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List of mathematical symbols
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This is a listing of common symbols found within all branches of mathematics. Symbols are used
in mathematical notation to express a formula or to replace a constant.

It is important to recognize that a mathematical concept is independent of the symbol chosen to
represent it when reading the list. The symbols below are usually synonymous with the
corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history
of mathematics) but in some situations a different convention may be used. For example, the
meaning of "≡" may represent congruence or a definition depending on context. Further, in
mathematical logic, the concept of numerical equality is sometimes represented by "≡" instead of
"=", with the latter taking the duty of representing equality of well-formed formulas. In short,
convention rather than the symbol dictates the meaning.

Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in
TEX, as an image.

Contents
[hide]

        1 Symbols
        2 Variations
        3 See also
     4 References
     5 External links



[edit] Symbols
           Name
Sy Sy
             Read as
mb mb
ol ol                                  Explanation                       Examples
in in           Category
HT TE
ML X
           equality

             is equal to;    x = y means x and y represent the    2=2
=               equals       same thing or value.                 1+1=2

              everywhere
                             x ≠ y means that x and y do not
           inequality
                             represent the same thing or value.

≠          is not equal to;
           does not equal
                            (The forms !=, /= or <> are
                            generally used in programming
                                                               2+2≠5

                            languages where ease of typing and
                everywhere
                            use of ASCII text is preferred.)
           strict
           inequality
                            x < y means x is less than y.
                                                               3<4
              is less than,
                                                               5>4
            is greater than x > y means x is greater than y.
<
              order theory
           proper
           subgroup
>                        H < G means H is a proper subgroup 5Z < Z
             is a proper
                         of G.                              A3 < S3
            subgroup of

             group theory
           (very) strict  x ≪ y means x is much less than y.
≪          inequality
                          x ≫ y means x is much greater than
                                                                  0.003 ≪ 1000000
     is much less y.
          than,

≫       is much
     greater than

       order theory
    asymptotic
    comparison
                      f ≪ g means the growth of f is
      is of smaller asymptotically bounded by g.
       order than,
                                                             x ≪ ex
      is of greater (This is I. M. Vinogradov's notation.
       order than Another notation is the Big O
                      notation, which looks like f = O(g).)
             analytic
     number theory
                      x ≤ y means x is less than or equal to
    inequality        y.

     is less than or x ≥ y means x is greater than or equal
        equal to,    to y.                                  3 ≤ 4 and 5 ≤ 5
     is greater than                                        5 ≥ 4 and 5 ≥ 5
       or equal to (The forms <= and >= are generally
                     used in programming languages
        order theory where ease of typing and use of
≤   subgroup
                     ASCII text is preferred.)


    is a subgroup                                           Z≤Z
                  H ≤ G means H is a subgroup of G.
≥         of                                                A3 ≤ S3

      group theory
    reduction                                               If
                  A ≤ B means the problem A can be
    is reducible to
                  reduced to the problem B. Subscripts
                  can be added to the ≤ to indicate
    computational                                      then
                  what kind of reduction.
      complexity
          theory
    congruence       7k ≡ 28 (mod 2) is only true if k is an
≦   relation         even integer. Assume that the
                     problem requires k to be non-
                                                             10a ≡ 5 (mod 5) for 1 ≦ a ≦
                                                             10
     ...is less than negative; the domain is defined as 0
      ... is greater ≦ k ≦ ∞.
≧       than...

           modular
         arithmetic
                      x ≦ y means that each component of
                      vector x is less than or equal to each
                      corresponding component of vector
    vector            y.
    inequality
                     x ≧ y means that each component of
    ... is less than vector x is greater than or equal to
     or equal... is each corresponding component of
    greater than or vector y.
         equal...
                     It is important to note that x ≦ y
       order theory 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.
    Karp reduction

         is Karp
      reducible to;
    is polynomial-
                    L ≺ L2 means that the problem L1 is If L1 ≺ L2 and L2 ∈ P, then L1
≺   time many-one 1
      reducible to
                    Karp reducible to L2.[1]            ∈ P.

     computational
       complexity
             theory
    proportionality

    is proportional
                    y ∝ x means that y = kx for some
           to;                                                 if y = 2x, then y ∝ x.
                    constant k.
        varies as


∝
       everywhere
    Karp
    reduction[2]
                    A ∝ B means the problem A can be
                                                        If L1 ∝ L2 and L2 ∈ P, then L1
         is Karp    polynomially reduced to the problem
                                                        ∈ P.
      reducible to; B.
    is polynomial-
    time many-one
     reducible to

     computational
       complexity
             theory
    addition

         plus;
                      4 + 6 means the sum of 4 and 6.     2+7=9
          add

         arithmetic
+   disjoint union
                                                          A1 = {3, 4, 5, 6} ∧ A2 = {7, 8,
     the disjoint
                  A1 + A2 means the disjoint union of     9, 10} ⇒
     union of ...
                  sets A1 and A2.                         A1 + A2 = {(3,1), (4,1), (5,1),
        and ...
                                                          (6,1), (7,2), (8,2), (9,2), (10,2)}
         set theory
    subtraction

        minus;
                      9 − 4 means the subtraction of 4
         take;                                            8−3=5
                      from 9.
       subtract

        arithmetic
    negative sign

       negative;
                    −3 means the negative of the number
−       minus;
    the opposite of
                    3.
                                                        −(−5) = 5


          arithmetic
    set-theoretic
    complement A − B means the set that contains all
                     the elements of A that are not in B.
         minus;                                            {1,2,4} − {1,3,4} = {2}
        without      (∖ can also be used for set-theoretic
                     complement as described below.)
          set theory
    plus-minus
                                                          The equation x = 5 ± √4, has
±   plus or minus 6 ± 3 means both 6 + 3 and 6 − 3.
                                                          two solutions, x = 7 and x = 3.
         arithmetic
    plus-minus
                  10 ± 2 or equivalently 10 ± 20%
                                                      If a = 100 ± 1 mm, then a ≥ 99
    plus or minus means the range from 10 − 2 to 10 +
                                                      mm and a ≤ 101 mm.
                  2.
     measurement
    minus-plus
                      6 ± (3 ∓ 5) means both 6 + (3 − 5)    cos(x ± y) = cos(x) cos(y) ∓
∓   minus or plus
                      and 6 − (3 + 5).                      sin(x) sin(y).
         arithmetic
                      3 × 4 means the multiplication of 3
    multiplication
                      by 4.
        times;
                    (The symbol * is generally used in 7 × 8 = 56
     multiplied by
                    programming languages, where ease
                    of typing and use of ASCII text is
         arithmetic
                    preferred.)
    Cartesian
    product

     the Cartesian
                      X×Y means the set of all ordered
     product of ...
                      pairs with the first element of each {1,2} × {3,4} =
        and ...;
                      pair selected from X and the second {(1,3),(1,4),(2,3),(2,4)}
       the direct
                      element selected from Y.
×    product of ...
         and ...

         set theory
    cross product
                      u × v means the cross product of      (1,2,5) × (3,4,−1) =
         cross
                      vectors u and v                       (−22, 16, − 2)
     linear algebra
    group of units R× consists of the set of units of the
                    ring R, along with the operation of
     the group of multiplication.
        units of
                    This may also be written R* as
        ring theory described below, or U(R).
    multiplication a * b means the product of a and b.

       times;      (Multiplication can also be denoted 4 * 3 means the product of 4
*   multiplied by with × or ⋅, or even simple          and 3, or 12.
                   juxtaposition. * is generally used
        arithmetic where ease of typing and use of
                     ASCII text is preferred, such as
                     programming languages.)
    convolution

     convolution;
      convolved f * g means the convolution of f and
         with     g.
                                                            .
         functional
            analysis
    complex
    conjugate        z* means the complex conjugate of
                     z.
       conjugate                                                                     .
                     ( can also be used for the
           complex conjugate of z, as described below.)
           numbers
    group of units R* consists of the set of units of the
                     ring R, along with the operation of
     the group of multiplication.
        units of
                     This may also be written R× as
        ring theory described above, or U(R).
    hyperreal
    numbers
                     *R means the set of hyperreal
      the (set of)                                          *N is the hypernatural
                     numbers. Other sets can be used in
      hyperreals                                            numbers.
                     place of R.
     non-standard
          analysis
    Hodge dual
                    *v means the Hodge dual of a vector
                                                        If  are the standard basis
     Hodge dual; v. If v is a k-vector within an n-
      Hodge star dimensional oriented inner product vectors of     ,
                    space, then *v is an (n−k)-vector.
     linear algebra
    multiplication

       times;     3 · 4 means the multiplication of 3
                                                            7 · 8 = 56
    multiplied by by 4.
·
         arithmetic
    dot product     u · v means the dot product of
                                                            (1,2,5) · (3,4,−1) = 6
                    vectors u and v
          dot

     linear algebra
    placeholder
                      A · means a placeholder for an
                      argument of a function. Indicates the
         (silent)
                      functional nature of an expression
                      without assigning a specific symbol
          functional
                      for an argument.
             analysis
    tensor product,
    tensor product
    of modules                 means the tensor product of
                                                            {1, 2, 3, 4} ⊗ {1, 1, 2} =
⊗                     V and U.[3]           means the
     tensor product tensor product of modules V and U
                                                            {{1, 2, 3, 4}, {1, 2, 3, 4}, {2,
                                                            4, 6, 8}}
            of        over the ring R.

     linear algebra
    division
    (Obelus)
                                                         2 ÷ 4 = 0.5
                  6 ÷ 3 or 6 ⁄ 3 means the division of 6
      divided by;
                  by 3.
         over                                            12 ⁄ 4 = 3

÷        arithmetic
    quotient group
                                                          {0, a, 2a, b, b+a, b+2a} / {0,
                      G / H means the quotient of group G
         mod                                              b} = {{0, b}, {a, b+a}, {2a,
⁄     group theory
                      modulo its subgroup H.
                                                          b+2a}}

    quotient set
                                                               If we define ~ by x ~ y ⇔ x −
                      A/~ means the set of all ~               y ∈ ℤ, then
         mod
                      equivalence classes in A.                ℝ/~ = { {x + n : n ∈ ℤ } : x ∈
                                                               [0,1) }
         set theory
    square root

    the (principal)   means the nonnegative number
    square root of whose square is .

√     real numbers
    complex           if                   is represented in
    square root       polar coordinates with
                                      , then
    the (complex)
                                                   .
    square root of

           complex
           numbers
    mean

       overbar;          (often read as “x bar”) is the mean
        … bar          (average value of ).                                                   .

          statistics
    complex
    conjugate
                        means the complex conjugate of z.
      conjugate                                                                    .
                       (z* can also be used for the
                       conjugate of z, as described above.)
           complex
           numbers
    finite
    sequence,
    tuple
                        means the finite sequence/tuple
        finite                                                                         .
x     sequence,                           .
        tuple

      model theory
    algebraic
    closure                                                     The field of algebraic numbers
                                                                is sometimes denoted as
      algebraic           is the algebraic closure of the field
                                                                because it is the algebraic
      closure of       F.
                                                                closure of the rational numbers
                                                                   .
       field theory
    topological
    closure            is the topological closure of the set In the space of the real
                    S.
     (topological)                                           numbers,           (the rational
       closure of This may also be denoted as cl(S) or       numbers are dense in the real
                                                             numbers).
                    Cl(S).
           topology
    unit vector
                       (pronounced "a hat") is the
â         hat       normalized version of vector ,
                    having length 1.
          geometry
     absolute value;                                                |3| = 3
     modulus
                    |x| means the distance along the real |–5| = |5| = 5
     absolute value line (or across the complex plane)
     of; modulus of between x and zero.                   |i|=1

            numbers                                                 | 3 + 4i | = 5
     Euclidean
     norm or
     Euclidean
     length or
     magnitude      |x| means the (Euclidean) length of             For x = (3,-4)

|       Euclidean
                    vector x.

…        norm of

|         geometry
     determinant
                         |A| means the determinant of the
     determinant of
                         matrix A
      matrix theory
     cardinality
                     |X| means the cardinality of the set X.
     cardinality of;
        size of;                                             |{3, 5, 7, 9}| = 4.
                     (# may be used instead as described
        order of
                     below.)
            set theory
     norm

        norm of;         || x || means the norm of the element
                                                               || x + y || ≤ || x || + || y ||
        length of        x of a normed vector space.[4]
||    linear algebra
…    nearest integer
     function
                         ||x|| means the nearest integer to x.
||   nearest integer
                                                                    ||1|| = 1, ||1.6|| = 2, ||−2.4|| = −2,
                     (This may also be written [x], ⌊x⌉,            ||3.49|| = 3
           to
                     nint(x) or Round(x).)
           numbers
     divisor,      a|b means a divides b.
∣    divides       a∤b means a does not divide b.
                                                                    Since 15 = 3×5, it is true that
                                                                    3|15 and 5|15.
         divides   (This symbol can be difficult to type,
                   and its negation is rare, so a regular

∤    number theory but slightly shorter vertical bar |
                   character can be used.)
     conditional
     probability
                   P(A|B) means the probability of the if X is a uniformly random day
                   event a occurring given that b         of the year P(X is May 25 | X
         given
                   occurs.                                is in May) = 1/31
          probability
     restriction
                     f| means the function f restricted to
      restriction of A                                      The function f : R → R
                     the set A, that is, it is the function
         … to …;                                            defined by f(x) = x2 is not
                     with domain A ∩ dom(f) that agrees
       restricted to                                        injective, but f|R+ is injective.
                     with f.
          set theory
     such that
                                                              S = {(x,y) | 0 < y < f(x)}
       such that;       | means “such that”, see ":"          The set of (x,y) such that y is
        so that         (described below).                    greater than 0 and less than
                                                              f(x).
        everywhere
     parallel

      is parallel to x || y means x is parallel to y.         If l || m and m ⊥ n then l ⊥ n.

          geometry
     incomparabilit
     y

          is                                                  {1,2} || {2,3} under set
                  x || y means x is incomparable to y.
||   incomparable
          to
                                                              containment.


        order theory
     exact
     divisibility
                     pa || n means pa exactly divides n
                                                              23 || 360.
     exactly divides (i.e. pa divides n but pa+1 does not).

     number theory
     cardinality     #X means the cardinality of the set
#    cardinality of;
                     X.                                       #{4, 6, 8} = 3
       size of;       (|…| may be used instead as
       order of       described above.)

        set theory
    connected sum

    connected sum
                   A#B is the connected sum of the
          of;
                   manifolds A and B. If A and B are          A#Sm is homeomorphic to A,
     knot sum of;
                   knots, then this denotes the knot          for any manifold A, and the
         knot
                   sum, which has a slightly stronger         sphere Sm.
    composition of
                   condition.
    topology, knot
            theory
    primorial
                      n# is product of all prime numbers      12# = 2 × 3 × 5 × 7 × 11 =
      primorial
                      less than or equal to n.                2310
    number theory
    aleph number
                      ℵα represents an infinite cardinality
                                                              |ℕ| = ℵ0, which is called aleph-
ℵ       aleph         (specifically, the α-th one, where α is
                      an ordinal).
                                                              null.
         set theory
    beth number
                      ℶα represents an infinite cardinality
                      (similar to ℵ, but ℶ does not
ℶ        beth
                      necessarily index all of the numbers
                      indexed by ℵ. ).
         set theory
    cardinality of
    the continuum

     cardinality of
    the continuum; The cardinality of is denoted by
           c;          or by the symbol (a lowercase
     cardinality of Fraktur letter C).
        the real
       numbers

         set theory
    such that
                      : means “such that”, and is used in
:     such that;
       so that
                      proofs and the set-builder notation
                      (described below).
                                                              ∃ n ∈ ℕ: n is even.
        everywhere
    field extension
                       K : F means the field K extends the
       extends;        field F.
                                                              ℝ:ℚ
         over
                       This may also be written as K ≥ F.
       field theory
                  A : B means the Frobenius inner
    inner product
                  product of the matrices A and B.
    of matrices
                     The general inner product is denoted
     inner product
                     by ⟨u, v⟩, ⟨u | v⟩ or (u | v), as
            of
                     described below. For spatial vectors,
                     the dot product notation, x·y is
      linear algebra
                     common. See also Bra-ket notation.
    index of a
    subgroup
                     The index of a subgroup H in a
                     group G is the "relative size" of H in
         index of
                     G: equivalently, the number of
        subgroup
                     "copies" (cosets) of H that fill up G
      group theory
    factorial

       factorial       n! means the product 1 × 2 × ... × n. 4! = 1 × 2 × 3 × 4 = 24

    combinatorics
                       The statement !A is true if and only
                       if A is false.
    logical
!   negation           A slash placed through another
                       operator is the same as "!" placed in
                                                             !(!A) ⇔ A
          not          front.
                                                             x ≠ y ⇔ !(x = y)
      propositional (The symbol ! is primarily from
               logic computer science. It is avoided in
                     mathematical texts, where the
                     notation ¬A is preferred.)
    probability
    distribution
                  X ~ D, means the random variable X X ~ N(0,1), the standard
~         has
     distribution
                  has the probability distribution D. normal distribution

          statistics
    row
    equivalence
                   A~B means that B can be generated
        is row     by using a series of elementary row
     equivalent to operations on A

      matrix theory
    same order of
    magnitude
                   m ~ n means the quantities m and n
                   have the same order of magnitude, or 2 ~ 5
       roughly
                   general size.
       similar;
                                                        8 × 9 ~ 100
        poorly
                   (Note that ~ is used for an
     approximates
                   approximation that is poor,          but π2 ≈ 10
                   otherwise use ≈ .)
    approximation
            theory
    asymptotically
    equivalent

          is
    asymptotically                                            x ~ x+1
     equivalent to f ~ g means                      .

         asymptotic
            analysis
    equivalence
    relation

    are in the same a ~ b means            (and
                                                              1 ~ 5 mod 4
      equivalence equivalently             ).
          class

       everywhere
    approximately
    equal
                       x ≈ y means x is approximately equal
                       to y.
          is
                                                      π ≈ 3.14159
    approximately
                  This may also be written ≃, ≅, ~, ♎
≈      equal to
                  (Libra Symbol), or ≒.
         everywhere
    isomorphism G ≈ H means that group G is                Q / {1, −1} ≈ V,
                    isomorphic (structurally identical) to where Q is the quaternion
     is isomorphic group H.                                group and V is the Klein four-
          to                                           group.
                   (≅ can also be used for isomorphic,
      group theory as described below.)
    wreath product
                   A ≀ H means the wreath product of            is isomorphic to the

≀   wreath product the group A by the group H.
      of … by …
                                                       automorphism group of the
                                                       complete bipartite graph on
                   This may also be written A wr H.    (n,n) vertices.
      group theory
    normal
    subgroup
                 N ◅ G means that N is a normal
     is a normal                                             Z(G) ◅ G
     subgroup of subgroup of group G.

      group theory
◅   ideal
                      I ◅ R means that I is an ideal of ring
     is an ideal of                                          (2) ◅ Z
                      R.
▻       ring theory
    antijoin
                     R ▻ S means the antijoin of the
     the antijoin of relations R and S, the tuples in R for
                     which there is not a tuple in S that is R S = R - R S
                     equal on their common attribute
          relational
                     names.
             algebra
                     N ⋊φ H is the semidirect product of
    semidirect
                     N (a normal subgroup) and H (a
    product
                     subgroup), with respect to φ. Also, if
                     G = N ⋊φ H, then G is said to split
     the semidirect
⋉
                     over N.
       product of
                      (⋊ may also be written the other way
      group theory
                      round, as ⋉, or as ×.)

⋊   semijoin
                     R ⋉ S is the semijoin of the relations
                     R and S, the set of all tuples in R for
    the semijoin of
                     which there is a tuple in S that is     R S=       a1,..,an(R   S)
                     equal on their common attribute
          relational
                     names.
            algebra
    natural join     R ⋈ S is the natural join of the
⋈                    relations R and S, the set of all
    the natural join combinations of tuples in R and S
          of           that are equal on their common
                       attribute names.
          relational
            algebra
    therefore

      therefore;                                                  All humans are mortal.
∴        so;
        hence
                       Sometimes used in proofs before
                       logical consequences.
                                                                  Socrates is a human. ∴
                                                                  Socrates is mortal.

       everywhere
    because
                                                                  3331 is prime ∵ it has no
∵      because;
        since
                       Sometimes used in proofs before
                       reasoning.
                                                                  positive integer factors other
                                                                  than itself and one.
       everywhere

■   end of proof

         QED;
                 Used to mark the end of a proof.
      tombstone;

□       Halmos
        symbol
                 (May also be written Q.E.D.)

       everywhere

∎                   It is the generalisation of the Laplace
    D'Alembertian operator in the sense that it is the
                    differential operator which is
▮   non-Euclidean invariant under the isometry group
      Laplacian of the underlying space and it
                    reduces to the Laplace operator if
    vector calculus restricted to time independent

‣                   functions.

                       A ⇒ B means if A is true then B is
⇒   material
    implication
                       also true; if A is false then nothing is
                       said about B.
       implies;                                           x = 2 ⇒ x2 = 4 is true, but x2 =
                       (→ may mean the same as ⇒, or it
→     if … then
                       may have the meaning for functions
                                                          4 ⇒ x = 2 is in general false
                                                          (since x could be −2).
                       given below.)
     propositional
    logic, Heyting
                   (⊃ may mean the same as ⇒,[5] or it
⊃          algebra
                   may have the meaning for superset
                     given below.)
    material

⇔   equivalence

    if and only if; A ⇔ B means A is true if B is true
                                                            x+5=y+2⇔x+3=y
          iff       and A is false if B is false.

↔    propositional
             logic
                     The statement ¬A is true if and only
                     if A is false.
    logical
                      A slash placed through another
¬   negation
                      operator is the same as "¬" placed in
                      front.                                ¬(¬A) ⇔ A
           not
                                                            x ≠ y ⇔ ¬(x = y)

˜     propositional
                logic
                      (The symbol ~ has many other uses,
                      so ¬ or the slash notation is
                      preferred. Computer scientists will
                      often use ! but this is avoided in
                      mathematical texts.)
    logical
    conjunction or
    meet in a
    lattice           The statement A ∧ B is true if A and
                      B are both true; else it is false.
          and;                                              n < 4 ∧ n >2 ⇔ n = 3 when n
          min;        For functions A(x) and B(x), A(x) ∧ is a natural number.
          meet        B(x) is used to mean min(A(x),
                      B(x)).
      propositional
       logic, lattice
              theory
∧   wedge product
                     u ∧ v means the wedge product of
        wedge
                     any multivectors u and v. In three
       product;
                     dimensional Euclidean space the
       exterior
                     wedge product and the cross product
       product
                     of two vectors are each other's
                     Hodge dual.
           exterior
           algebra
    exponentiation a ^ b means a raised to the power of
                    b                                   2^3 = 23 = 8
     … (raised) to
     the power of (a ^ b is more commonly written ab.
            …         The symbol ^ is generally used in
                      programming languages where ease
        everywhere of typing and use of plain ASCII text
                      is preferred.)
    logical
    disjunction or
    join in a
                      The statement A ∨ B is true if A or B
    lattice
                      (or both) are true; if both are false,
                      the statement is false.
                                                             n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n
∨           or;
          max;
                      For functions A(x) and B(x), A(x) ∨
                                                             is a natural number.
           join
                      B(x) is used to mean max(A(x),
                      B(x)).
      propositional
       logic, lattice
               theory
    exclusive or

          xor        The statement A ⊕ B is true when
                                                              (¬A) ⊕ A is always true, A ⊕
                     either A or B, but not both, are true.
                                                              A is always false.
      propositional A ⊻ B means the same.
⊕   logic, Boolean
             algebra
                     The direct sum is a special way of
    direct sum
⊻    direct sum of
                     combining several objects into one
                     general object.
                                                              Most commonly, for vector
                                                              spaces U, V, and W, the
                                                              following consequence is used:
                    (The bun symbol ⊕, or the                 U = V ⊕ W ⇔ (U = V + W) ∧
           abstract
                    coproduct symbol ∐, is used; ⊻ is         (V ∩ W = {0})
            algebra
                    only for logic.)
    universal
    quantification


∀       for all;
       for any;
                      ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n.
       for each

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

       there is;
        there are

     predicate logic
     uniqueness
     quantification

∃!    there exists
      exactly one
                   ∃! x: P(x) means there is exactly one
                   x such that P(x) is true.
                                                         ∃! n ∈ ℕ: n + 5 = 2n.


     predicate logic

=:

:=

≡                      x := y, y =: x or x ≡ y means x is
                       defined to be another name for y,
     definition
                       under certain assumptions taken in
                       context.
:    is defined as;
       is equal by
⇔     definition to
                    (Some writers use ≡ to mean
                    congruence).
        everywhere
                       P :⇔ Q means P is defined to be

≜                      logically equivalent to Q.




≝

≐
     congruence
                     △ABC ≅ △DEF means triangle
     is congruent to ABC is congruent to (has the same
≅          geometry
                     measurements as) triangle DEF.

     isomorphic      G ≅ H means that group G is
                     isomorphic (structurally identical) to        .
      is isomorphic group H.
            to
                      (≈ can also be used for isomorphic,
             abstract as described above.)
              algebra
     congruence
     relation

     ... is congruent
                      a ≡ b (mod n) means a − b is
≡      to ... modulo
              ...
                      divisible by n
                                                              5 ≡ 2 (mod 3)


            modular
          arithmetic
     set brackets
{,    the set of …
                        {a,b,c} means the set consisting of
                                                              ℕ = { 1, 2, 3, …}
                        a, b, and c.[6]
}          set theory

{:
}
     set builder
     notation
{|                 {x : P(x)} means the set of all x for
                                      [6]
      the set of … which P(x) is true. {x | P(x)} is the
                                                         {n ∈ ℕ : n2 < 20} = { 1, 2, 3,
                                                         4}
}       such that same as {x : P(x)}.

           set theory

{;
}
∅    empty set
                        ∅ means the set with no elements.[6]
     the empty set                                           {n ∈ ℕ : 1 < n2 < 4} = ∅
                        { } means the same.
{          set theory
}
                        a ∈ S means a is an element of the    (1/2)−1 ∈ ℕ
∈    set
                        set S;[6] a ∉ S means a is not an
    membership         element of S.[6]                       2−1 ∉ ℕ


∉    is an element
           of;
        is not an
       element of

       everywhere,
         set theory
                       (subset) A ⊆ B means every element

⊆   subset
                       of A is also an element of B.[7]
                                                              (A ∩ B) ⊆ A
                       (proper subset) A ⊂ B means A ⊆ B
     is a subset of                                      ℕ⊂ℚ
                       but A ≠ B.

⊂        set theory
                       (Some writers use the symbol ⊂ as if
                                                              ℚ⊂ℝ
                       it were the same as ⊆.)
                       A ⊇ B means every element of B is
⊇   superset           also an element of A.
                                                              (A ∪ B) ⊇ B
    is a superset of A ⊃ B means A ⊇ B but A ≠ B.
                                                              ℝ⊃ℚ
⊃         set theory (Some writers use the symbol ⊃ as if
                     it were the same as ⊇.)
    set-theoretic
    union
                  A ∪ B means the set of those
∪    the union of
       … or …;
                  elements which are either in A, or in A ⊆ B ⇔ (A ∪ B) = B
                  B, or in both.[7]
        union

          set theory
    set-theoretic
    intersection
                       A ∩ B means the set that contains all
∩     intersected
         with;
                       those elements that A and B have in {x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
                       common.[7]
       intersect

         set theory
                       A ∆ B means the set of elements in
    symmetric
                       exactly one of A or B.
∆   difference
                       (Not to be confused with delta, Δ,
                                                              {1,5,6,8} ∆ {2,5,8} = {1,2,6}
      symmetric
                       described below.)
      difference

          set theory
    set-theoretic
                     A ∖ B means the set that contains all
    complement
                     those elements of A that are not in

∖        minus;
        without
                     B.[7]
                                                           {1,2,3,4} ∖ {3,4,5,6} = {1,2}
                     (− can also be used for set-theoretic
                     complement as described above.)
          set theory
    function arrow

      from … to       f: X → Y means the function f maps Let f: ℤ → ℕ∪{0} be defined
→                     the set X into the set Y.          by f(x) := x2.
        set theory,
       type theory
    function arrow
                      f: a ↦ b means the function f maps      Let f: x ↦ x+1 (the successor
↦      maps to
                      the element a to the element b.         function).
         set theory
    function
    composition

∘     composed
        with
                      f∘g is the function, such that (f∘g)(x) if f(x) := 2x, and g(x) := x + 3,
                      = f(g(x)).[8]                           then (f∘g)(x) = 2(x + 3).

         set theory
                      For two matrices (or vectors) of the
                      same dimensions
    Hadamard          the Hadamard product is a matrix of
    product           the same dimensions
                                        with elements
o     entrywise
       product
                      given by
                                                        .
     linear algebra This is often used in matrix based
                    programming such as MATLAB
                    where the operation is done by A.*B

ℕ   natural
    numbers
                    N means either { 0, 1, 2, 3, ...} or {
                    1, 2, 3, ...}.
                                                              ℕ = {|a| : a ∈ ℤ} or ℕ = {|a| >
           N;         The choice depends on the area of       0: a ∈ ℤ}
      the (set of)    mathematics being studied; e.g.
N       natural       number theorists prefer the latter;
        numbers    analysts, set theorists and computer
                   scientists prefer the former. To avoid
           numbers 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 ≤.
     integers
ℤ           Z;
                      ℤ means {..., −3, −2, −1, 0, 1, 2, 3,
                      ...}.
       the (set of)                                            ℤ = {p, −p : p ∈ ℕ ∪ {0}}
         integers     ℤ or ℤ means {1, 2, 3, ...} . ℤ or
                       +     >                         *

Z                     ℤ≥ means {0, 1, 2, 3, ...} .
            numbers
     integers mod n ℤn means {[0], [1], [2], ...[n−1]}
ℤn                   with addition and multiplication
            Zn;      modulo n.
       the (set of)
                                                        ℤ3 = {[0], [1], [2]}
ℤp
         integers    Note that any letter may be used
        modulo n instead of n, such as p. To avoid
                     confusion with p-adic numbers, use
            numbers ℤ/pℤ or ℤ/(p) instead.
     p-adic integers
Zn
      the (set of) p-
       adic integers Note that any letter may be used
                      instead of p, such as n or l.
Zp           numbers
     projective
     space

            P;

ℙ
     the projective
         space;     ℙ means a space with a point at
     the projective infinity.                                    ,
          line;
     the projective
P        plane

          topology
     probability   ℙ(X) means the probability of the           If a fair coin is flipped,
                   event X occurring.                          ℙ(Heads) = ℙ(Tails) = 0.5.
    the probability
           of        This may also be written as P(X),
                     Pr(X), P[X] or Pr[X].
         probability
              theory
    rational
    numbers

ℚ          Q;
                                                           3.14000... ∈ ℚ
      the (set of)
                   ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.
        rational
                                                           π∉ℚ
       numbers;
Q    the rationals

           numbers
    real numbers

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

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

          numbers
    Big O notation The Big O notation describes the        If f(x) = 6x4 − 2x3 + 5 and g(x)
                   limiting behavior of a function,        = x4 , then
O     big-oh of when the argument tends towards a
                   particular value or infinity.
     Computational
         complexity
              theory
     infinity
                          ∞ is an element of the extended

∞        infinity
                          number line that is greater than all
                          real numbers; it often occurs in
                          limits.
             numbers
     floor

⌊        floor;
                          ⌊x⌋ means the floor of x, i.e. the
                          largest integer less than or equal to x.
                                                                     ⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2,
…       greatest
        integer;
                          (This may also be written [x],
                                                                     ⌊−2.6⌋ = −3

⌋
         entier
                          floor(x) or int(x).)
             numbers

⌈    ceiling
                     ⌈x⌉ means the ceiling of x, i.e. the
                     smallest integer greater than or equal
                                                            ⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3,
…        ceiling
                     to x.
                                                            ⌈−2.6⌉ = −2

⌉           numbers
                     (This may also be written ceil(x) or
                     ceiling(x).)
     nearest integer
⌊    function
                     ⌊x⌉ means the nearest integer to x.
                                                            ⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊-3.4⌉ = -3,
…    nearest integer
            to
                     (This may also be written [x], ||x||,  ⌊4.49⌉ = 4

⌉           numbers
                     nint(x) or Round(x).)


     degree of a
                                                                     [ℚ(√2) : ℚ] = 2
     field extension
[:                  [K : F] means the degree of the
                                                                     [ℂ : ℝ] = 2
      the degree of extension K : F.
]                                                                    [ℝ : ℚ] = ∞
         field theory
     equivalence
[]   class
                          [a] means the equivalence class of a,
                                                                Let a ~ b be true iff a ≡ b (mod
                          i.e. {x : x ~ a}, where ~ is an
          the                                                   5).
                          equivalence relation.
      equivalence
[,      class of
                          [a]R means the same, but with R as
                                                                     Then [2] = {…, −8, −3, 2, 7,
                                                                     …}.
                          the equivalence relation.
]              abstract
               algebra
     floor
                       [x] means the floor of x, i.e. the
                       largest integer less than or equal to x.
         floor;
[,      greatest
                       (This may also be written ⌊x⌋,
                                                              [3] = 3, [3.5] = 3, [3.99] = 3,
        integer;                                              [−3.7] = −4
,]       entier
                       floor(x) or int(x). Not to be confused
                       with the nearest integer function, as
                       described below.)
            numbers
     nearest integer
                     [x] means the nearest integer to x.
     function
                     (This may also be written ⌊x⌉, ||x||,        [2] = 2, [2.6] = 3, [-3.4] = -3,
     nearest integer
                     nint(x) or Round(x). Not to be               [4.49] = 4
           to
                     confused with the floor function, as
                     described above.)
           numbers
     Iverson
     bracket

       1 if true, 0    [S] maps a true statement S to 1 and [0=5]=0, [7>0]=1, [2 ∈
       otherwise       a false statement S to 0.            {2,3,4}]=1, [5 ∈ {2,3,4}]=0

       propositional
               logic
                       f[X] means { f(x) : x ∈ X }, the image
                       of the function f under the set X ⊆
     image             dom(f).

       image of … (This may also be written as f(X) if
        under … there is no risk of confusing the
                     image of f under X with the function
         everywhere application f of X. Another notation
                     is Im f, the image of f under its
                     domain.)
     closed interval
                                                                  0 and 1/2 are in the interval
     closed interval                                            . [0,1].

       order theory
     commutator
                       [g, h] = g−1h−1gh (or ghg−1h−1), if g,
                                                                  xy = x[x, y] (group theory).
         the           h ∈ G (a group).
     commutator of
                                                             [AB, C] = A[B, C] + [A, C]B
                    [a, b] = ab − ba, if a, b ∈ R (a ring or
                                                             (ring theory).
      group theory, commutative algebra).
        ring theory
     triple scalar
     product

       the triple [a, b, c] = a × b · c, the scalar
                                                                [a, b, c] = [b, c, a] = [c, a, b].
     scalar product product of a × b with c.
           of

     vector calculus
     function
     application
                       f(x) means the value of the function f
                                                              If f(x) := x2, then f(3) = 32 = 9.
             of        at the element x.

          set theory
                       f(X) means { f(x) : x ∈ X }, the image
                       of the function f under the set X ⊆
     image             dom(f).

      image of … (This may also be written as f[X] if
       under … there is a risk of confusing the image
                   of f under X with the function
        everywhere application f of X. Another notation
                   is Im f, the image of f under its
()                 domain.)
     combinations
                            means the number of
        (from) n
(,      choose r
                       combinations of r elements drawn
                       from a set of n elements.
)     combinatorics (This may also be written as nC .)
                                                        r
     precedence
     grouping
                  Perform the operations inside the             (8/4)/2 = 2/2 = 1, but 8/(4/2) =
      parentheses parentheses first.                            8/2 = 4.

          everywhere
     tuple             An ordered list (or sequence, or         (a, b) is an ordered pair (or 2-
                       horizontal vector, or row vector) of     tuple).
      tuple; n-tuple; values.
          ordered                                               (a, b, c) is an ordered triple (or
      pair/triple/etc; (Note that the notation (a,b) is         3-tuple).
        row vector; ambiguous: it could be an ordered
         sequence pair or an open interval. Set                 ( ) is the empty tuple (or 0-
                       theorists and computer scientists        tuple).
         everywhere often use angle brackets ⟨ ⟩ instead
                    of parentheses.)
     highest
     common
     factor

         highest      (a, b) means the highest common
        common        factor of a and b.
                                                             (3, 7) = 1 (they are coprime);
         factor;
                                                             (15, 25) = 5.
        greatest      (This may also be written hcf(a, b)
        common        or gcd(a, b).)
      divisor; hcf;
           gcd

     number theory

(,
)    open interval    .                                      4 is not in the interval (4, 18).
      open interval (Note that the notation (a,b) is
                                                         (0, +∞) equals the set of
                    ambiguous: it could be an ordered positive real numbers.
],     order theory pair or an open interval. The
                    notation ]a,b[ can be used instead.)
[
     multichoose
((    multichoose
                          means n multichoose k.
))   combinatorics

(,   left-open
     interval
]      half-open
        interval;                                           . (−1, 7] and (−∞, −1]
       left-open
],       interval

]      order theory
     right-open
[,   interval
                                                            . [4, 18) and [1, +∞)
)      half-open
        interval;
        right-open
          interval

[,      order theory

[
                       ⟨u,v⟩ means the inner product of u
                       and v, where u and v are members of
                       an inner product space.

                       Note that the notation ⟨u, v⟩ may be
                       ambiguous: it could mean the inner
                       product or the linear span.
      inner product
                                                               The standard inner product
                        There are many variants of the
       inner product                                           between two vectors x = (2, 3)
                        notation, such as ⟨u | v⟩ and (u | v),
             of                                                and y = (−1, 5) is:
                        which are described below. For
                                                               ⟨x, y⟩ = 2 × −1 + 3 × 5 = 13
                        spatial vectors, the dot product
        linear algebra
                        notation, x·y is common. For
                        matrices, the colon notation A : B
                        may be used. As ⟨ and ⟩ can be hard
                        to type, the more “keyboard
                        friendly” forms < and > are
⟨⟩                      sometimes seen. These are avoided
                        in mathematical texts.
                                                               for a time series :g(t) (t = 1,
      average                                                  2,...)
⟨,⟩      average of
                        let S be a subset of N for example,
                             represents the average of all the we can define the structure
                        element in S.                          functions Sq( ):
             statistics

                       ⟨S⟩ means the span of S ⊆ V. That is,
                       it is the intersection of all subspaces
                       of V which contain S.
      linear span      ⟨u1, u2, …⟩is shorthand for ⟨{u1, u2,
                       …}⟩.
       (linear) span
             of;
       linear hull of Note that the notation ⟨u, v⟩ may be .
                       ambiguous: it could mean the inner
        linear algebra product or the linear span.

                       The span of S may also be written as
                       Sp(S).
      subgroup
      generated by a     means the smallest subgroup of In S ,
      set            G (where S ⊆ G, a group) containing
                                                            3

                     every element of S.                                             and
       the subgroup
       generated by                 is shorthand for
                                                         .
                                  .
        group theory
      tuple                                                    is an ordered pair (or 2-
                       An ordered list (or sequence, or         tuple).
      tuple; n-tuple;
                       horizontal vector, or row vector) of
          ordered
                       values.                                              is an ordered triple (or
      pair/triple/etc;
        row vector;                                             3-tuple).
                       (The notation (a,b) is often used as
         sequence
                       well.)                                     is the empty tuple (or 0-
         everywhere                                             tuple).
                        ⟨u | v⟩ means the inner product of u
                        and v, where u and v are members of
                        an inner product space.[9] (u | v)
                        means the same.

⟨|⟩   inner product
                       Another variant of the notation is ⟨u,
                       v⟩ which is described above. For
       inner product
                       spatial vectors, the dot product
             of
                       notation, x·y is common. For
(|)     linear algebra
                       matrices, the colon notation A : B
                       may be used. As ⟨ and ⟩ can be hard
                       to type, the more “keyboard
                       friendly” forms < and > are
                       sometimes seen. These are avoided
                       in mathematical texts.
      ket vector
                                                                A qubit's state can be
        the ket …; |φ⟩ means the vector with label φ,           represented as α|0⟩+ β|1⟩,
|⟩     the vector … which is in a Hilbert space.                where α and β are complex
                                                                numbers s.t. |α|2 + |β|2 = 1.
       Dirac notation
      bra vector
                    ⟨φ| means the dual of the vector |φ⟩,
⟨|      the bra …;
      the dual of …
                    a linear functional which maps a ket
                    |ψ⟩ onto the inner product ⟨φ|ψ⟩.
      Dirac notation
    summation

     sum over …
∑   from … to …
         of                means a1 + a2 + … + an.
                                                                  = 12 + 22 + 32 + 42

                                                                  = 1 + 4 + 9 + 16 = 30
        arithmetic
    product

    product over
                                                                       =
    … from … to
                                                           (1+2)(2+2)(3+2)(4+2)
       … of                means a1a2···an.
                                                                  = 3 × 4 × 5 × 6 = 360
         arithmetic

∏   Cartesian
    product

     the Cartesian       means the set of all (n+1)-
      product of; tuples
       the direct
       product of        (y0, …, yn).
         set theory
                    A general construction which
    coproduct
                    subsumes the disjoint union of sets
                    and of topological spaces, the free
    coproduct over
                    product of groups, and the direct
∐    … from … to
          … of
                    sum of modules and vector spaces.
                    The coproduct of a family of objects
                    is essentially the "least specific"
           category
                    object to which each object in the
             theory
                    family admits a morphism.
                    Δx means a (non-infinitesimal)
    delta
                    change in x.
        delta;                                               is the gradient of a straight
                   (If the change becomes infinitesimal,
      change in                                          line
                   δ and even d are used instead. Not to
                   be confused with the symmetric
Δ
          calculus
                   difference, written ∆, above.)
    Laplacian
                                                         If ƒ is a twice-differentiable
                   The Laplace operator is a second      real-valued function, then the
       Laplace
                   order differential operator in n-     Laplacian of ƒ is defined by
       operator
                   dimensional Euclidean space
    vector calculus
    Dirac delta
    function
                                                              δ(x)
    Dirac delta of

     hyperfunction
    Kronecker
    delta

      Kronecker                                               δij
δ      delta of

     hyperfunction
    Functional
    derivative

      Functional
     derivative of

       Differential
         operators
    partial
    derivative
                      ∂f/∂xi means the partial derivative of
        partial;      f with respect to xi, where f is a     If f(x,y) := x2y, then ∂f/∂x = 2xy
           d          function on (x1, …, xn).

          calculus
    boundary

∂    boundary of ∂M means the boundary of M                   ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}

          topology
    degree of a
    polynomial      ∂f means the degree of the
                    polynomial f.
                                                            ∂(x2 − 1) = 2
       degree of
                    (This may also be written deg f.)
            algebra
    gradient
                    ∇f (x1, …, xn) is the vector of partial If f (x,y,z) := 3xy + z², then ∇f =
∇         del;      derivatives (∂f / ∂x1, …, ∂f / ∂xn).    (3y, 3x, 2z)
        nabla;
      gradient of

    vector calculus
    divergence

       del dot;                                              If
    divergence of                                            , then                                .
    vector calculus
    curl
                                                             If
           curl of                                           , then
                                                                                               .
    vector calculus
                      f ′(x) means the derivative of the
    derivative
                      function f at the point x, i.e., the
                      slope of the tangent to f at x.
      … prime;
′    derivative of
                    (The single-quote character ' is
                                                             If f(x) := x2, then f ′(x) = 2x

                    sometimes used instead, especially in
           calculus
                    ASCII text.)
    derivative
                      means the derivative of x with
        … dot;
•                   respect to time. That is
    time derivative                                       If x(t) := t2, then                          .
           of
                                      .
           calculus
    indefinite
    integral or
    antiderivative

      indefinite
                   ∫ f(x) dx means a function whose
     integral of                                             ∫x2 dx = x3/3 + C
                   derivative is f.
         the

∫
    antiderivative
          of

           calculus
    definite
                    ∫ab f(x) dx means the signed area
    integral
                    between the x-axis and the graph of
                                                         ∫ b x2 dx = b3/3 − a3/3;
                    the function f between x = a and x = a
     integral from
                    b.
     … to … of …
    with respect to

           calculus
    line integral      ∫C f ds means the integral of f along
                     the curve C,                         ,
       line/ path/ where r is a parametrization of C.
    curve/ integral
     of… along… (If the curve is closed, the symbol ∮
                     may be used instead, as described
            calculus below.)
                     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
    Contour          the enclosing surface. Instances
    integral;        where the latter requires
    closed line      simultaneous double integration, the
    integral         symbol ∯ would be more                 If C is a Jordan curve about 0,
∮        contour
                     appropriate. A third related symbol
                     is the closed volume integral,         then                    .
       integral of denoted by the symbol ∰.

           calculus 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.
    projection


π    Projection of                   restricts to the
                                       attribute set.
          relational
            algebra
     Pi
                   Used in various formulas involving
                   circles; π is equivalent to the amount
          pi;
                   of area a circle would take up in a
      3.1415926;
                   square of equal width with an area of A=πR2=314.16→R=10
        ≈22÷7
                   4 square units, roughly 3.14/4. It is
                   also the ratio of the circumference to
      mathematical
                   the diameter of a circle.
          constant
                       The selection         selects all
     selection
                       those tuples in for which holds
                       between the and the attribute. The
σ     Selection of
                        selection        selects all those
             relational tuples in for which holds
               algebra between the attribute and the value
                          .
     cover
                                                               {1, 8} <• {1, 3, 8} among the
     is covered by x <• y means that x is covered by y.        subsets of {1, 2, …, 10}
<:                                                             ordered by containment.
        order theory
     subtype
<·   is a subtype of
                       T1 <: T2 means that T1 is a subtype of If S <: T and T <: U then S <:
                       T2.                                    U (transitivity).
         type theory
     conjugate
     transpose

        conjugate
                       A† means the transpose of the
        transpose;
                       complex conjugate of A.[10]
†         adjoint;
                                                               If A = (aij) then A† = (aji).
        Hermitian
                       This may also be written A*T, AT*,
     adjoint/conjug
                       A*, AT or AT.
      ate/transpose

             matrix
          operations
     transpose
                       AT means A, but with its rows
T         transpose    swapped for columns.
                                                               If A = (aij) then AT = (aji).
              matrix This may also be written A', At or Atr.
           operations
    top element

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


⊤
      lattice theory
    top type
                   ⊤ means the top or universal type;
     the top type;
                   every type in the type system of          ∀ types T, T <: ⊤
          top
                   interest is a subtype of top.
       type theory
    perpendicular

         is       x ⊥ y means x is perpendicular to y;
                                                       If l ⊥ m and m ⊥ n in the plane,
    perpendicular or more generally x is orthogonal to
                                                       then l || n.
         to       y.

         geometry
    orthogonal
    complement
                       W⊥ means the orthogonal
     orthogonal/
                       complement of W (where W is a
    perpendicular
                       subspace of the inner product space Within       ,              .
     complement
                       V), the set of all vectors in V
          of;
                       orthogonal to every vector in W.
         perp

⊥    linear algebra
    coprime
                       x ⊥ y means x has no factor greater
     is coprime to                                           34 ⊥ 55.
                       than 1 in common with y.
     number theory
    independent
                   A ⊥ B means A is an event whose
    is independent
                   probability is independent of event       If A ⊥ B, then P(A|B) = P(A).
           of
                   B.
        probability
    bottom
    element         ⊥ means the smallest element of a
                                                             ∀x : x ∧ ⊥ = ⊥
                    lattice.
      the bottom
       element

      lattice theory
    bottom type
                       ⊥ means the bottom type (a.k.a. the
      the bottom
                       zero type or empty type); bottom is
         type;                                               ∀ types T, ⊥ <: T
                       the subtype of every type in the type
          bot
                       system.
       type theory
    comparability

    is comparable x ⊥ y means that x is comparable to         {e, π} ⊥ {1, 2, e, 3, π} under
          to      y.                                          set containment.

       order theory
    entailment
                       A ⊧ B means the sentence A entails

⊧       entails
                       the sentence B, that is in every
                       model in which A is true, B is also
                                                              A ⊧ A ∨ ¬A
                       true.
      model theory
    inference

        infers;
      is derived
         from          x ⊢ y means y is derivable from x.     A → B ⊢ ¬B → ¬A.

      propositional
⊢             logic,
    predicate logic
    partition

     is a partition
                    p ⊢ n means that p is a partition of n. (4,3,1,1) ⊢ 9,
           of
                                                            .
    number theory
    vertical
    ellipsis
                       Denotes that certain constants and
                       terms are missing out (i.e. for
        vertical
                       clarity) and that only the important
        ellipsis
                       terms are being listed.
       everywhere
[edit] Variations
In mathematics written in Arabic, some symbols may be reversed to make right-to-left writing
and reading easier. [11]

[edit] See also
      Greek letters used in mathematics, science, and engineering
      ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology)
      List of mathematical abbreviations
      Mathematical alphanumeric symbols
      Mathematical notation
      Notation in probability and statistics
      Physical constants
      Latin letters used in mathematics
      Table of logic symbols
      Table of mathematical symbols by introduction date
      Unicode mathematical operators

[edit] References
   1. ^ Rónyai, Lajos (1998), Algoritmusok(Algorithms), TYPOTEX, ISBN 963-9132-16-0
   2. ^ Berman, Kenneth A; Paul, Jerome L. (2005), Algorithms: Sequential, Parallel, and Distributed,
       Boston: Course Technology, p. 822, ISBN 0-534-42057-5
   3. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum
       Information, New York: Cambridge University Press, pp. 71–72, ISBN 0-521-63503-9, OCLC
       43641333
   4. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum
       Information, New York: Cambridge University Press, p. 66, ISBN 0-521-63503-9, OCLC
       43641333
   5. ^ Copi, Irving M.; Cohen, Carl (1990) [1953], "Chapter 8.3: Conditional Statements and Material
       Implication", Introduction to Logic (8th ed.), New York: Macmillan, pp. 268–269, ISBN 0-02-
       325035-6, LCCN 8937742
   6. ^ a b c d e Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 3, ISBN 0-
       412-60610-0
   7. ^ a b c d Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 4, ISBN 0-412-
       60610-0
   8. ^ Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 5, ISBN 0-412-
       60610-0
   9. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum
       Information, New York: Cambridge University Press, p. 62, ISBN 0-521-63503-9, OCLC
       43641333
   10. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum
       Information, New York: Cambridge University Press, pp. 69–70, ISBN 0-521-63503-9, OCLC
       43641333
   11. ^ M. Benatia, A. Lazrik, and K. Sami, "Arabic mathematical symbols in Unicode", 27th
       Internationalization and Unicode Conference, 2005.
[edit] External links
      The complete set of mathematics Unicode characters
      Jeff Miller: Earliest Uses of Various Mathematical Symbols
      Numericana: Scientific Symbols and Icons
      TCAEP - Institute of Physics
      GIF and PNG Images for Math Symbols
      Mathematical Symbols in Unicode
      Using Greek and special characters from Symbol font in HTML
      Unicode Math Symbols - a quick form for using unicode math symbols.
      DeTeXify handwritten symbol recognition — doodle a symbol in the box, and the
       program will tell you what its name is

Some Unicode charts of mathematical operators:

      Index of Unicode symbols
      Range 2100–214F: Unicode Letterlike Symbols
      Range 2190–21FF: Unicode Arrows
      Range 2200–22FF: Unicode Mathematical Operators
      Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A
      Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B
      Range 2A00–2AFF: Unicode Supplementary Mathematical Operators

Some Unicode cross-references:

      Short list of commonly used LaTeX symbols and Comprehensive LaTeX Symbol List
      MathML Characters - sorts out Unicode, HTML and MathML/TeX names on one page
      Unicode values and MathML names
      Unicode values and Postscript names from the source code for Ghostscript

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