User:Alksentrs/Sandbox

Relations

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
<·	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
∝	proportionality	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x
	is proportional to; varies as
	everywhere
|	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
~	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 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
◅	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
:= ≡ :⇔	definition	x := y or x ≡ y means x is defined to be another name for y (Some writers use ≡ to mean congruence). P :⇔ Q means P is defined to be logically equivalent to Q.	cosh x := (1/2)(exp(x)+exp(-x))
	is defined as
	everywhere
≅	congruence	△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
	is congruent to
	geometry
≡	congruence relation	a ≡ b (mod n) means a − b is divisible by n	5 ≡ 11 (mod 3)
	... is congruent to ... modulo ...
	modular arithmetic
∈ ∉	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 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
||	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	pf || n means pf exactly divides n.	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 or is derived from
	propositional logic, predicate logic
$\triangleq$	delta equal to	$\triangleq$ means equal by definition. When $\triangleq$ is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡.	$p(x_{1},x_{2},...,x_{n})\triangleq \prod _{i=1}^{n}p(x_{i}|x_{\pi _{i}})$.
	equal by definition
	everywhere

Nullary functions

Symbol	Name	Explanation	Examples
	Read as
	Category
∅ { } ||empty set	∅ means the set with no elements. { } means the same.	{n ∈  ℕ : 1 < n2 < 4} = ∅
			the empty set
			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.	$x^{2}\in \mathbb {C} \,\forall x\in \mathbb {K}$ because $x^{2}\in \mathbb {C} \,\forall x\in \mathbb {R}$ and $x^{2}\in \mathbb {C} \,\forall x\in \mathbb {C}$.
	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

Binary functions

Symbol	Name	Explanation	Examples
	Read as
	Category
+	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
×	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
∪	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 … 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
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
〈,〉 ( | ) < , > · :	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.	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
⊗	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, convoluted with
	functional analysis

Unary functions

Symbol	Name	Explanation	Examples
	Read as
	Category
±	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 $-\pi <\phi \leq \pi$, 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.	|{3, 5, 7, 9}| = 4.
	cardinality of
	Set theory
!	factorial	n! is the product 1 × 2 × ... × n.	4! = 1 × 2 × 3 × 4 = 24
	factorial
	combinatorics
T	transpose	Swap rows for columns	$A_{ij}=(A^{T})_{ji}$
	transpose
	matrix operations
||…||	norm	|| x || is the norm of the element x of a normed vector space.	|| x  + y || ≤  || x ||  +  || y ||
	norm of length of
	linear algebra
${\bar {x}}$ 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
${\overline {z}}$ $z^{\ast }$	complex conjugate	${\overline {z}}=z^{\ast }$ is the complex conjugate of z.	${\overline {3+4i}}=(3+4i)^{\ast }=3-4i$
	conjugate
	complex numbers

The rest

Symbol	Name	Explanation	Examples
	Read as
	Category
−	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
⇒ → ⊃	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
{ , }	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
( )	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
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,type theory
∑	summation	$\sum _{k=1}^{n}{a_{k}}$ means a1 + a2 + … + an.	$\sum _{k=1}^{4}{k^{2}}$ = 12 + 22 + 32 + 42  = 1 + 4 + 9 + 16 = 30
	sum over … from … to … of
	arithmetic
∏	product	$\prod _{k=1}^{n}a_{k}$ means a1a2···an.	$\prod _{k=1}^{4}(k+2)$ = (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, …, yn).	$\prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}$
	the Cartesian product of; the direct product of
	set theory
∐	coproduct
	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	∫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.	∫ab 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.
	contour integral of
	calculus
∇	gradient	∇f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn).	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 differential	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, d
	calculus
	boundary	∂M means the boundary of M	∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
	boundary of
	topology
δ	Dirac delta function	$\delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}$	δ(x)
	Dirac delta of
	hyperfunction
	Kronecker delta	$\delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}$	δij
	Kronecker delta of
	hyperfunction
⊥	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
	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
	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