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# Inequality (mathematics)

In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality). [1]

79 relations: Addition, Additive inverse, Arithmetic mean, Azuma's inequality, Bell's theorem, Bernoulli's inequality, Binary relation, Boole's inequality, Bracket, Bracket (mathematics), Cauchy–Schwarz inequality, Chebyshev's inequality, Chernoff bound, Complex number, Conic section, Converse relation, Cramér–Rao bound, Division (mathematics), Domain of a function, Ed Pegg Jr., Elsevier, Equality (mathematics), Field (mathematics), Fourier–Motzkin elimination, Function (mathematics), Geometric mean, Harmonic mean, Hölder's inequality, Hoeffding's inequality, Inequality of arithmetic and geometric means, Inequation, Integer, Interval (mathematics), Jensen's inequality, Kissing number problem, Kolmogorov's inequality, Lexicographical order, List of inequalities, List of mathematics competitions, List of order structures in mathematics, List of triangle inequalities, Logical conjunction, Logical equivalence, Markov's inequality, Material conditional, Mathematician, Mathematics, Measure (mathematics), Minkowski inequality, Monotonic function, ... Expand index (29 more) »

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

In mathematics, the additive inverse of a number is the number that, when added to, yields zero.

## Arithmetic mean

In mathematics and statistics, the arithmetic mean (stress on third syllable of "arithmetic"), or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection.

## Azuma's inequality

In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences.

## Bell's theorem

Bell's theorem is a "no-go theorem" that draws an important distinction between quantum mechanics and the world as described by classical mechanics.

## Bernoulli's inequality

In real analysis, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + x.

## Binary relation

In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

## Boole's inequality

In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events.

## Bracket

A bracket is a tall punctuation mark typically used in matched pairs within text, to set apart or interject other text.

## Bracket (mathematics)

In mathematics, various typographical forms of brackets are frequently used in mathematical notation such as parentheses, square brackets, braces, and angle brackets ⟨.

## Cauchy–Schwarz inequality

In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas.

## Chebyshev's inequality

In probability theory, Chebyshev's inequality (also spelled as Tchebysheff's inequality, Нера́венство Чебышёва, also called Bienaymé-Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean.

## Chernoff bound

In probability theory, the Chernoff bound, named after Herman Chernoff but due to Herman Rubin, gives exponentially decreasing bounds on tail distributions of sums of independent random variables.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Conic section

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

## Converse relation

In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation.

## Cramér–Rao bound

In estimation theory and statistics, the Cramér–Rao bound (CRB), Cramér–Rao lower bound (CRLB), Cramér–Rao inequality, Frechet–Darmois–Cramér–Rao inequality, or information inequality expresses a lower bound on the variance of unbiased estimators of a deterministic (fixed, though unknown) parameter.

## Division (mathematics)

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Ed Pegg Jr.

Ed Pegg Jr. (born December 7, 1963) is an expert on mathematical puzzles and is a self-described recreational mathematician.

## Elsevier

Elsevier is an information and analytics company and one of the world's major providers of scientific, technical, and medical information.

## Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Fourier–Motzkin elimination

Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Geometric mean

In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum).

## Harmonic mean

In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average, and in particular one of the Pythagorean means.

## Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of ''Lp'' spaces.

## Hoeffding's inequality

In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount.

## Inequality of arithmetic and geometric means

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.

## Inequation

In mathematics, an inequation is a statement that an inequality holds between two values.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

## Jensen's inequality

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.

## Kissing number problem

In geometry, a kissing number is defined as the number of non-overlapping unit spheres that can be arranged such that they each touch another given unit sphere.

## Kolmogorov's inequality

In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound.

## Lexicographical order

In mathematics, the lexicographic or lexicographical order (also known as lexical order, dictionary order, alphabetical order or lexicographic(al) product) is a generalization of the way words are alphabetically ordered based on the alphabetical order of their component letters.

## List of mathematics competitions

Mathematics competitions or mathematical olympiads are competitive events where participants sit a mathematics test.

## List of order structures in mathematics

In mathematics, and more particularly in order theory, several different types of ordered set have been studied.

## List of triangle inequalities

In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions.

## Logical conjunction

In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true.

## Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.

## Markov's inequality

In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant.

## Material conditional

The material conditional (also known as material implication, material consequence, or simply implication, implies, or conditional) is a logical connective (or a binary operator) that is often symbolized by a forward arrow "→".

## Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

## Minkowski inequality

In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces.

## Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.

## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

## Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x&minus;1, is a number which when multiplied by x yields the multiplicative identity, 1.

## Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to.

## Negative number

In mathematics, a negative number is a real number that is less than zero.

## Nesbitt's inequality

In mathematics, Nesbitt's inequality is a special case of the Shapiro inequality.

## Number line

In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by \mathbb.

## Order of magnitude

An order of magnitude is an approximate measure of the number of digits that a number has in the commonly-used base-ten number system.

## Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations.

## Partially ordered group

In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b.

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## Pedoe's inequality

In geometry, Pedoe's inequality (also Neuberg-Pedoe inequality), named after Daniel Pedoe (1910-1998) and Joseph Jean Baptiste Neuberg (1840-1926), states that if a, b, and c are the lengths of the sides of a triangle with area &fnof;, and A, B, and C are the lengths of the sides of a triangle with area F, then with equality if and only if the two triangles are similar with pairs of corresponding sides (A, a), (B, b), and (C, c).

## Poincaré inequality

In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré.

## Programming language

A programming language is a formal language that specifies a set of instructions that can be used to produce various kinds of output.

## Property (philosophy)

In philosophy, mathematics, and logic, a property is a characteristic of an object; a red object is said to have the property of redness.

## Python (programming language)

Python is an interpreted high-level programming language for general-purpose programming.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Relational operator

In computer science, a relational operator is a programming language construct or operator that tests or defines some kind of relation between two entities.

## Root mean square

In statistics and its applications, the root mean square (abbreviated RMS or rms) is defined as the square root of the mean square (the arithmetic mean of the squares of a set of numbers).

## Row and column vectors

In linear algebra, a column vector or column matrix is an m &times; 1 matrix, that is, a matrix consisting of a single column of m elements, Similarly, a row vector or row matrix is a 1 &times; m matrix, that is, a matrix consisting of a single row of m elements Throughout, boldface is used for the row and column vectors.

## Samuelson's inequality

In statistics, Samuelson's inequality, named after the economist Paul Samuelson, also called the Laguerre&ndash;Samuelson inequality, after the mathematician Edmond Laguerre, states that every one of any collection x1,..., xn, is within uncorrected sample standard deviations of their sample mean.

## Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Subtraction

Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.

## Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

## Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

## Trichotomy (mathematics)

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.

## Unicode

Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems.

## Universal quantification

In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all".

## Wolfram Demonstrations Project

The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields.

## References

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