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∀ ∀ {\displaystyle \forall }

\forall universal quantification for all;

for any;

for each;

for every predicate logic ∀ x, P(x) means P(x) is true for all x. ∀ n ∈ ℕ, n2 ≥ n .

𝔹



B B {\displaystyle \mathbb {B} }

\mathbb{B}



B {\displaystyle \mathbf {B} }

\mathbf{B} boolean domain B;

the (set of) boolean values;

the (set of) truth values; set theory, boolean algebra 𝔹 means either {0, 1}, {false, true}, {F, T}, or { ⊥ , ⊤ } {\displaystyle \left\{\bot ,\top \right\}} (¬False) ∈ 𝔹

ℂ



C C {\displaystyle \mathbb {C} }

\mathbb{C}



C {\displaystyle \mathbf {C} }

\mathbf{C} complex numbers C;

the (set of) complex numbers numbers ℂ means {a + b i : a,b ∈ ℝ} . i = √ −1 ∈ ℂ

𝔠 c {\displaystyle {\mathfrak {c}}}

\mathfrak c cardinality of the continuum cardinality of the continuum;

c;

cardinality of the real numbers set theory The cardinality of R {\displaystyle \mathbb {R} } | R | {\displaystyle |\mathbb {R} |} c {\displaystyle {\mathfrak {c}}} Fraktur letter C). c = ℶ 1 {\displaystyle {\mathfrak {c}}={\beth }_{1}}



∂ ∂ {\displaystyle \partial }

\partial partial derivative partial;

d calculus ∂f/∂x i means the partial derivative of f with respect to x i , where f is a function on (x 1 , ..., x n ). 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

𝔼



E E {\displaystyle \mathbb {E} }

\mathbb E



E {\displaystyle \mathrm {E} }

\mathrm{E} expected value expected value probability theory the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained E [ X ] = x 1 p 1 + x 2 p 2 + ⋯ + x k p k p 1 + p 2 + ⋯ + p k {\displaystyle \mathbb {E} [X]={\frac {x_{1}p_{1}+x_{2}p_{2}+\dotsb +x_{k}p_{k}}{p_{1}+p_{2}+\dotsb +p_{k}}}}

∃ ∃ {\displaystyle \exists }

\exists existential quantification there exists;

there is;

there are predicate logic ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ ℕ: n is even.

∃! ∃ ! {\displaystyle \exists !}

\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.

∈



∉ ∈ {\displaystyle \in }

\in



∉ {\displaystyle

otin }



otin 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;[15] a ∉ S means a is not an element of S.[15] (1/2)−1 ∈ ℕ



2−1 ∉ ℕ

∌ ∌ {\displaystyle

ot

i }



ot

i set membership does not contain as an element set theory S ∌ e means the same thing as e ∉ S, where S is a set and e is not an element of S.

∋ ∋ {\displaystyle

i }



i such that symbol such that mathematical logic often abbreviated as "s.t."; : and | are also used to abbreviate "such that". The use of ∋ goes back to early mathematical logic and its usage in this sense is declining. The symbol ∍ {\displaystyle \backepsilon } Choose x {\displaystyle x} x {\displaystyle x} x {\displaystyle x}

set membership contains as an element set theory S ∋ e means the same thing as e ∈ S, where S is a set and e is an element of S.

ℍ



H H {\displaystyle \mathbb {H} }

\mathbb{H}



H {\displaystyle \mathbf {H} }

\mathbf{H} quaternions or Hamiltonian quaternions H;

the (set of) quaternions numbers ℍ means {a + b i + c j + d k : a,b,c,d ∈ ℝ}.

𝕀



I I {\displaystyle \mathbb {I} }

\mathbb{I}



I {\displaystyle \mathbf {I} }

\mathbf{I} Indicator function the indicator of Boolean algebra The indicator function of a subset A of a set X is a function 1 A : X → { 0 , 1 } {\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\}} 1 A ( x ) := { 1 if x ∈ A , 0 if x ∉ A . {\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1&{\text{if }}x\in A,\\0&{\text{if }}x

otin A.\end{cases}}} Note that the indicator function is also sometimes denoted 1.

ℕ



N N {\displaystyle \mathbb {N} }

\mathbb{N}



N {\displaystyle \mathbf {N} }

\mathbf{N} natural numbers the (set of) natural numbers numbers N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.



The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analysts, set theorists and computer scientists prefer the former. To avoid confusion, always check an author's definition of N.



Set theorists often use the notation ω (for least infinite ordinal) to denote the set of natural numbers (including zero), along with the standard ordering relation ≤. ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| > 0: a ∈ ℤ}

○ ∘ {\displaystyle \circ }

\circ Hadamard product entrywise product linear algebra For two matrices (or vectors) of the same dimensions A , B ∈ R m × n {\displaystyle A,B\in {\mathbb {R} }^{m\times n}} A ∘ B ∈ R m × n {\displaystyle A\circ B\in {\mathbb {R} }^{m\times n}} ( A ∘ B ) i , j = ( A ) i , j ⋅ ( B ) i , j {\displaystyle (A\circ B)_{i,j}=(A)_{i,j}\cdot (B)_{i,j}} [ 1 2 2 4 ] ∘ [ 1 2 0 0 ] = [ 1 4 0 0 ] {\displaystyle {\begin{bmatrix}1&2\\2&4\\\end{bmatrix}}\circ {\begin{bmatrix}1&2\\0&0\\\end{bmatrix}}={\begin{bmatrix}1&4\\0&0\\\end{bmatrix}}}

∘ ∘ {\displaystyle \circ }

\circ function composition composed with set theory f ∘ g is the function such that (f ∘ g)(x) = f(g(x)).[22] if f(x) := 2x, and g(x) := x + 3, then (f ∘ g)(x) = 2(x + 3).

∅



{ } ∅ {\displaystyle \emptyset }



\emptyset

∅ {\displaystyle \varnothing }

\varnothing

{ } {\displaystyle \{\}}

\{\} empty set the empty set null set set theory ∅ means the set with no elements.[15] { } means the same. {n ∈ ℕ : 1 < n2 < 4} = ∅

ℙ



P P {\displaystyle \mathbb {P} }

\mathbb{P}



P {\displaystyle \mathbf {P} }

\mathbf{P} set of primes P;

the set of prime numbers arithmetic ℙ is often used to denote the set of prime numbers. 2 ∈ P , 3 ∈ P , 8 ∉ P {\displaystyle 2\in \mathbb {P} ,3\in \mathbb {P} ,8

otin \mathbb {P} }

projective space P;

the projective space;

the projective line;

the projective plane topology ℙ means a space with a point at infinity. P 1 {\displaystyle \mathbb {P} ^{1}} P 2 {\displaystyle \mathbb {P} ^{2}}

polynomials the space of all possible polynomials vector space ℙ means a n xn + a n-1 xn-1...a 1 x+a 0

ℙ n means the space of all polynomials of degree less than or equal to n 2 x 3 − 3 x 2 + 2 ∈ P 3 {\displaystyle 2x^{3}-3x^{2}+2\in \mathbb {P} _{3}}

probability the probability of probability theory ℙ(X) means the probability of the event X occurring.



This may also be written as P(X), Pr(X), P[X] or Pr[X]. If a fair coin is flipped, ℙ(Heads) = ℙ(Tails) = 0.5.

Power set the Power set of Powerset Given a set S, the power set of S is the set of all subsets of the set S. The power set of S0 is denoted by P(S). The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence, P({0, 1, 2}) = {∅, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2} }.

ℚ



Q Q {\displaystyle \mathbb {Q} }

\mathbb{Q}



Q {\displaystyle \mathbf {Q} }

\mathbf{Q} rational numbers Q;

the (set of) rational numbers;

the rationals numbers ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. 3.14000... ∈ ℚ



π ∉ ℚ



ℝ



R R {\displaystyle \mathbb {R} }

\mathbb{R}



R {\displaystyle \mathbf {R} }

\mathbf{R} real numbers R;

the (set of) real numbers;

the reals numbers ℝ means the set of real numbers. π ∈ ℝ



√(−1) ∉ ℝ



† † {\displaystyle {}^{\dagger }}

{}^\dagger conjugate transpose conjugate transpose;

adjoint;

Hermitian adjoint/conjugate/transpose/dagger matrix operations A† means the transpose of the complex conjugate of A.[23]



This may also be written A∗T, AT∗, A∗, A T or AT . If A = (a ij ) then A† = ( a ji ).

T T {\displaystyle {}^{\mathsf {T}}}

{}^{\mathsf{T}} transpose transpose matrix operations AT means A, but with its rows swapped for columns.



This may also be written A′, At or Atr. If A = (a ij ) then AT = (a ji ).

⊤ ⊤ {\displaystyle \top }

\top top element the top element lattice theory ⊤ means the largest element of a lattice. ∀x : x ∨ ⊤ = ⊤

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 <: ⊤

⊥ ⊥ {\displaystyle \bot }

\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.

orthogonal complement orthogonal/ perpendicular complement of;

perp linear algebra W⊥ means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W. Within R 3 {\displaystyle \mathbb {R} ^{3}} ( R 2 ) ⊥ ≅ R {\displaystyle (\mathbb {R} ^{2})^{\perp }\cong \mathbb {R} }

coprime is coprime to number theory x ⊥ y means x has no factor greater than 1 in common with y. 34 ⊥ 55

independent is independent of probability A ⊥ B means A is an event whose probability is independent of event B. The double perpendicular symbol ( ⊥ ⊥ {\displaystyle \perp \!\!\!\perp } A ⊥ ⊥ B {\displaystyle A\perp \!\!\!\perp B} If A ⊥ B, then P(A|B) = P(A).

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.



𝕌



U U {\displaystyle \mathbb {U} }

\mathbb{U}



U {\displaystyle \mathbf {U} }

\mathbf{U} all numbers being considered U;

the universal set;

the set of all numbers;

all numbers considered set theory 𝕌 means "the set of all elements being considered."

It may represent all numbers both real and complex, or any subset of these—hence the term "universal". 𝕌 = {ℝ,ℂ} includes all numbers.



If instead, 𝕌 = {ℤ,ℂ}, then π ∉ 𝕌.

∪ ∪ {\displaystyle \cup }

\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.[13] A ⊆ B ⇔ (A ∪ B) = B

∩ ∩ {\displaystyle \cap }

\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.[13] {x ∈ ℝ : x2 = 1} ∩ ℕ = {1}

∨ ∨ {\displaystyle \lor }

\lor join in a logical disjunction orin 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.

wedge product wedge product;

exterior product exterior algebra u ∧ v means the wedge product of any multivectors u and v. In three-dimensional Euclidean space the wedge product and the cross product of two vectors are each other's Hodge dual. u ∧ v = ∗ ( u × v ) if u , v ∈ R 3 {\displaystyle u\wedge v=*(u\times v)\ {\text{ if }}u,v\in \mathbb {R} ^{3}}

× × {\displaystyle \times }

\times multiplication times;

multiplied by arithmetic 3 × 4 means the multiplication of 3 by 4.



(The symbol * is generally used in programming languages, where ease of typing and use of ASCII text is preferred.) 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 linear 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). ( Z / 5 Z ) × = { [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] } ≅ C 4 {\displaystyle {\begin{aligned}(\mathbb {Z} /5\mathbb {Z} )^{\times }&=\{[1],[2],[3],[4]\}\\&\cong \mathrm {C} _{4}\\\end{aligned}}}

⋉



⋊ ⋉ {\displaystyle \ltimes }

\ltimes



⋊ {\displaystyle \rtimes }

\rtimes semidirect product the semidirect product of group theory N ⋊ φ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ. Also, if G = N ⋊ φ H, then G is said to split over N.



(⋊ may also be written the other way round, as ⋉, or as ×.) D 2 n ≅ C n ⋊ C 2 {\displaystyle D_{2n}\cong \mathrm {C} _{n}\rtimes \mathrm {C} _{2}}

semijoin the semijoin of relational algebra R ⋉ S is the semijoin of the relations R and S, the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names. R ⋉ {\displaystyle \ltimes } S = Π {\displaystyle \Pi } a 1 ,..,a n (R ⋈ {\displaystyle \bowtie } S)

⋈ ⋈ {\displaystyle \bowtie }

\bowtie natural join the natural join of relational algebra R ⋈ S is the natural join of the relations R and S, the set of all combinations of tuples in R and S that are equal on their common attribute names.



ℤ



Z Z {\displaystyle \mathbb {Z} }

\mathbb{Z}



Z {\displaystyle \mathbf {Z} }

\mathbf{Z} integers the (set of) integers numbers ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}. ℤ+ or ℤ> means {1, 2, 3, ...} .

ℤ≥ means {0, 1, 2, 3, ...} .

ℤ* is used by some authors to mean {0, 1, 2, 3, ...}[25] and others to mean {... -2, -1, 1, 2, 3, ... }[26] . ℤ = {p, −p : p ∈ ℕ ∪ {0}}

ℤ n



ℤ p



Z n



Z p Z n {\displaystyle \mathbb {Z} _{n}}

\mathbb{Z}_n



Z p {\displaystyle \mathbb {Z} _{p}}

\mathbb{Z}_p



Z n {\displaystyle \mathbf {Z} _{n}}

\mathbf{Z}_n



Z p {\displaystyle \mathbf {Z} _{p}} integers mod n the (set of) integers modulo n numbers ℤ n means {[0], [1], [2], ...[n−1]} with addition and multiplication modulo n.



Note that any letter may be used instead of n, such as p. To avoid confusion with p-adic numbers, use ℤ/pℤ or ℤ/(p) instead. ℤ 3 = {[0], [1], [2]}