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Symbol in HTML Symbol in TEX Name Explanation Examples Read as Category = = equality is equal to; equals everywhere x = y means x and y represent the same thing or value. 2 = 2 1 + 1 = 2 ≠ \ne inequality is not equal to; does not equal everywhere x \ne y means that x and y do not represent the same thing or value. (The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.) 2 + 2 \ne 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 A_3 < S_3 ≪ ≫ \ll \!\, \gg \!\, significant (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 asymptotic comparison is of smaller order than, is of 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 ≪ ex ≤ ≥ \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. (The forms <= and >= are generally used in programming languages, where ease of typing and use of ASCII text is preferred.) 3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5 subgroup is a subgroup of group theory H ≤ G means H is a subgroup of G. Z ≤ Z A3 ≤ S3 reduction is reducible to computational complexity theory A ≤ B means the problem A can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction. If \exists f \in F \mbox{ . } \forall x \in \mathbb{N} \mbox{ . } x \in A \Leftrightarrow f(x) \in B then A \leq_{F} B ≦ ≧ \leqq \!\, \geqq \!\, congruence relation ...is less than ... is greater than... modular arithmetic 7k ≡ 28 (mod 2) is only true if k is an even integer. Assume that the problem requires k to be non-negative; the domain is defined as 0 ≦ k ≦ ∞. 10a ≡ 5 (mod 5) for 1 ≦ a ≦ 10 vector inequality ... is less than or equal... is greater than or equal... order theory x ≦ y means that each component of vector x is less than or equal to each corresponding component of vector y. x ≧ y means that each component of vector x is greater than or equal to each corresponding component of vector y. It is important to note that x ≦ y remains true if every element is equal. However, if the operator is changed, x ≤ y is true if and only if x ≠ y is also true. ≺ \prec \!\, Karp reduction is Karp reducible to; is polynomial-time many-one reducible to computational complexity theory L1 ≺ L2 means that the problem L1 is Karp reducible to L2.[1] If L1 ≺ L2 and L2 ∈ P, then L1 ∈ P. ∝ \propto \!\, proportionality is proportional to; varies as everywhere y ∝ x means that y = kx for some constant k. if y = 2x, then y ∝ x. Karp reduction[2] is Karp reducible to; is polynomial-time many-one reducible to computational complexity theory A ∝ B means the problem A can be polynomially reduced to the problem B. If L1 ∝ L2 and L2 ∈ P, then L1 ∈ P. + + \!\, 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} ± \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. 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 6 + (3 − 5) and 6 − (3 + 5). cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y). × \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). \begin{align} (\mathbb{Z} / 5\mathbb{Z})^\times & = \{ [1], [2], [3], [4] \} \\ & \cong C_4 \\ \end{align} * * \!\, multiplication times; multiplied by arithmetic a * b means the product of a and b. (Multiplication can also be denoted with × or ⋅, or even simple juxtaposition. * is generally used where ease of typing and use of ASCII text is preferred, such as programming languages.) 4 * 3 means the product of 4 and 3, or 12. convolution convolution; convolved with functional analysis f * g means the convolution of f and g. (f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau)\, d\tau. complex conjugate conjugate complex numbers z* means 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× as described above, or U(R). \begin{align} (\mathbb{Z} / 5\mathbb{Z})^\ast & = \{ [1], [2], [3], [4] \} \\ & \cong C_4 \\ \end{align} hyperreal numbers the (set of) hyperreals non-standard analysis *R means the set of hyperreal numbers. Other sets can be used in place of R. *N is the hypernatural numbers. Hodge dual Hodge dual; Hodge star linear algebra *v means the Hodge dual of a vector v. If v is a k-vector within an n-dimensional oriented inner product space, then *v is an (n−k)-vector. If \{e_i\} are the standard basis vectors of \mathbb{R}^5, *(e_1\wedge e_2\wedge e_3)= e_4\wedge e_5 · \cdot \!\, multiplication times; multiplied by arithmetic 3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56 dot product dot linear algebra u · v means the dot product of vectors u and v (1,2,5) · (3,4,−1) = 6 placeholder (silent) functional analysis A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. \|\cdot\| ⊗ \otimes \!\, tensor product, tensor product of modules tensor product of linear algebra V \otimes U means the tensor product of V and U.[3] 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}} {~\wedge\!\!\!\!\!\!\bigcirc~} Kulkarni–Nomizu product Kulkarni–Nomizu product tensor algebra Derived from the tensor product of two symmetric type (0,2) tensors; it has the algebraic symmetries of the Riemann tensor. f=g{\,\wedge\!\!\!\!\!\!\bigcirc\,}h has components f_{\alpha\beta\gamma\delta}=g_{\alpha\gamma}h_{\beta\delta}+g_{\beta\delta}h_{\alpha\gamma}-g_{\alpha\delta}h_{\beta\gamma}-g_{\beta\gamma}h_{\alpha\delta}. ÷ ⁄ \div \!\, / \!\, division (Obelus) divided by; over arithmetic 6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = 0.5 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) } √ \surd \!\, \sqrt{\ } \!\, square root the (principal) square root of real numbers \sqrt{x} means the nonnegative number whose square is x. \sqrt{4}=2 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 x \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. complex conjugate conjugate complex numbers \overline{z} means the complex conjugate of z. (z* can also be used for the conjugate of z, as described above.) \overline{3+4i} = 3-4i. finite sequence, tuple finite sequence, tuple model theory \overline{a} means the finite sequence/tuple (a_1,a_2, ... ,a_n).. \overline{a}:=(a_1,a_2, ... ,a_n). algebraic closure algebraic closure of field theory \overline{F} is the algebraic closure of the field F. The field of algebraic numbers is sometimes denoted as \overline{\mathbb{Q}} because it is the algebraic closure of the rational numbers {\mathbb{Q}}. topological closure (topological) closure of topology \overline{S} is the topological closure of the set S. This may also be denoted as cl(S) or Cl(S). In the space of the real numbers, \overline{\mathbb{Q}} = \mathbb{R} (the rational numbers are dense in the real numbers). â \hat a unit vector hat geometry \mathbf{\hat a} (pronounced "a hat") is the normalized version of vector \mathbf a, having length 1. estimator estimator for statistics \hat \theta is the estimator or the estimate for the parameter \theta. The estimator \mathbf{\hat \mu} = \frac {\sum_i x_i} {n} produces a sample estimate \mathbf{\hat \mu} (\mathbf x) for the mean \mu. |…| | \ldots | \!\, absolute value; 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 norm or Euclidean length or magnitude Euclidean norm of geometry |x| means the (Euclidean) length of vector x. For x = (3,-4) |\textbf{x}| = \sqrt{3^2 + (-4)^2} = 5 determinant determinant of matrix theory |A| means the determinant of the matrix A \begin{vmatrix} 1&2 \\ 2&9 \\ \end{vmatrix} = 5 cardinality cardinality of; size of; order of set theory |X| means the cardinality of the set X. (# 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.[4] ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ nearest integer function nearest integer to numbers ‖x‖ means the nearest integer to x. (This may also be written [x], ⌊x⌉, nint(x) or Round(x).) ‖1‖ = 1, ‖1.6‖ = 2, ‖−2.4‖ = −2, ‖3.49‖ = 3 | | \!\, conditional event given probability P(A|B) means the probability of the event a occurring given that b occurs. if X is a uniformly random day of the year P(X is May 25 | X is in May) = 1/31 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. such that such that; so that everywhere | means "such that", see ":" (described below). S = {(x,y) | 0 < y < f(x)} The set of (x,y) such that y is greater than 0 and less than f(x). ∣ ∤ \mid \!\, \nmid \!\, divisor, divides divides number theory a ∣ b means a divides b. a ∤ b means a does not divide b. (The symbol ∣ can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar | character is often used instead.) Since 15 = 3×5, it is true that 3 ∣ 15 and 5 ∣ 15. ∣∣ \mid\mid \!\, exact divisibility exactly divides number theory pa ∣∣ n means pa exactly divides n (i.e. pa divides n but pa+1 does not). 23 ∣∣ 360. ∥ ∦ ⋕ \| \!\, parallel is parallel to geometry x ∥ y means x is parallel to y. x ∦ y means x is not parallel to y. x ⋕ y means x is equal and parallel to y. (The symbol ∥ can be difficult to type, and its negation is rare, so two regular but slightly longer vertical bar || characters are often used instead.) If l ∥ m and m ⊥ n then l ⊥ n. incomparability is incomparable to order theory x ∥ y means x is incomparable to y. {1,2} ∥ {2,3} under set containment. # \# \!\, 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 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#Sm is homeomorphic to A, for any manifold A, and the sphere Sm. primorial primorial number theory n# is product of all prime numbers less than or equal to n. 12# = 2 × 3 × 5 × 7 × 11 = 2310 ℵ \aleph \!\, aleph number aleph set theory ℵα represents an infinite cardinality (specifically, the α-th one, where α is an ordinal). |ℕ| = ℵ0, which is called aleph-null. ℶ \beth \!\, beth number beth set theory ℶα represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ). \beth_1 = |P(\mathbb{N})| = 2^{\aleph_0}. |
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