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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.
Basic mathematical symbols
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 also
External links
- Jeff Miller: Earliest Uses of Various Mathematical Symbols
- TCAEP - Institute of Physics
- GIF and PNG Images for Math Symbols
Special characters
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