Table of mathematical symbols

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This is a listing of common symbols found within all branches of the science of mathematics.

Symbol	Name	Explanation	Examples
	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. (The symbols != and <> are primarily from computer science. They are avoided in mathematical texts.)	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 ≪ 1000000
	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. (The symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.)	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; varies as
	everywhere
+	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; the opposite of
	arithmetic
	set-theoretic complement	A − B means the set that contains all the elements of A that are not in B. (∖ can also be used for set-theoretic complement as described below.)	{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
·	multiplication	3 · 4 means the multiplication of 3 by 4.	7 · 8 = 56
	times
	arithmetic
	dot product	u · v means the dot product of vectors u and v	(1,2,5) · (3,4,−1) = 6
	dot
	vector algebra
÷ ⁄	division	6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.	2 ÷ 4 = .5 12 ⁄ 4 = 3
	divided by
	arithmetic
	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
	quotient set	A/~ means the set of all ~ equivalence classes in A.	If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then ℝ/~ = {x + n : n ∈ ℤ : x ∈ (0,1]}
	mod
	set theory
±	plus-minus	6 ± 3 means both 6 + 3 and 6 − 3.	The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
	plus or minus
	arithmetic
	plus-minus	10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.	If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
	plus or minus
	measurement
∓	minus-plus	6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5).	cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
	minus or plus
	arithmetic
√	square root	$\sqrt{x}$ means the positive number whose square is $x$ .	$\sqrt{4}=2$
	the principal square root of; square root
	real numbers
	complex square root	if $z=r\,\exp(i\phi)$ is represented in polar coordinates with $-\pi < \phi \le \pi$ , then $\sqrt{z} = \sqrt{r} \exp(i \phi/2)$ .	$\sqrt{-1}=i$
	the complex square root of …; square root
	complex numbers
\|…\|	absolute value or modulus	\|x\| means the distance along the real line (or across the complex plane) between x and zero.	\|3\| = 3 \|–5\| = \|5\| = 5 \| i \| = 1 \| 3 + 4i \| = 5
	absolute value (modulus) of
	numbers
	Euclidean distance	\|x – y\| means the Euclidean distance between x and y.	For x = (1,1), and y = (4,5), \|x – y\| = √([1–4]² + [1–5]²) = 5
	Euclidean distance between; Euclidean norm of
	geometry
	determinant	\|A\| means the determinant of the matrix A	$\begin{vmatrix} 1&2 \\ 2&4 \\ \end{vmatrix} = 0$
	determinant of
	matrix theory
	cardinality	\|X\| means the cardinality of the set X. (# or ♯ may be used instead as described below.)	\|{3, 5, 7, 9}\| = 4.
	cardinality of; size of
	set theory
# ♯	cardinality	#X means the cardinality of the set X. (\|…\| may be used instead as described above.)	#{4, 6, 8} = 3
	cardinality of; size of
	set theory
\|	divides	A single vertical bar is used to denote divisibility. a\|b means a divides b.	Since 15 = 3×5, it is true that 3\|15 and 5\|15.
	divides
	number theory
	conditional probability	A single vertical bar is used to describe the probability of an event given another event happening. P(A\|B) means a given b.	If P(A)=0.4 and P(B)=0.5, P(A\|B)=((0.4)(0.5))/(0.5)=0.4
	given
	probability
!	factorial	n! is the product 1 × 2 × ... × n.	4! = 1 × 2 × 3 × 4 = 24
	factorial
	combinatorics
^T ^tr	transpose	Swap rows for columns	If $A = (a i j)$ then $A T = (a j i)$ .
	transpose
	matrix operations
~	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
	row equivalence	A~B means that B can be generated by using a series of elementary row operations on A	$\begin{bmatrix} 1&2 \\ 2&4 \\ \end{bmatrix} \sim \begin{bmatrix} 1&2 \\ 0&0 \\ \end{bmatrix}$
	is row equivalent to
	matrix theory
	same order of magnitude	m ~ n means the quantities m and n have the same order of magnitude, or general size. (Note that ~ is used for an approximation that is poor, otherwise use ≈ .)	2 ~ 5 8 × 9 ~ 100 but π² ≈ 10
	roughly similar; poorly approximates
	approximation theory
	asymptotically equivalent	f ~ g means $\lim_{n\to\infty} \frac{f(n)}{g(n)} = 1$ .	x ~ x+1
	is asymptotically equivalent to
	asymptotic analysis
	equivalence relation	a ~ b means $b \in [a]$ (and equivalently $a \in [b]$ ).	1 ~ 5 mod 4
	are in the same equivalence class
	everywhere
≈	approximately equal	x ≈ y means x is approximately equal to y.	π ≈ 3.14159
	is approximately equal to
	everywhere
	isomorphism	G ≈ H means that group G is isomorphic (structurally identical) to group H. (≅ can also be used for isomorphic, as described below.)	Q / {1, −1} ≈ V, where Q is the quaternion group and V is the Klein four-group.
	is isomorphic to
	group theory
◅	normal subgroup	N ◅ G means that N is a normal subgroup of group G.	Z(G) ◅ G
	is a normal subgroup of
	group theory
	ideal	I ◅ R means that I is an ideal of ring R.	(2) ◅ Z
	is an ideal of
	ring theory
∴	therefore	Sometimes used in proofs before logical consequences.	All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
	therefore; so; hence
	everywhere
∵	because	Sometimes used in proofs before reasoning.	3331 is prime ∵ it has no positive integer factors other than itself and one.
	because; since
	everywhere
⇒ → ⊃	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, Heyting algebra
⇔ ↔	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. (The symbol ~ has many other uses, so ¬ or the slash notation is preferred.)	¬(¬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. For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)). (Old notation) u ∧ v means the cross product of vectors u and v.	n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.
	and; min
	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. For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).	n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number.
	or; max
	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
	direct sum	The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).	Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0})
	direct sum of
	abstract 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
:= ≡ :⇔ $\triangleq$ $\overset{\underset{\mathrm{def}}{}}{=}$	definition	x := y or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context. (Some writers use ≡ to mean congruence). P :⇔ Q means P is defined to be logically equivalent to Q.	$\cosh x := \frac{e^x + e^{-x}}{2}$
	is defined as; equal by definition
	everywhere
≅	congruence	△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
	is congruent to
	geometry
	isomorphic	G ≅ H means that group G is isomorphic (structurally identical) to group H. (≈ can also be used for isomorphic, as described above.)	$\mathbb{R}^2 \cong \mathbb{C}$ .
	is isomorphic to
	abstract algebra
≡	congruence relation	a ≡ b (mod n) means a − b is divisible by n	5 ≡ 11 (mod 3)
	... is congruent to ... modulo ...
	modular arithmetic
{ , }	set brackets	{a,b,c} means the set consisting of a, b, and c.	ℕ = { 1, 2, 3, …}
	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} = { 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	(subset) A ⊆ B means every element of A is also an element of B. (proper subset) A ⊂ B means A ⊆ B but A ≠ B. (Some writers use the symbol ⊂ as if it were the same as ⊆.)	(A ∩ B) ⊆ A ℕ ⊂ ℚ 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. (Some writers use the symbol ⊃ as if it were the same as ⊇.)	(A ∪ B) ⊇ B ℝ ⊃ ℚ
	is a superset of
	set theory
∪	set-theoretic union	A ∪ B means the set of those elements which are either in A, or in B, or in both.	A ⊆ B ⇔ (A ∪ B) = B
	the union of … or …; 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
∆	symmetric difference	A ∆ B means the set of elements in exactly one of A or B.	{1,5,6,8} ∆ {2,5,8} = {1,2,6}
	symmetric difference
	set theory
∖	set-theoretic complement	A ∖ B means the set that contains all those elements of A that are not in B. (− can also be used for set-theoretic complement as described above.)	{1,2,3,4} ∖ {3,4,5,6} = {1,2}
	minus; without
	set theory
→	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, type theory
↦	function arrow	f: a ↦ b means the function f maps the element a to the element b.	Let f: x ↦ x+1 (the successor function).
	maps 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 { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.	ℕ = {\|a\| : a ∈ ℤ, a ≠ 0}
	N
	numbers
ℤ Z	integers	ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ₊ means {1, 2, 3, ...} = ℕ.	ℤ = {p, −p : p ∈ ℕ ∪ {0}
	Z
	numbers
ℚ Q	rational numbers	ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.	3.14000... ∈ 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 + b i : a,b ∈ ℝ}.	i = √(−1) ∈ ℂ
	C
	numbers
	arbitrary constant	C can be any number, most likely unknown; usually occurs when calculating antiderivatives.	if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C, where F'(x) = f(x)
	C
	integral calculus
𝕂 K	real or complex numbers	K means the statement holds substituting K for R and also for C.	$\begin{array}{l} \quad x^2\in\mathbb{C}\,\forall x\in \mathbb{K} \\ \text{because} \\ \quad x^2\in\mathbb{C}\,\forall x\in \mathbb{R} \\ \text{and} \\ \quad x^2\in\mathbb{C}\,\forall x\in \mathbb{C}. \end{array}$
	K
	linear algebra
∞	infinity	∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.	$\lim_{x\to 0} \frac{1}{\|x\|} = \infty$
	infinity
	numbers
[…]	equivalence class	a is the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation. a_R is the same, but with R as the equivalence relation.	Let a ~ b be true iff a ≡ b (mod 5). Then [2] = {…, −8, −3, 2, 7, …}.
	the equivalence class of
	abstract algebra
[ , ]	closed interval	$[a,b] = \{x \in \mathbb{R} : a \le x \le b \}$ .	[0,1]
	closed interval
	order theory
	commutator	If g, h ∈ G (a group), then g, h = g⁻¹h⁻¹gh (or ghg⁻¹h⁻¹). If a, b ∈ R (a ring or commutative algebra), then a, b = ab − ba.	x^y = xx, y (group theory). AB, C = AB, C + A, CB (ring theory).
	the commutator of
	group theory, ring 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.
	parentheses
	everywhere
	tuple	An ordered list (or sequence, or horizontal vector, or row vector) of values. (Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval.)	(a, b) is an ordered pair (or 2-tuple). (a, b, c) is an ordered triple (or 3-tuple). ( ) is the empty tuple (or 0-tuple).
	tuple; n-tuple; ordered pair/triple/etc; row vector
	everywhere
( , ) ] ,	open interval	$(a,b) = \{x \in \mathbb{R} : a < x < b \}$ . (Note that the notation (a,b) is ambiguous: it could be an ordered pair or an open interval. The notation ]a,b[ can be used instead.)	(4,18)
	open interval
	order theory
( , ] ] , ]	left-open interval	$(a,b] = \{x \in \mathbb{R} : a < x \le b \}$ .	(−1, 7] and (−∞, −1]
	half-open interval; left-open interval
	order theory
[ , ) [ ,	right-open interval	$[a,b) = \{x \in \mathbb{R} : a \le x < b \}$ .	[4, 18) and [1, +∞)
	half-open interval; right-open interval
	order theory
\|\|…\|\|	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	$\sum_{k=1}^{n}{a_k}$ means a₁ + a₂ + … + a_n.	$\sum_{k=1}^{4}{k^2}$ = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
	sum over … from … to … of
	arithmetic
∏	product	$\prod_{k=1}^na_k$ means a₁a₂···a_n.	$\prod_{k=1}^4(k+2)$ = (1+2)(2+2)(3+2)(4+2) = 3 × 4 × 5 × 6 = 360
	product over … from … to … of
	arithmetic
	Cartesian product	$\prod_{i=0}^{n}{Y_i}$ means the set of all (n+1)-tuples (y₀, …, y_n).	$\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3$
	the Cartesian product of; the direct product of
	set theory
∐	coproduct	A general construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism.
	coproduct over … from … to … of
	category 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. The dot notation indicates a time derivative. That is $\dot{x}(t)=\frac{\partial}{\partial t}x(t)$ .	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	∫_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.	∫_a^b x² dx = b³/3 − a³/3;
	integral from … to … of … with respect to
	calculus
∮	contour integral or closed line integral	Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∰. The contour integral can also frequently be found with a subscript capital letter C, ∮_C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮_S, is used to denote that the integration is over a closed surface.	If C is a Jordan curve about 0, then $\oint_C {1 \over z}\,dz = 2\pi i$ .
	contour integral of
	calculus
∇	gradient	∇f (x₁, …, x_n) is the vector of partial derivatives (∂f / ∂x₁, …, ∂f / ∂x_n).	If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
	del, nabla, gradient of
	vector calculus
	divergence	$\nabla \cdot \vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z}$	If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$ , then $\nabla \cdot \vec v = 3y + 2yz$ .
	del dot, divergence of
	vector calculus
	curl	$\nabla \times \vec v = \left( {\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \mathbf{i}$ $+ \left( {\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \mathbf{j} + \left( {\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \mathbf{k}$	If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$ , then $\nabla\times\vec v = -y^2\mathbf{i} - 3x\mathbf{k}$ .
	curl of
	vector calculus
∂	partial derivative	With f (x₁, …, 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) := x²y, then ∂f/∂x = 2xy
	partial, d
	calculus
	boundary	∂M means the boundary of M	∂{x : \|\|x\|\| ≤ 2} = {x : \|\|x\|\| = 2}
	boundary of
	topology
	degree of a polynomial	∂f means the degree of the polynomial f. This may also be written deg f.	∂(x² − 1) = 2
	degree of
	algebra
δ	Dirac delta function	$\delta(x) = \begin{cases} \infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$	δ(x)
	Dirac delta of
	hyperfunction
	Kronecker delta	$\delta_{ij} = \begin{cases} 1, & i = j \\ 0, & i \ne j \end{cases}$	δ_ij
	Kronecker delta of
	hyperfunction
<: <·	cover	x <• y means that x is covered by y.	{1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment.
	is covered by
	order theory
	subtype	T₁ <: T₂ means that T₁ is a subtype of T₂.	If S <: T and T <: U then S <: U (transitivity).
	is a subtype of
	type theory
⊤	top element	x = ⊤ means x is the largest element.	∀x : x ∨ ⊤ = ⊤
	the top element
	lattice theory
	top type	The top or universal type; every type in the type system of interest is a subtype of top.	∀ types T, T <: ⊤
	the top type; top
	type theory
⊥	perpendicular	x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.	If l ⊥ m and m ⊥ n in the plane then l \|\| n.
	is perpendicular to
	geometry
	orthogonal complement	If W is a subspace of the inner product space V, then W^⊥ is the set of all vectors in V orthogonal to every vector in W.	Within $\mathbb{R}^3$ , $(\mathbb{R}^2)^{\perp} \cong \mathbb{R}$ .
	orthogonal/perpendicular complement of; perp
	linear algebra
	coprime	x ⊥ y means x has no factor in common with y.	34 ⊥ 55.
	is coprime to
	number theory
	bottom element	x = ⊥ means x is the smallest element.	∀x : x ∧ ⊥ = ⊥
	the bottom element
	lattice theory
	bottom type	The bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system.	∀ types T, ⊥ <: T
	the bottom type; bot
	type theory
	comparability	x ⊥ y means that x is comparable to y.	{e, π} ⊥ {1, 2, e, 3, π} under set containment.
	is comparable to
	order theory
\|\|	parallel	x \|\| y means x is parallel to y.	If l \|\| m and m ⊥ n then l ⊥ n. In physics this is also used to express $x \\| y \Leftrightarrow \frac{1}{x^{-1} + y^{-1}}$ .
	is parallel to
	geometry, physics
	incomparability	x \|\| y means x is incomparable to y.	{1,2} \|\| {2,3} under set containment.
	is incomparable to
	order theory
	exact divisibility	p^a \|\| n means p^a exactly divides n (i.e. p^a divides n but p^a+1 does not).	2³ \|\| 360.
	exactly divides
	number theory
⊧	entailment	A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true.	A ⊧ A ∨ ¬A
	entails
	model theory
⊢	inference	x ⊢ y means y is derivable from x.	A → B ⊢ ¬B → ¬A.
	infers; is derived from
	propositional logic, predicate logic
〈,〉 ( \| ) < , > · :	inner product	〈x,y〉 means the inner product of x and y as defined in an inner product space. For spatial vectors, the dot product notation, x·y is common. For matrices, the colon notation may be used. (Note that the notation 〈x, y〉 may be ambiguous: it could mean the inner product or the linear span.)	The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is: 〈x, y〉 = 2 × −1 + 3 × 5 = 13 $A:B = \sum_{i,j} A_{ij}B_{ij}\!\,$
	inner product of
	linear algebra
〈 , 〉 < , > Sp	linear span	If u,v,w ∈ V then 〈u, v, w〉 means the span of u, v and w, as does Sp(u, v, w). That is, it is the intersection of all subspaces of V which contain u, v and w. (Note that the notation 〈u, v〉 may be ambiguous: it could mean the inner product or the span.)	$\left\lang \left( \begin{smallmatrix} 1 \\ 0 \\ 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} 0 \\ 1 \\ 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} 0 \\ 0 \\ 1 \end{smallmatrix} \right) \right\rang = \mathbb{R}^3$ .
	(linear) span of; linear hull of
	linear algebra
⊗	tensor product, tensor product of modules	$V \otimes U$ means the tensor product of V and U. $V \otimes_R U$ means the tensor product of modules V and U over the ring R.	{1, 2, 3, 4} ⊗ {1, 1, 2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
	tensor product of
	linear algebra
*	convolution	f * g means the convolution of f and g.	$(f * g)(t) = \int f(\tau) g(t - \tau)\, d\tau$ .
	convolution, convolved with
	functional analysis
	complex conjugate	z* is the complex conjugate of z. ( $\bar{z}$ can also be used for the conjugate of z, as described below.)	$(3+4i)^\ast = 3-4i$ .
	conjugate
	complex numbers
	group of units	R* consists of the set of units of the ring R, along with the operation of multiplication. (This may also be written R^× or U(R).)	$(\mathbb{Z} / 5\mathbb{Z})^\ast = \{ [1], [2], [3], [4] \}$ .
	the group of units of
	ring theory
$\bar{x}$ x̄	mean	$\bar{x}$ (often read as "x bar") is the mean (average value of $x i$ ).	$x = \{1,2,3,4,5\}; \bar{x} = 3$ .
	overbar, … bar
	statistics
	complex conjugate	$\overline{z}$ is the complex conjugate of z. (z* can also be used for the conjugate of z, as described above.)	$\overline{3+4i} = 3-4i$ .
	conjugate
	complex numbers

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