Mathematical symbols..

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shaista farheen

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Common symbols

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

This list is incomplete; you can help by expanding it.

Symbol	Symbol	Name	Explanation	Examples
		Read as
		Category
=	$= \!\,$	equality is equal to; equals everywhere	x = y means x and y represent the same thing or value.	1 + 1 = 2
≠	$\ne \!\,$	inequation is not equal to; does not equal everywhere	x ≠ y means that x and y do not represent the same thing or value. (As ≠ can be hard to type, the more "keyboard friendly" forms !=, /= or <> may be seen. These are avoided in mathematical texts.)	2 + 2 ≠ 5
< >	$< \!\,$ $> \!\,$	strict inequality is less than, is greater than order theory	x < y means x is less than y. x > y means x is greater than y.	3 < 4 5 > 4
< >	$< \!\,$ $> \!\,$	proper subgroup is a proper subgroup of group theory	H < G means H is a proper subgroup of G.	5Z < Z A₃ <S₃
≪ ≫	$\ll \!\,$ $\gg \!\,$	(very) strict inequality is much less than, is much greater than order theory	x ≪ y means x is much less than y. x ≫ y means x is much greater than y.	0.003 ≪ 1000000
≪ ≫	$\ll \!\,$ $\gg \!\,$	asymptotic comparison of smaller (greater) order than analytic number theory	f ≪ g means the growth of f is asymptotically bounded by g. (This is I. M. Vinogradov's notation. Another notation is the Big O notation, which looks like f = O(g).)	x ≪ e^x
≤ ≥	$\le \!\,$ $\ge \!\,$	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. (As ≤ and ≥ can be hard to type, the more "keyboard friendly" forms <= and >= may be seen. These are avoided in mathematical texts.)	3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5
≤ ≥	$\le \!\,$ $\ge \!\,$	subgroup is a subgroup of group theory	H ≤ G means H is a subgroup of G.	Z ≤ Z A₃ ≤S₃
∝	$\propto \!\,$	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; add 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 = {3, 4, 5, 6} ∧ A₂ = {7, 8, 9, 10} ⇒ A₁ + A₂ = {(3,1), (4,1), (5,1), (6,1), (7,2), (8,2), (9,2), (10,2)}
−	$- \!\,$	subtraction minus; take; subtract 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
		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 as described below.)	{1,2,4} − {1,3,4} = {2}
×	$\times \!\,$	multiplication times; multiplied by arithmetic	3 × 4 means the multiplication of 3 by 4.	7 × 8 = 56
		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)}
		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)
		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* as described below, or U(R).	$\begin{align} (\mathbb{Z} / 5\mathbb{Z})^\times & = \{ [1], [2], [3], [4] \} \\ & \cong C_4 \\ \end{align}$
·	$\cdot \!\,$	multiplication times; multiplied by arithmetic	3 · 4 means the multiplication of 3 by 4.	7 · 8 = 56
·	$\cdot \!\,$	dot product dot vector algebra	u · v means the dot product of vectors u and v	(1,2,5) · (3,4,−1) = 6
÷ ⁄	$\div \!\,$ $/ \!\,$	division divided by; over 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).
√	$\surd \!\,$ $\sqrt{\ } \!\,$	square root the (principal) square root of real numbers	$\sqrt{x}$ means the positive number whose square is $x$ .	$\sqrt{4}=2$
√	$\surd \!\,$ $\sqrt{\ } \!\,$	complex square root the (complex) square root of complex numbers	if $z=r\,\exp(i\phi)$ is represented in polar coordinates with $-\pi < \phi \le \pi$ , then $\sqrt{z} = \sqrt{r} \exp(i \phi/2)$ .	$\sqrt{-1}=i$
\|…\|	$\| \ldots \| \!\,$	absolute value or modulus absolute value of; 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
		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
		determinant determinant of matrix theory	\|A\| means the determinant of the matrix A	$\begin{vmatrix} 1&2 \\ 2&4 \\ \end{vmatrix} = 0$
		cardinality cardinality of; size of; order of set theory	\|X\| means the cardinality of the set X. (# or ♯ may be used instead as described below.)	\|{3, 5, 7, 9}\| = 4.
\|\|…\|\|	$\\| \ldots \\| \!\,$	norm norm of; length of linear algebra	\|\| x \|\| means the norm of the element x of a normed vector space.^[1]	\|\| x + y \|\| ≤ \|\| x \|\| + \|\| y \|\|
\|\|…\|\|	$\\| \ldots \\| \!\,$	nearest integer function nearest integer to numbers	\|\|x\|\| means the nearest integer to x, with half-integers being rounded to even. (This may also be written [x], ⌊x⌉, nint(x) or Round(x).)	\|\|1\|\| = 1, \|\|1.5\|\| = 2, \|\|−2.5\|\| = 2, \|\|3.49\|\| = 3
∣ ∤	$\mid \!\,$ $\nmid \!\,$	divisor, divides divides number theory	a\|b means a divides b. a∤b means a does not divide b. (This symbol can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar \| character can be used.)	Since 15 = 3×5, it is true that 3\|15 and 5\|15.
		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
		restriction restriction of … to …; restricted to set theory	f\|_A means the function f restricted to the set A, that is, it is the function with domain A ∩ dom(f) that agrees with f.	The function f : R → R defined by f(x) = x² is not injective, but f\|_R⁺ is injective.
\|\|	$\\| \!\,$	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.
		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.
# ♯	$\# \!\,$ $\sharp \!\,$	cardinality cardinality of; size of; order of set theory	#X means the cardinality of the set X. (\|…\| may be used instead as described above.)	#{4, 6, 8} = 3
# ♯	$\# \!\,$ $\sharp \!\,$	connected sum connected sum of; knot sum of; knot composition of topology, knot theory	A#B is the connected sum of the manifolds A and B. If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition.	A#S^m is homeomorphic to A, for any manifold A, and the sphere S^m.
ℵ	$\aleph \!\,$	aleph number aleph set theory	ℵ_α represents an infinite cardinality (specifically, the α-th one, where α is an ordinal).	\|ℕ\| = ℵ₀, which is called aleph-null.
:	$: \!\,$	such that such that; so that everywhere	: means "such that", and is used in proofs and the set-builder notation (described below).	∃ n ∈ ℕ: n is even.
		field extension extends; over field theory	K : F means the field K extends the field F. This may also be written as K ≥ F.	ℝ : ℚ
		inner product of matrices inner product of linear algebra	A : B means the inner product of the matrices A and B. The general inner product is denoted by 〈u, v〉, 〈u \| v〉 or (u \| v), as described below. For spatial vectors, the dot product notation, x·y is common. See also Bra-ket notation.	$A:B = \sum_{i,j} A_{ij}B_{ij}\!\,$
!	$! \!\,$	factorial factorial combinatorics	n! means the product 1 × 2 × ... × n.	4! = 1 × 2 × 3 × 4 = 24
!	$! \!\,$	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 ! is primarily from computer science. It is avoided in mathematical texts, where the notation ¬A is preferred.)	!(!A) ⇔ A x ≠ y ⇔ !(x = y)
~	$\sim \!\,$	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