... to ... of ... with respect to | |||
calculus | |||
∇
| gradient | ∇f (x_{1}, …, x_{n}) is the vector of partial derivatives (df / dx_{1}, …, df / dx_{n}). | If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z) |
del, nabla, gradient of | |||
calculus | |||
∂
| partial derivative | With f (x_{1}, …, x_{n}), ∂f/∂x_{i} is the derivative of f with respect to x_{i}, with all other variables kept constant. | If f(x,y) := x^{2}y, then ∂f/∂x = 2xy |
partial derivative of | |||
calculus | |||
boundary | ∂M means the boundary of M | ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2} | |
boundary of | |||
topology | |||
⊥
| perpendicular | x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y. | If l⊥m and m⊥n then l || n. |
is perpendicular to | |||
geometry | |||
bottom element | x = ⊥ means x is the smallest element. | ∀x : x ∧ ⊥ = ⊥ | |
the bottom element | |||
lattice theory | |||
⊧
| entailment | A ⊧ B means the sentence A entails the sentence B, that is every model in which A is true, B is also true. | A ⊧ A ∨ ¬A |
entails | |||
model theory | |||
⊢
| inference | x ⊢ y means y is derived from x. | A → B ⊢ ¬B → ¬A |
infers or is derived from | |||
propositional logic, predicate logic | |||
◅
| normal subgroup | N ◅ G 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 | |||
≈
| isomorphism | G ≈ H 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 | x ≈ y means x is approximately equal to y | π ≈ 3.14159 | |
is approximately equal to | |||
everywhere
| |||
⊗
| tensor product | V ⊗ U 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 |
See alsoEdit
External linksEdit
- Jeff Miller: Earliest Uses of Various Mathematical Symbols
- TCAEP - Institute of Physics
- GIF and PNG Images for Math Symbols
Special charactersEdit
Technical note: Due to technical limitations, many computers cannot display some of the special characters in this article. Such characters may be rendered as boxes, question marks, or other nonsense symbols, depending on your browser, operating system, and installed fonts. Even if you have ensured that your browser is interpreting the article as UTF-8 encoded and you have installed a font that supports a wide range of Unicode, such as Code2000, Arial Unicode MS, Lucida Sans Unicode or one of the free Unicode fonts, you may still need to use a different browser, as browser capabilities in this regard tend to vary.de:Wikipedia:Tabelle mathematischer Symbole es:Tabla de símbolos matemáticos fr:Table des symboles mathématiques id:Daftar simbol matematika nl:Lijst van wiskundige symbolenpt:Tabela de símbolos matemáticos ru:Таблица математических символов su:Tabel lambang matematis sv:Tabell över matematiska symboler zh:数学符号表
This page uses Creative Commons Licensed content from Wikipedia (view authors). |