The following table lists many specialized symbols commonly used in mathematics. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Basic mathematical symbols
Symbol  Name  Explanation  Examples  Read as  Category  =  equality  x = y means x and y represent the same thing or value.  1 + 1 = 2  is equal to; equals  everywhere  ≠ <> !=  inequation  x ≠ y means that x and y do not represent the same thing or value. (The symbols != and <> are primarily from computer science. They are avoided in mathematical texts.)  1 ≠ 2  is not equal to; does not equal  everywhere  < > ≪ ≫  strict inequality  x < y means x is less than y. x > y means x is greater than y. x ≪ y means x is much less than y. x ≫ y means x is much greater than y.  3 < 4 5 > 4. 0.003 ≪ 1000000 In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...
In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. ...
For the use of the < and > signs in punctuation, see Bracket. ...
 is less than, is greater than, is much less than, is much greater than  order theory  ≤ <= ≥ >=  inequality  x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y. (The symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.)  3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5  is less than or equal to, is greater than or equal to  order theory  ∝  proportionality  y ∝ x means that y = kx for some constant k.  if y = 2x, then y ∝ x  is proportional to; varies as  everywhere  +  addition  4 + 6 means the sum of 4 and 6.  2 + 7 = 9  plus  arithmetic  disjoint union  A_{1} + A_{2} means the disjoint union of sets A_{1} and A_{2}.  A_{1} = {1, 2, 3, 4} ∧ A_{2} = {2, 4, 5, 7} ⇒ A_{1} + A_{2} = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}  the disjoint union of ... and ...  set theory  −  subtraction  9 − 4 means the subtraction of 4 from 9.  8 − 3 = 5  minus  arithmetic  negative sign  −3 means the negative of the number 3.  −(−5) = 5  negative; minus  arithmetic  settheoretic complement  A − B means the set that contains all the elements of A that are not in B. ∖ can also be used for settheoretic complement as described below.  {1,2,4} − {1,3,4} = {2}  minus; without  set theory  ×  multiplication  3 × 4 means the multiplication of 3 by 4.  7 × 8 = 56  times  arithmetic  Cartesian product  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)}  the Cartesian product of ... and ...; the direct product of ... and ...  set theory  cross product  u × v means the cross product of vectors u and v  (1,2,5) × (3,4,−1) = (−22, 16, − 2)  cross  vector algebra  ·  multiplication  3 · 4 means the multiplication of 3 by 4.  7 · 8 = 56  times  arithmetic  dot product  u · v means the dot product of vectors u and v  (1,2,5) · (3,4,−1) = 6  dot  vector algebra  ÷ ⁄  division  6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.  2 ÷ 4 = .5 12 ⁄ 4 = 3  divided by  arithmetic  ±  plusminus  6 ± 3 means both 6 + 3 and 6  3.  The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.  plus or minus  arithmetic  plusminus  10 ± 2 or eqivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.  If a = 100 ± 1 mm, then a is ≥ 99 mm and ≤ 101 mm.
 plus or minus  measurement  ∓  minusplus  6 ± (3 ∓ 5) means both 6 + (3  5) and 6  (3 + 5).  cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).  minus or plus  arithmetic  √  square root  √x means the positive number whose square is x.  √4 = 2  the principal square root of; square root  real numbers  complex square root  if z = r exp(iφ) is represented in polar coordinates with π < φ ≤ π, then √z = √r exp(i φ/2).  √(1) = i  the complex square root of … square root  complex numbers  …  absolute value or modulus  x means the distance along the real line (or across the complex plane) between x and zero.  3 = 3 –5 = 5  i  = 1  3 + 4i  = 5  absolute value (modulus) of  numbers  Euclidean distance  x – y means the Euclidean distance between x and y.  For x = (1,1), and y = (4,5), x – y = √([1–4]^{2} + [1–5]^{2}) = 5  Euclidean distance between; Euclidean norm of  Geometry  Determinant  A means the determinant of the matrix A   determinant of  Matrix theory    divides  A single vertical bar is used to denote divisibility. ab means a divides b.  Since 15 = 3×5, it is true that 315 and 515.  divides  Number Theory  !  factorial  n ! is the product 1 × 2× ... × n.  4! = 1 × 2 × 3 × 4 = 24  factorial  combinatorics  T  transpose  Swap rows for columns  A_{ij} = (A^{T})_{ji}  transpose  matrix operations  ~  probability distribution  X ~ D, means the random variable X has the probability distribution D.  X ~ N(0,1), the standard normal distribution  has distribution  statistics  Row equivalence  A~B means that B can be generated by using a series of elementary row operations on A   is row equivalent to  Matrix theory  ⇒ → ⊃  material implication  A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B. → may mean the same as ⇒, or it may have the meaning for functions given below. ⊃ may mean the same as ⇒, or it may have the meaning for superset given below.  x = 2 ⇒ x^{2} = 4 is true, but x^{2} = 4 ⇒ x = 2 is in general false (since x could be −2).  implies; if … then  propositional logic, Heyting algebra  ⇔ ↔  material equivalence  A ⇔ B means A is true if B is true and A is false if B is false.  x + 5 = y +2 ⇔ x + 3 = y  if and only if; iff  propositional logic  ¬ ˜  logical negation  The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front. (The symbol ~ has many other uses, so ¬ or the slash notation is preferred.)  ¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y)  not  propositional logic  ∧  logical conjunction or meet in a lattice  The statement A ∧ B is true if A and B are both true; else it is false. For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).  n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.  and; min  propositional logic, lattice theory  ∨  logical disjunction or join in a lattice  The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).  n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number.  or; max  propositional logic, lattice theory  ⊕ ⊻  exclusive or  The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.  (¬A) ⊕ A is always true, A ⊕ A is always false.  xor  propositional logic, Boolean algebra  direct sum  The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).
 Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = ∅)  direct sum of  Abstract algebra  ∀  universal quantification  ∀ x: P(x) means P(x) is true for all x.  ∀ n ∈ ℕ: n^{2} ≥ n.  for all; for any; for each  predicate logic  ∃  existential quantification  ∃ x: P(x) means there is at least one x such that P(x) is true.  ∃ n ∈ ℕ: n is even.  there exists  predicate logic  ∃!  uniqueness quantification  ∃! x: P(x) means there is exactly one x such that P(x) is true.  ∃! n ∈ ℕ: n + 5 = 2n.  there exists exactly one  predicate logic  := ≡ :⇔  definition  x := y or x ≡ y means x is defined to be another name for y (Some writers use ≡ to mean congruence). P :⇔ Q means P is defined to be logically equivalent to Q.  cosh x := (1/2)(exp x + exp (−x)) A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B)  is defined as  everywhere  ≅  congruence  △ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.   is congruent to  geometry  ≡  congruence relation  a ≡ b (mod n) means a − b is divisible by n  5 ≡ 11 (mod 3)  ... is congruent to ... modulo ...  modular arithmetic  { , }  set brackets  {a,b,c} means the set consisting of a, b, and c.  ℕ = { 1, 2, 3, …}  the set of …  set theory  { : } {  }  set builder notation  {x : P(x)} means the set of all x for which P(x) is true. {x  P(x)} is the same as {x : P(x)}.  {n ∈ ℕ : n^{2} < 20} = { 1, 2, 3, 4}  the set of … such that  set theory  ∅ { }  empty set  ∅ means the set with no elements. { } means the same.  {n ∈ ℕ : 1 < n^{2} < 4} = ∅  the empty set  set theory  ∈ ∉  set membership  a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S.  (1/2)^{−1} ∈ ℕ 2^{−1} ∉ ℕ  is an element of; is not an element of  everywhere, set theory  ⊆ ⊂  subset  (subset) A ⊆ B means every element of A is also element of B. (proper subset) A ⊂ B means A ⊆ B but A ≠ B. (Some writers use the symbol ⊂ as if it were the same as ⊆.)  (A ∩ B) ⊆ A ℕ ⊂ ℚ ℚ ⊂ ℝ  is a subset of  set theory  ⊇ ⊃  superset  A ⊇ B means every element of B is also element of A. A ⊃ B means A ⊇ B but A ≠ B. (Some writers use the symbol ⊃ as if it were the same as ⊇.)  (A ∪ B) ⊇ B ℝ ⊃ ℚ  is a superset of  set theory  ∪  settheoretic union  (exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both. "A or B, but not both." (inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B. "A or B or both".  A ⊆ B ⇔ (A ∪ B) = B (inclusive)  the union of … and … union  set theory  ∩  settheoretic intersection  A ∩ B means the set that contains all those elements that A and B have in common.  {x ∈ ℝ : x^{2} = 1} ∩ ℕ = {1}  intersected with; intersect  set theory  Δ  symmetric difference  AΔB means the set of elements in exactly one of A or B.  {1,5,6,8} Δ {2,5,8} = {1,2,6}  symmetric difference  set theory  ∖  settheoretic complement  A ∖ B means the set that contains all those elements of A that are not in B. − can also be used for settheoretic complement as described above.  {1,2,3,4} ∖ {3,4,5,6} = {1,2}  minus; without  set theory  ( )  function application  f(x) means the value of the function f at the element x.  If f(x) := x^{2}, then f(3) = 3^{2} = 9.  of  set theory  precedence grouping  Perform the operations inside the parentheses first.  (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.  parentheses  everywhere  f:X→Y  function arrow  f: X → Y means the function f maps the set X into the set Y.  Let f: ℤ → ℕ be defined by f(x) := x^{2}.  from … to  set theory,type theory  o  function composition  fog is the function, such that (fog)(x) = f(g(x)).  if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).  composed with  set theory  ℕ N  natural numbers  N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.  ℕ = {a : a ∈ ℤ, a ≠ 0}  N  numbers  ℤ Z  integers  ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ^{+} means {1, 2, 3, ...} = ℕ.  ℤ = {p, p : p ∈ ℕ} ∪ {0}  Z  numbers  ℚ Q  rational numbers  ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.  3.14000... ∈ ℚ π ∉ ℚ  Q  numbers  ℝ R  real numbers  ℝ means the set of real numbers.  π ∈ ℝ √(−1) ∉ ℝ  R  numbers  ℂ C  complex numbers  ℂ means {a + b i : a,b ∈ ℝ}.  i = √(−1) ∈ ℂ  C  numbers  arbitrary constant  C can be any number, most likely unknown; usually occurs when calculating antiderivatives.  if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C, where F'(x) = f(x)  C  integral calculus  𝕂 K  real or complex numbers  K means the statement holds substituting K for R and also for C. 
because Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
For the use of the < and > signs in punctuation, see Bracket. ...
Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. ...
3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ...
The plus and minus signs (+ and âˆ’) are used to represent the notions of positive and negative as well as the operations of addition and subtraction. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
5  2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ...
The plus and minus signs (+ and âˆ’) are used to represent the notions of positive and negative as well as the operations of addition and subtraction. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
A negative number is a number that is less than zero, such as âˆ’3. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In mathematics, multiplication is an elementary arithmetic operation. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
In mathematics, the Cartesian product is a direct product of sets. ...
An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
For the crossed product in algebra and functional analysis, see crossed product. ...
A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...
This article or section should be merged with Vector (spatial) Vector (spatial) addition follows the parallelogram law, is commutative and associative. ...
In mathematics, multiplication is an elementary arithmetic operation. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a realvalued scalar quantity. ...
A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...
This article or section should be merged with Vector (spatial) Vector (spatial) addition follows the parallelogram law, is commutative and associative. ...
In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
The plusminus sign (Â±) is a mathematical symbol commonly used to indicate the precision of an approximation, or as a convenient shorthand for a quantity which has two possible values opposite in sign. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
The plusminus sign (Â±) is a mathematical symbol commonly used to indicate the precision of an approximation, or as a convenient shorthand for a quantity which has two possible values opposite in sign. ...
A millimetre (American spelling: millimeter, symbol mm) is an SI unit of length that is equal to one thousandth of a metre. ...
Various meters Measurement is the estimation of a physical quantity such as length, temperature, or time. ...
The plusminus sign (Â±) is a mathematical symbol commonly used to indicate the precision of an approximation, or as a convenient shorthand for a quantity which has two possible values opposite in sign. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...
Please refer to Real vs. ...
In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...
A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a twodimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...
The complex numbers are an extension of the real numbers, in which all nonconstant polynomials have roots. ...
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...
Mathematical meanings Especially in British/European usage, the modulus of a number is its absolute value. ...
In mathematics, the real line is simply the set of real numbers. ...
In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...
0 (zero) is both a number and a numerical digit used to represent that number in numerals. ...
A number is an abstract idea used in counting and measuring. ...
In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...
CalabiYau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
Matrix theory is a branch of mathematics which focuses on the study of matrices. ...
In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...
For factorial rings in mathematics, see unique factorisation domain. ...
Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ...
In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A...
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...
In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...
A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...
Probability density function of Gaussian distribution (bell curve). ...
A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ...
Matrix theory is a branch of mathematics which focuses on the study of matrices. ...
In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...
Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ...
In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ...
In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ...
IFF, Iff or iff can stand for: Interchange File Format  a computer file format introduced by Electronic Arts Identification, friend or foe  a radio based identification system utilizing transponders iff  the mathematics concept if and only if International Flavors and Fragrances  a company producing flavors and fragrances International Freedom Foundation...
Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ...
Negation, in its most basic sense, changes the truth value of a statement to its opposite. ...
Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ...
AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a twoplace logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ...
The name lattice is suggested by the form of the Hasse diagram depicting it. ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ...
The name lattice is suggested by the form of the Hasse diagram depicting it. ...
OR logic gate. ...
The name lattice is suggested by the form of the Hasse diagram depicting it. ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ...
The name lattice is suggested by the form of the Hasse diagram depicting it. ...
Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ...
Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ...
In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ...
In abstract algebra, the direct sum is a construction which combines several modules into a new, bigger one. ...
Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
In predicate logic, universal quantification is an attempt to formalize the notion that something (a logical predicate) is true for everything, or every relevant thing. ...
...
In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. ...
...
In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ...
...
A definition is a concise statement explaining the meaning of a term, word or phrase. ...
As an abstract term, congruence means similarity between objects. ...
An example of congruence. ...
CalabiYau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...
In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ...
Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In set theory and its applications to logic, mathematics, and computer science, setbuilder notation is a mathematical notation for describing a set by indicating the properties that its members must satisfy. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
The empty set is the set containing no elements. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
Intuitionistic Type Theory, or Constructive Type Theory, or MartinLÃ¶f Type Theory or just Type Theory (with capital letters) is at the same time a functional programming language, a logic and a set theory based on the principles of mathematical constructivism. ...
In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
A number is an abstract idea used in counting and measuring. ...
The integers are commonly denoted by the above symbol. ...
A number is an abstract idea used in counting and measuring. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
A number is an abstract idea used in counting and measuring. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
A number is an abstract idea used in counting and measuring. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...
A number is an abstract idea used in counting and measuring. ...
In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...
In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...
This article deals with the concept of an integral in calculus. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...
and
 .
 K  linear algebra  ∞  infinity  ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.  lim_{x→0} 1/x = ∞  infinity  numbers  …  norm   x  is the norm of the element x of a normed vector space.   x + y  ≤  x  +  y   norm of length of  linear algebra  ∑  summation  means a_{1} + a_{2} + … + a_{n}. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
The infinity symbol âˆž in several typefaces. ...
The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...
Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...
A number is an abstract idea used in counting and measuring. ...
In mathematics, with 2 or 3dimensional vectors with realvalued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ...
In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
In mathematics, with 2 or 3dimensional vectors with realvalued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
For evaluation of sums in closed form see evaluating sums. ...
 = 1^{2} + 2^{2} + 3^{2} + 4^{2}

 = 1 + 4 + 9 + 16 = 30
 sum over … from … to … of  arithmetic  ∏  product  means a_{1}a_{2}···a_{n}. Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
 = (1+2)(2+2)(3+2)(4+2) 
 = 3 × 4 × 5 × 6 = 360
 product over … from … to … of  arithmetic  Cartesian product  means the set of all (n+1)tuples Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
In mathematics, the Cartesian product is a direct product of sets. ...
In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...

 (y_{0}, …, y_{n}).
  the Cartesian product of; the direct product of  set theory  ∐  coproduct    coproduct over … from … to … of  category theory  ′ ^{•}  derivative  f ′(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. The dot notation indicates a time derivative. That is . In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ...
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
For a nontechnical overview of the subject, see Calculus. ...
Look up Slope in Wiktionary, the free dictionary. ...
In mathematics, the word tangent has two distinct but etymologicallyrelated meanings: one in geometry and one in trigonometry. ...
 If f(x) := x^{2}, then f ′(x) = 2x  … prime derivative of  calculus  ∫  indefinite integral or antiderivative  ∫ f(x) dx means a function whose derivative is f.  ∫x^{2} dx = x^{3}/3 + C  indefinite integral of the antiderivative of  calculus  definite integral  ∫_{a}^{b} f(x) dx means the signed area between the xaxis and the graph of the function f between x = a and x = b.  ∫_{0}^{b} x^{2} dx = b^{3}/3;  integral from … to … of … with respect to  calculus  ∮  contour integral or closed line integral  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 this formula involves a closed surface integral, the representation describes only the first or initial integration of the volume, over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol ∯ would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol ∰. This symbol can also frequently be found with a subscript capital letter C, ∮_{C}, denoting that the closed loop integral is 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. Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ...
In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...
In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...
Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ...
This article deals with the concept of an integral in calculus. ...
Area is a physical quantity expressing the size of a part of a surface. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ...
This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...
This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...
In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ...
In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ...
In mathematics â€” in particular, in multivariable calculus â€” a volume integral refers to an integral over a 3dimensional domain. ...
  contour integral of  calculus  ∇  gradient  ∇f (x_{1}, …, x_{n}) is the vector of partial derivatives (∂f / ∂x_{1}, …, ∂f / ∂x_{n}).  If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)  del, nabla, gradient of  vector calculus  divergence   If , then .  del dot, divergence of  vector calculus  curl   If , then .  curl of  vector calculus  ∂  partial differential  With f (x_{1}, …, x_{n}), ∂f/∂x_{i} is the derivative of f with respect to x_{i}, with all other variables kept constant.  If f(x,y) := x^{2}y, then ∂f/∂x = 2xy  partial, d  calculus  boundary  ∂M means the boundary of M  ∂{x : x ≤ 2} = {x : x = 2}  boundary of  topology  ⊥  perpendicular  x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.  If l ⊥ m and m ⊥ n then l  n.  is perpendicular to  geometry  bottom element  x = ⊥ means x is the smallest element.  ∀x : x ∧ ⊥ = ⊥  the bottom element  lattice theory    parallel  x  y means x is parallel to y.  If l  m and m ⊥ n then l ⊥ n.  is parallel to  geometry  ⊧  entailment  A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true.  A ⊧ A ∨ ¬A  entails  model theory  ⊢  inference  x ⊢ y means y is derived from x.  A → B ⊢ ¬B → ¬A  infers or is derived from  propositional logic, predicate logic  ◅  normal subgroup  N ◅ G means that N is a normal subgroup of group G.  Z(G) ◅ G  is a normal subgroup of  group theory  /  quotient group  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}}  mod  group theory  quotient set  A/~ means the set of all ~ equivalence classes in A.  If we define ~ by x~y ⇔ xy∈Z, then R/~ = {{x+n : n∈Z} : x ∈ (0,1]}  mod  set theory  ≈  isomorphism  G ≈ H means that group G is isomorphic to group H  Q / {1, −1} ≈ V, where Q is the quaternion group and V is the Klein fourgroup.  is isomorphic to  group theory  approximately equal  x ≈ y means x is approximately equal to y  π ≈ 3.14159  is approximately equal to  everywhere  ~  same order of magnitude  m ~ n, means the quantities m and n have the general size. (Note that ~ is used for an approximation that is poor, otherwise use ≈ .)  2 ~ 5 8 × 9 ~ 100 but π^{2} ≈ 10  roughly similar poorly approximates  Approximation theory  〈,〉 (  ) < , > · :  inner product  〈x,y〉 means the inner product of x and y as defined in an inner product space. For spatial vectors, the dot product notation, x·y is common. For matricies, the colon notation may be used. Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ...
For other uses, see Gradient (disambiguation). ...
In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ...
Nabla is a symbol, shown as . ...
For other uses, see Gradient (disambiguation). ...
Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...
In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...
In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ...
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...
Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and power series and constitutes a major part of modern university curriculum. ...
In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...
A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...
Fig. ...
CalabiYau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ...
The name lattice is suggested by the form of the Hasse diagram depicting it. ...
Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ...
CalabiYau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ...
Implication or entailment is used in propositional logic and predicate logic to describe a relationship between two sentences or sets of sentences. ...
In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...
Inference is the act or process of deriving a conclusion based solely on what one already knows. ...
Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ...
...
In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
Group theory is that branch of mathematics concerned with the study of groups. ...
In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...
The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ...
Group theory is that branch of mathematics concerned with the study of groups. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X  x ~ a } The notion of equivalence classes is useful for constructing sets out...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ...
This article is about the mathematical group. ...
Group theory is that branch of mathematics concerned with the study of groups. ...
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ...
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ...
It has been suggested that this article or section be merged with estimation. ...
In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ...
In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a realvalued scalar quantity. ...
 The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is: 〈x, y〉 = 2×−1 + 3×5 = 13  inner product of  linear algebra  ⊗  tensor product  V ⊗ U means the tensor product of V and U.  {1, 2, 3, 4} ⊗ {1,1,2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}  tensor product of  linear algebra  *  convolution  f * g means the convolution of f and g.   convolution, convoluted with  functional analysis   mean  (often read as "x bar") is the mean (average value of x_{i}).  .  overbar, … bar  statistics   complex conjugate  is the complex conjugate of z.   conjugate  complex numbers   delta equal to  means equal by definition. When is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡.  .  equal by definition  everywhere  In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a realvalued scalar quantity. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ...
In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ...
A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ...
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...
The complex numbers are an extension of the real numbers, in which all nonconstant polynomials have roots. ...
See also Mathematical alphanumeric symbols are modifications of Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles (one example is blackboard bold, or doublestruck (in Unicode terminology)). Unicode now includes many such symbols (in the range U+1D400 . ...
In logic, a set of symbols is frequently used to express logical constructs. ...
Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...
ISO 3111 is the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology. ...
Many Roman letters, both capital and small, are used in mathematics, science and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, physical entities. ...
Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. ...
In probability theory and statistics, some special forms of mathematical notation are of interest : Random variables (for example, the height of students) are written in upper case. ...
In physics, a physical constant is a physical quantity of a value that is generally believed to be both universal in nature and not believed to change in time. ...
ISO 31 Category: ...
External links 