User:Alksentrs/Table of mathematical symbols (testing)

Test version of the article Table of mathematical symbols.

Symbol (HTML)	Symbol (TeX)	Name	Explanation	Examples
		Read as
		Category
\|…\|	$\|\ldots \|\!\,$	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
# ♯	$\#\!\,$ $\sharp \!\,$	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	${}^{\mathrm {T} }\!\,$ ${}^{\mathrm {tr} }\!\,$	transpose	Swap rows for columns	If $A=(a_{ij})$ then $A^{\mathrm {T} }=(a_{ji})$ .
		transpose
		matrix operations
~	$\sim \!\,$	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
≈	$\approx \!\,$	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
◅	$\triangleleft \!\,$	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
∴	$\therefore \!\,$	therefore	Sometimes used in proofs before logical consequences.	All humans are mortal. Socrates is a human. ∴ Socrates is mortal.
		therefore; so; hence
		everywhere
∵	$\because \!\,$	because	Sometimes used in proofs before reasoning.	3331 is prime ∵ it has no positive integer factors other than itself and one.
		because; since
		everywhere
⇒ → ⊃	$\Rightarrow \!\,$ $\rightarrow \!\,$ $\supset \!\,$	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
⇔ ↔	$\Leftrightarrow \!\,$ $\leftrightarrow \!\,$	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
¬ ˜	$\neg \!\,$ $\sim \!\,$	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
∧	$\land \!\,$	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
∨	$\lor \!\,$	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
⊕ ⊻	$\oplus \!\,$ $\veebar \!\,$	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
∀	$\forall \!\,$	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
∃	$\exists \!\,$	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
∃!	$\exists !\!\,$	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
:= ≡ :⇔	$:=\!\,$ $\equiv \!\,$ $:\Leftrightarrow \!\,$ $\triangleq \!\,$ ${\overset {\underset {\mathrm {def} }{}}{=}}\!\,$	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
≅	$\cong \!\,$	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