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
| |||
|
∝ |
Proportionality | y ∝ x means that y = kx for some constant k. | if y = 2x, then y ∝ x |
| is proportional to | |||
| everywhere | |||
|
< > |
strict inequality | x < y means x is less than
y. x > y means x is greater than y. |
3 < 4 5 > 4 |
| is less than, is 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 | |||
|
+ |
addition | 4 + 6 means the sum of 4 and 6. | 2 + 7 = 9 |
| plus | |||
| arithmetic | |||
| disjoint union | A1 + A2 means the disjoint union of sets A1 and A2. | A1={1,2,3,4} ∧
A2={2,4,5,7} ⇒ A1 + A2 = {(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 ⇒ x2 = 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 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: n2 ≥ 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 : n2 < 20} = {0,1,2,3,4} |
| the set of ... such that ... | |||
| set theory | |||
∅ {} |
empty set | ∅ means the set with no elements. {} means the same. | {n ∈ N : 1 < n2 < 4} = ∅ |
| the 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 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 | 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 : x2 = 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) := 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 Y. | Let f: Z → N be defined by f(x) = x2. |
| 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. | limx→0 1/|x| = ∞ |
| infinity | |||
| numbers | |||
|
π |
pi | π means 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=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 | |||
|
' |
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) = x2, then f '(x) = 2x |
| … prime; derivative of … | |||
| calculus | |||
|
∫ |
indefinite integral or antiderivative | ∫ f(x) dx means a function whose derivative is f. | ∫x2 dx = x3/3 + C |
| indefinite integral of …; the antiderivative of … | |||
| calculus | |||
| definite integral | ∫ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. | ∫0b x2 dx = b3/3; | |
| integral from ... to ... of ... with respect to | |||
| calculus | |||
|
∇ |
gradient | ∇f (x1, …, 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 derivative | With f (x1, …, 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 | |||
| 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 equaly 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} ⊗ {1,1,1} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {1, 2, 3, 4}} |
| tensor product of | |||
| linear algebra |
