User:Alksentrs/Table of mathematical symbols (grouped like in German version)

This is an experimental version of Table of mathematical symbols. (Structure is based on the article Mathematische Symbole on the German Wikipedia.)

Algebra

Linear algebra

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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A^T A^tr	${\color {Gray}A}^{\mathrm {T} }\!\,$ ${\color {Gray}A}^{\mathrm {tr} }\!\,$	transpose transpose matrix operations	A^T means A, but with its rows swapped for columns.	If A = (a_ij) then A^T = (a_ji).
\|A\| det(A)	$\|{\color {Gray}A}\|\!\,$ $\det({\color {Gray}A})\!\,$	determinant determinant of matrix theory	\|A\| means the determinant of the matrix A	${\begin{vmatrix}1&2\\2&4\\\end{vmatrix}}=0$
W^⊥	${\color {Gray}W}^{\bot }\!\,$	orthogonal complement orthogonal/perpendicular complement of; perp linear algebra	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 $\mathbb {R} ^{3}$ , $(\mathbb {R} ^{2})^{\perp }\cong \mathbb {R}$ .
V ⊕ W	${\color {Gray}V}\oplus {\color {Gray}W}\!\,$	direct sum direct sum of abstract algebra	The direct sum is a special way of combining several modules into one general module.	Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = {0})
〈,〉 ( \| ) < , > · :	$\langle \ ,\ \rangle \!\,$ $(\ \|\ )\!\,$ $<\ ,\ >\!\,$ $\cdot \!\,$ $:\!\,$	inner product inner product of linear algebra	〈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 $A:B=\sum _{i,j}A_{ij}B_{ij}\!\,$
〈 , 〉 < , > Sp	$\langle \ ,\ \rangle \!\,$ $<\ ,\ >\!\,$ $\mathrm {Sp} \!\,$	linear span (linear) span of; linear hull of linear algebra	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.)	$\left\langle \left({\begin{smallmatrix}1\\0\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\1\\0\end{smallmatrix}}\right),\left({\begin{smallmatrix}0\\0\\1\end{smallmatrix}}\right)\right\rangle =\mathbb {R} ^{3}$ .
⊗	$\otimes \!\,$	tensor product, tensor product of modules tensor product of linear algebra	$V\otimes U$ 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}}

Group and ring theory

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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Elementary mathematics

Elementary functions

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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\|…\|	$\|\ldots \|\!\,$	absolute value or modulus absolute value (modulus) of numbers	\|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

Intervals

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Trigonometric functions

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Complex numbers

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Geometry

Elementary geometry

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Vector calculus

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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·	$\cdot \!\,$	dot product dot vector algebra	u · v means the dot product of vectors u and v	(1,2,5) · (3,4,−1) = 6
×	$\times \!\,$	cross product cross vector algebra	u × v means the cross product of vectors u and v	(1,2,5) × (3,4,−1) = (−22, 16, − 2)
∧	$\land \!\,$	wedge product wedge product; exterior product linear algebra	u ∧ v means the wedge product of vectors u and v. This generalizes the cross product to higher dimensions. (For vectors in R³, × can also be used.)	$u\wedge v=u\times v,{\mbox{ if }}u,v\in \mathbb {R} ^{3}$

Set theory

Set functions

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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\|…\| # ♯	$\|\ldots \|\!\,$ $\#\!\,$ $\sharp \!\,$	cardinality cardinality of; size of set theory	\|X\| means the cardinality of the set X.	\|{3, 5, 7, 9}\| = 4.

Cardinal numbers

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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Set operations

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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×	$\times \!\,$	Cartesian product the Cartesian product of ... and ...; the direct product of ... and ... set theory	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)}
∏	$\prod \!\,$	Cartesian product the Cartesian product of; the direct product of set theory	$\prod _{i=0}^{n}{Y_{i}}$ means the set of all (n+1)-tuples (y₀, …, y_n).	$\prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}$
− ∖	$-\!\,$ $\setminus \!\,$	set-theoretic complement minus; without set theory	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.)	{1,2,4} − {1,3,4} = {2}
∪	$\cup \!\,$	set-theoretic union the union of … or …; union set theory	A ∪ B means the set of those elements which are either in A, or in B, or in both.	A ⊆ B ⇔ (A ∪ B) = B
∩	$\cap \!\,$	set-theoretic intersection intersected with; intersect set theory	A ∩ B means the set that contains all those elements that A and B have in common.	{x ∈ ℝ : x² = 1} ∩ ℕ = {1}
∆	$\vartriangle \!\,$	symmetric difference symmetric difference set theory	A ∆ B means the set of elements in exactly one of A or B.	{1,5,6,8} ∆ {2,5,8} = {1,2,6}

Set relations

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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∈ ∉	$\in \!\,$ $\notin \!\,$	set membership is an element of; is not an element of everywhere, set theory	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
⊆ ⊂	$\subseteq \!\,$ $\subset \!\,$	subset is a subset of set theory	(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
⊇ ⊃	$\supseteq \!\,$ $\supset \!\,$	superset is a superset of set theory	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

Ordinal numbers and types

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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Special functions

Error functions

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Number theory

Sets of numbers

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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ℕ N	$\mathbb {N} \!\,$ $\mathbf {N} \!\,$	natural numbers N numbers	N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.	ℕ = {\|a\| : a ∈ ℤ, a ≠ 0}
ℤ Z	$\mathbb {Z} \!\,$ $\mathbf {Z} \!\,$	integers Z numbers	ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ₊ means {1, 2, 3, ...} = ℕ.	ℤ = {p, −p : p ∈ ℕ ∪ {0}
ℚ Q	$\mathbb {Q} \!\,$ $\mathbf {Q} \!\,$	rational numbers Q numbers	ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.	3.14000... ∈ ℚ π ∉ ℚ
ℝ R	$\mathbb {R} \!\,$ $\mathbf {R} \!\,$	real numbers R numbers	ℝ means the set of real numbers.	π ∈ ℝ √(−1) ∉ ℝ
ℂ C	$\mathbb {C} \!\,$ $\mathbf {C} \!\,$	complex numbers C numbers	ℂ means {a + b i : a,b ∈ ℝ}.	i = √(−1) ∈ ℂ
𝕂 K	$\mathbb {K} \!\,$ $\mathbf {K} \!\,$	real or complex numbers K linear algebra	K means the statement holds substituting K for R and also for C.	${\begin{array}{l}\quad x^{2}\in \mathbb {C} \,\forall x\in \mathbb {K} \\{\text{because}}\\\quad x^{2}\in \mathbb {C} \,\forall x\in \mathbb {R} \\{\text{and}}\\\quad x^{2}\in \mathbb {C} \,\forall x\in \mathbb {C} .\end{array}}$

Divisibility

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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\|	$\|\!\,$	divides divides number theory	a\|b means a divides b.	Since 15 = 3×5, it is true that 3\|15 and 5\|15.
\|\|	$\\|\!\,$	exact divisibility exactly divides number theory	p^a \|\| n means p^a exactly divides n (i.e. p^a divides n but p^a+1 does not).	2³ \|\| 360.
⊥	$\bot \!\,$	coprime is coprime to number theory	x ⊥ y means x has no factor in common with y.	34 ⊥ 55.

Elementary arithmetic functions

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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⌊…⌋	$\lfloor \ldots \rfloor \!\,$	floor floor; greatest integer; entier numbers	⌊x⌋ means the floor of x, i.e. the largest integer less than or equal to x. (This may also be written [x], floor(x) or int(x).)	⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3
⌈…⌉	$\lceil \ldots \rceil \!\,$	ceiling ceiling numbers	⌈x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x. (This may also be written ceil(x) or ceiling(x).)	⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈−2.6⌉ = −2

Multiplicative number-theoretic functions

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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Further functions from analytical number theory

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Logic and Boolean algebra

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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∴	$\therefore \!\,$	therefore therefore; so; hence everywhere	Sometimes used in proofs before logical consequences.	All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
∵	$\because \!\,$	because because; since everywhere	Sometimes used in proofs before reasoning.	3331 is prime ∵ it has no positive integer factors other than itself and one.
⇒ → ⊃	$\Rightarrow \!\,$ $\rightarrow \!\,$ $\supset \!\,$	material implication implies; if … then propositional logic, Heyting algebra	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).
⇔ ↔	$\Leftrightarrow \!\,$ $\leftrightarrow \!\,$	material equivalence if and only if; iff propositional logic	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
¬ ˜	$\neg \!\,$ $\sim \!\,$	logical negation not propositional logic	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)
∧	$\land \!\,$	logical conjunction or meet in a lattice and; min; meet propositional logic, lattice theory	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)).	n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.
∨	$\lor \!\,$	logical disjunction or join in a lattice or; max; join propositional logic, lattice theory	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.
⊕ ⊻	$\oplus \!\,$ $\veebar \!\,$	exclusive or xor propositional logic, Boolean algebra	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.
∀	$\forall \!\,$	universal quantification for all; for any; for each predicate logic	∀ x: P(x) means P(x) is true for all x.	∀ n ∈ ℕ: n² ≥ n.
∃	$\exists \!\,$	existential quantification there exists predicate logic	∃ x: P(x) means there is at least one x such that P(x) is true.	∃ n ∈ ℕ: n is even.
∃!	$\exists !\!\,$	uniqueness quantification there exists exactly one predicate logic	∃! x: P(x) means there is exactly one x such that P(x) is true.	∃! n ∈ ℕ: n + 5 = 2n.
⊧	$\vDash \!\,$	entailment entails model theory	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
⊢	$\vdash \!\,$	inference infers; is derived from propositional logic, predicate logic	x ⊢ y means y is derivable from x.	A → B ⊢ ¬B → ¬A.

Misc.

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
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=	$=\!\,$	equality is equal to; equals everywhere	x = y means x and y represent the same thing or value.	1 + 1 = 2
≠ <> !=	$\neq \!\,$	inequation is not equal to; does not equal everywhere	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
< > ≪ ≫	$<\!\,$ $>\!\,$ $\ll \!\,$ $\gg \!\,$	strict inequality is less than, is greater than, is much less than, is much greater than order theory	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
≤ <= ≥ >=	$\leq \!\,$ $\geq \!\,$	inequality is less than or equal to, is greater than or equal to order theory	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
∝	$\propto \!\,$	proportionality is proportional to; varies as everywhere	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x
+	$+\!\,$	addition plus arithmetic	4 + 6 means the sum of 4 and 6.	2 + 7 = 9
+	$+\!\,$	disjoint union the disjoint union of ... and ... set theory	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)}
−	$-\!\,$	subtraction minus arithmetic	9 − 4 means the subtraction of 4 from 9.	8 − 3 = 5
−	$-\!\,$	negative sign negative; minus; the opposite of arithmetic	−3 means the negative of the number 3.	−(−5) = 5
×	$\times \!\,$	multiplication times arithmetic	3 × 4 means the multiplication of 3 by 4.	7 × 8 = 56
·	$\cdot \!\,$	multiplication times arithmetic	3 · 4 means the multiplication of 3 by 4.	7 · 8 = 56
÷ ⁄	$\div \!\,$ $/\!\,$	division divided by arithmetic	6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.	2 ÷ 4 = .5 12 ⁄ 4 = 3
		quotient group mod group theory	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}}
		quotient set mod set theory	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]}
±	$\pm \!\,$	plus-minus plus or minus arithmetic	6 ± 3 means both 6 + 3 and 6 − 3.	The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
±	$\pm \!\,$	plus-minus plus or minus measurement	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.
∓	$\mp \!\,$	minus-plus minus or plus arithmetic	6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5).	cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
√	${\sqrt {\ }}\!\,$	square root the principal square root of; square root real numbers	${\sqrt {x}}$ means the positive number whose square is $x$ .	${\sqrt {4}}=2$
√	${\sqrt {\ }}\!\,$	complex square root the complex square root of …; square root complex numbers	if $z=r\,\exp(i\phi )$ is represented in polar coordinates with $-\pi <\phi \leq \pi$ , then ${\sqrt {z}}={\sqrt {r}}\exp(i\phi /2)$ .	${\sqrt {-1}}=i$
\|…\|	$\|\ldots \|\!\,$	Euclidean distance Euclidean distance between; Euclidean norm of geometry	\|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
\|	$\|\!\,$	conditional probability given probability	P(A\|B) means the probability of the event a occurring given that b occurs.	If P(A)=0.4 and P(B)=0.5, P(A\|B)=((0.4)(0.5))/(0.5)=0.4
!	$!\!\,$	factorial factorial combinatorics	n! is the product 1 × 2 × ... × n.	4! = 1 × 2 × 3 × 4 = 24
~	$\sim \!\,$	probability distribution has distribution statistics	X ~ D, means the random variable X has the probability distribution D.	X ~ N(0,1), the standard normal distribution
		row equivalence is row equivalent to matrix theory	A~B means that B can be generated by using a series of elementary row operations on A	${\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\sim {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}$
		same order of magnitude roughly similar; poorly approximates approximation theory	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
		asymptotically equivalent is asymptotically equivalent to asymptotic analysis	f ~ g means $\lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1$ .	x ~ x+1
		equivalence relation are in the same equivalence class everywhere	a ~ b means $b\in [a]$ (and equivalently $a\in [b]$ ).	1 ~ 5 mod 4
≈	$\approx \!\,$	approximately equal is approximately equal to everywhere	x ≈ y means x is approximately equal to y.	π ≈ 3.14159
≈	$\approx \!\,$	isomorphism is isomorphic to group theory	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.
◅	$\triangleleft \!\,$	normal subgroup is a normal subgroup of group theory	N ◅ G means that N is a normal subgroup of group G.	Z(G) ◅ G
◅	$\triangleleft \!\,$	ideal is an ideal of ring theory	I ◅ R means that I is an ideal of ring R.	(2) ◅ Z
:= ≡ :⇔	$:=\!\,$ $\equiv \!\,$ $:\Leftrightarrow \!\,$ $\triangleq \!\,$ ${\overset {\underset {\mathrm {def} }{}}{=}}\!\,$	definition is defined as; equal by definition everywhere	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.	$\cosh x:={\frac {e^{x}+e^{-x}}{2}}$
≅	$\cong \!\,$	congruence is congruent to geometry	△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
≅	$\cong \!\,$	isomorphic is isomorphic to abstract algebra	G ≅ H means that group G is isomorphic (structurally identical) to group H. (≈ can also be used for isomorphic, as described above.)	$\mathbb {R} ^{2}\cong \mathbb {C}$ .
≡	$\equiv \!\,$	congruence relation ... is congruent to ... modulo ... modular arithmetic	a ≡ b (mod n) means a − b is divisible by n	5 ≡ 11 (mod 3)
{ , }	${\{\ ,\!\ \}}\!\,$	set brackets the set of … set theory	{a,b,c} means the set consisting of a, b, and c.	ℕ = { 1, 2, 3, …}
{ : } { \| }	$\{\ :\ \}\!\,$ $\{\ \|\ \}\!\,$	set builder notation the set of … such that set theory	{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}
∅ { }	$\emptyset \!\,$ $\varnothing \!\,$ $\{\}\!\,$	empty set the empty set set theory	∅ means the set with no elements. { } means the same.	{n ∈ ℕ : 1 < n² < 4} = ∅
→	$\to \!\,$	function arrow from … to set theory, type theory	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.
↦	$\mapsto \!\,$	function arrow maps to set theory	f: a ↦ b means the function f maps the element a to the element b.	Let f: x ↦ x+1 (the successor function).
o	$\circ \!\,$	function composition composed with set theory	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).
∞	$\infty \!\,$	infinity infinity numbers	∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.	$\lim _{x\to 0}{\frac {1}{\|x\|}}=\infty$
[ ] [ , ] [ , , ]	$[\ ]\!\,$ $[\ ,\ ]\!\,$ $[\ ,\ ,\ ]\!\,$	equivalence class the equivalence class of abstract algebra	[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, …}.
		closed interval closed interval order theory	$[a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}$ .	[0,1]
		commutator the commutator of group theory, ring theory	[g, h] = g⁻¹h⁻¹gh (or ghg⁻¹h⁻¹), if g, h ∈ G (a group). [a, b] = ab − ba, if a, b ∈ R (a ring or commutative algebra).	x^y = x[x, y] (group theory). [AB, C] = A[B, C] + [A, C]B (ring theory).
		triple scalar product the triple scalar product of vector calculus	[a, b, c] = a × b · c, the scalar product of a × b with c.	[a, b, c] = [b, c, a] = [c, a, b].
( ) ( , )	$(\ )\!\,$ $(\ ,\ )\!\,$	function application of set theory	f(x) means the value of the function f at the element x.	If f(x) := x², then f(3) = 3² = 9.
		precedence grouping parentheses everywhere	Perform the operations inside the parentheses first.	(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
		tuple tuple; n-tuple; ordered pair/triple/etc; row vector everywhere	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).
		highest common factor highest common factor; hcf number theory	(a, b) means the highest common factor of a and b. (This may also be written hcf(a, b).)	(3, 7) = 1 (they are coprime); (15, 25) = 5.
( , ) ] , [	$(\ ,\ )\!\,$ $]\ ,\ [\!\,$	open interval open interval order theory	$(a,b)=\{x\in \mathbb {R} :a<x<b\}$ . (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)
( , ] ] , ]	$(\ ,\ ]\!\,$ $]\ ,\ ]\!\,$	left-open interval half-open interval; left-open interval order theory	$(a,b]=\{x\in \mathbb {R} :a<x\leq b\}$ .	(−1, 7] and (−∞, −1]
[ , ) [ , [	$[\ ,\ )\!\,$ $[\ ,\ [\!\,$	right-open interval half-open interval; right-open interval order theory	$[a,b)=\{x\in \mathbb {R} :a\leq x<b\}$ .	[4, 18) and [1, +∞)
∑	$\sum \!\,$	summation sum over … from … to … of arithmetic	$\sum _{k=1}^{n}{a_{k}}$ means a₁ + a₂ + … + a_n.	$\sum _{k=1}^{4}{k^{2}}$ = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
∏	$\prod \!\,$	product product over … from … to … of arithmetic	$\prod _{k=1}^{n}a_{k}$ means a₁a₂···a_n.	$\prod _{k=1}^{4}(k+2)$ = (1+2)(2+2)(3+2)(4+2) = 3 × 4 × 5 × 6 = 360
∐	$\coprod \!\,$	coproduct coproduct over … from … to … of category theory	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.
′ ^•	$'\!\,$ ${\dot {\,}}\!\,$	derivative … prime derivative of calculus	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
∫	$\int \!\,$	indefinite integral or antiderivative indefinite integral of the antiderivative of calculus	∫ f(x) dx means a function whose derivative is f.	∫x² dx = x³/3 + C
∫	$\int \!\,$	definite integral integral from … to … of … with respect to calculus	∫_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;
∮	$\oint \!\,$	contour integral or closed line integral contour integral of calculus	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 $\oint _{C}{1 \over z}\,dz=2\pi i$ .
∇	$\nabla \!\,$	gradient del, nabla, gradient of vector calculus	∇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)
		divergence del dot, divergence of vector calculus	$\nabla \cdot {\vec {v}}={\partial v_{x} \over \partial x}+{\partial v_{y} \over \partial y}+{\partial v_{z} \over \partial z}$	If ${\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k}$ , then $\nabla \cdot {\vec {v}}=3y+2yz$ .
		curl curl of vector calculus	$\nabla \times {\vec {v}}=\left({\partial v_{z} \over \partial y}-{\partial v_{y} \over \partial z}\right)\mathbf {i}$ $+\left({\partial v_{x} \over \partial z}-{\partial v_{z} \over \partial x}\right)\mathbf {j} +\left({\partial v_{y} \over \partial x}-{\partial v_{x} \over \partial y}\right)\mathbf {k}$	If ${\vec {v}}:=3xy\mathbf {i} +y^{2}z\mathbf {j} +5\mathbf {k}$ , then $\nabla \times {\vec {v}}=-y^{2}\mathbf {i} -3x\mathbf {k}$ .
∂	$\partial \!\,$	partial derivative partial, d calculus	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
		boundary boundary of topology	∂M means the boundary of M	∂{x : \|\|x\|\| ≤ 2} = {x : \|\|x\|\| = 2}
		degree of a polynomial degree of algebra	∂f means the degree of the polynomial f. (This may also be written deg f.)	∂(x² − 1) = 2
δ	$\delta \!\,$	Dirac delta function Dirac delta of hyperfunction	$\delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}$	δ(x)
δ	$\delta \!\,$	Kronecker delta Kronecker delta of hyperfunction	$\delta _{ij}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}$	δ_ij
<: <·	$<:\!\,$ ${<}{\cdot }\!\,$	cover is covered by order theory	x <• y means that x is covered by y.	{1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment.
<: <·	$<:\!\,$ ${<}{\cdot }\!\,$	subtype is a subtype of type theory	T₁ <: T₂ means that T₁ is a subtype of T₂.	If S <: T and T <: U then S <: U (transitivity).
⊤	$\top \!\,$	top element the top element lattice theory	⊤ means the largest element of a lattice.	∀x : x ∨ ⊤ = ⊤
⊤	$\top \!\,$	top type the top type; top type theory	⊤ means the top or universal type; every type in the type system of interest is a subtype of top.	∀ types T, T <: ⊤
⊥	$\bot \!\,$	perpendicular is perpendicular to geometry	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.
		bottom element the bottom element lattice theory	⊥ means the smallest element of a lattice.	∀x : x ∧ ⊥ = ⊥
		bottom type the bottom type; bot type theory	⊥ means 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
		comparability is comparable to order theory	x ⊥ y means that x is comparable to y.	{e, π} ⊥ {1, 2, e, 3, π} under set containment.
\|\|…\|\|	$\\|\ldots \\|\!\,$	norm norm of; length of linear algebra	\|\| x \|\| is the norm of the element x of a normed vector space.	\|\| x + y \|\| ≤ \|\| x \|\| + \|\| y \|\|
\|\|	$\\|\!\,$	parallel is parallel to geometry, physics	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 $x\\|y\Leftrightarrow {\frac {1}{x^{-1}+y^{-1}}}$ .
\|\|	$\\|\!\,$	incomparability is incomparable to order theory	x \|\| y means x is incomparable to y.	{1,2} \|\| {2,3} under set containment.
*	$*\!\,$	convolution convolution, convolved with functional analysis	f * g means the convolution of f and g.	$(f*g)(t)=\int f(\tau )g(t-\tau )\,d\tau$ .
		complex conjugate conjugate complex numbers	z* is the complex conjugate of z. ( ${\bar {z}}$ can also be used for the conjugate of z, as described below.)	$(3+4i)^{\ast }=3-4i$ .
		group of units the group of units of ring theory	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).)	$(\mathbb {Z} /5\mathbb {Z} )^{\ast }=\{[1],[2],[3],[4]\}$ .
x̄	${\bar {x}}\!\,$	mean overbar, … bar statistics	${\bar {x}}$ (often read as "x bar") is the mean (average value of $x_{i}$ ).	$x=\{1,2,3,4,5\};{\bar {x}}=3$ .
x̄	${\bar {x}}\!\,$	complex conjugate conjugate complex numbers	${\overline {z}}$ is the complex conjugate of z. (z* can also be used for the conjugate of z, as described above.)	${\overline {3+4i}}=3-4i$ .