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Encyclopedia > Inequality
The feasible regions of linear programming are defined by a set of inequalities.
The feasible regions of linear programming are defined by a set of inequalities.

In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not (See also: equality) Look up inequality in Wiktionary, the free dictionary. ... For technical reasons, :) and some similar combinations starting with : redirect here. ... In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. ... A graph showing how a series of linear constraints on two variables produce a feasible region in a linear programming problem. ... A graph showing how a series of linear constraints on two variables produce a feasible region in a linear programming problem. ... In optimization (a branch of mathematics), a candidate solution is a member of a set of possible solutions to a given problem. ... In mathematics, linear programming (LP) problems involve the optimization of a linear objective function, subject to linear equality and inequality constraints. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...

  • The notation a < b ! means that a is less than b and
  • The notation a > b ! means that a is greater than b.
  • The notation a neq b means that a is not equal to b, but does not say that one is bigger than the other or even that they can be compared in size – they could be apples and oranges

In all these cases, a is not equal to b, hence, "inequality".


These relations are known as strict inequality; in contrast

  • a le b means that a is less than or equal to b (or, equivalently, not greater than b);
  • a ge b means that a is greater than or equal to b (or, equivalently, not smaller than b);

An additional use of the notation is to show that one quantity is much greater than another, normally by several orders of magnitude. 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. ...

  • The notation a ll b means that a is much less than b.
  • The notation a gg b means that a is much greater than b.

If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditional" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality.

Contents

Solving Inequalities

An inequality may appear unsolvable because it only states whether a number is larger or smaller than another number; but it is possible to apply the same operations for equalities to inequalities. For example, to find x for the inequality 10x > 23 one would divide 23 by 10.


Properties

Inequalities are governed by the following properties. Note that, for the transitivity, reversal, addition and subtraction, and multiplication and division properties, the property also holds if strict inequality signs (< and >) are replaced with their corresponding non-strict inequality sign (≤ and ≥). A property is an intrinsic or extrinsic quality of an object—where an object may be of any differing nature, depending on the context and field — be it computing, philosophy, etc. ...


Trichotomy

The trichotomy property states: For other uses, see trichotomy (disambiguation). ...

  • For any real numbers, a and b, exactly one of the following is true:
    • a < b
    • a = b
    • a > b

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Transitivity

The transitivity of inequalities states: In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ...

  • For any real numbers, a, b, c:
    • If a > b and b > c; then a > c
    • If a < b and b < c; then a < c

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Reversal

The inequality relations are inverse relations: In logic and mathematics, the inverse relation of a binary relation is the binary relation defined by . ...

  • For any real numbers, a and b:
    • If a > b then b < a
    • If a < b then b > a

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Addition and subtraction

The properties which deal with addition and subtraction state: 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ...

  • For any real numbers, a, b, c:
    • If a > b, then a + c > b + c and ac > bc
    • If a < b, then a + c < b + c and ac < bc

i.e., the real numbers are an ordered group. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In abstract algebra, an ordered group is a group G equipped with a partial order ≤ which is translation-invariant; in other words, ≤ has the property that, for all a, b, and g in G, if a ≤ b then ag ≤ bg and ga ≤ gb. ...


Multiplication and division

The properties which deal with multiplication and division state: In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...

  • For any real numbers, a, b, c:
    • If c is positive and a < b, then ac < bc
    • If c is negative and a < b, then ac > bc

More generally this applies for an ordered field, see below. A negative number is a number that is less than zero, such as −3. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, an ordered field is a field together with an ordering of its elements. ...


Additive inverse

The properties for the additive inverse state: The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ...

  • For any real numbers a and b
    • If a < b then −a > −b
    • If a > b then −a < −b

Multiplicative inverse

The properties for the multiplicative inverse state: The reciprocal function: y = 1/x. ...

  • For any real numbers a and b that are both positive or both negative
    • If a < b then 1/a > 1/b
    • If a > b then 1/a < 1/b

In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... Negative has meaning in several contexts: Look up negative in Wiktionary, the free dictionary. ...

Applying a function to both sides

We consider two cases of functions: monotonic and strictly monotonic.


Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold. Applying a strictly monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for additive and multiplicative inverses are both examples of applying a monotonically decreasing function. Monotonicity redirects here. ... This article is about functions in mathematics. ...


If you have a non-strict inequality (ab, ab) then:

  • Applying a monotonically increasing function preserves the relation (≤ remains ≤, ≥ remains ≥)
  • Applying a monotonically decreasing function reverses the relation (≤ becomes ≥, ≥ becomes ≤)

It will never become strictly unequal, since, for example, 3 ≤ 3 does not imply that 3 < 3.


Ordered fields

If F,+,* be a field and ≤ be a total order on F, then F,+,*,≤ is called an ordered field if and only if: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... In mathematics, an ordered field is a field together with an ordering of its elements. ...

  • if ab then a + cb + c
  • if 0 ≤ a and 0 ≤ b then 0 ≤ a b

Note that both mathbb{Q},+,*,≤ and mathbb{R},+,*,≤ are ordered fields. In mathematics, an ordered field is a field together with an ordering of its elements. ...


≤ cannot be defined in order to make mathbb{C},+,*,≤ an ordered field. In mathematics, an ordered field is a field together with an ordering of its elements. ...


The non-strict inequalities ≤ and ≥ on real numbers are total orders. The strict inequalities < and > on real numbers are strict total ordershttp://en.wikipedia.org/wiki/Total_order#Strict_total_order. In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ...


Chained notation

The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. But care must be taken so that you really use the same number in all cases, e.g. a < b + e < c is equivalent to ae < b < ce.


This notation can be generalized to any number of terms: for instance, a1a2 ≤ ... ≤ an means that aiai+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to aiaj for any 1 ≤ ijn.


When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance to solve the inequality 4x < 2x + 1 ≤ 3x + 2, you won't be able to isolate x in any one part of the inequality through addition or subtraction. Instead, you can solve 4x < 2x + 1 and 2x + 1 ≤ 3x + 2 independently, yielding x < 1/2 and x ≥ -1 respectively, which can be combined into the final solution -1 ≤ x < 1/2.


Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For instance, a < b > cd means that a < b, b > c, and cd. In addition to rare use in mathematics, this notation exists in a few programming languages such as Python. AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... Python is a general-purpose, high-level programming language. ...


Representing inequalities on the real number line

Every inequality (except those which involve imaginary numbers) can be represented on the real number line showing darkened regions on the line. In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is negative or zero. ... A number line, invented by John Wallis, is a one-dimensional picture in which the integers are shown as specially-marked points evenly spaced on a line. ...

Image File history File links This is a lossless scalable vector image. ...

Inequalities between means

There are many inequalities between means. For example, for any positive numbers a1, a2, ..., an

H le G le A le Q, where
H = frac{n}{frac{1}{a_1}+frac{1}{a_2}+...+frac{1}{a_n}} (harmonic mean),
G = sqrt[n]{a_1 cdot a_2 cdot ... cdot a_n} (geometric mean),
A = frac{a_1 + a_2 + ... + a_n}{n} (arithmetic mean),
Q = sqrt{frac{a_1^2 + a_2^2 + ... + a_n^2}{n}} (quadratic mean).

In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ...

Power inequalities

Sometimes with notation "power inequality" understand inequalities which contain ab type expressions where a and b are real positive numbers or expressions of some variables. They can appear in exercises of mathematical olympiads and some calculations.


Examples

  1. If x > 0, then x^x ge left( tfrac{1}{e}right)^ tfrac{1}{e}
  2. If x > 0, then x^{x^x} ge x
  3. If x,y,z > 0, then (x + y)z + (x + z)y + (y + z)x > 2.
  4. For any real distinct numbers a and b, tfrac{e^b-e^a}{b-a}>e^{frac{a+b}{2}}
  5. If x,y > 0 and 0 < p < 1, then (x + y)p < xp + yp
  6. If x, y and z are positive, then x^x y^y z^z ge (xyz)^ frac{x+y+z}{3}
  7. If a and b are positive, then ab + ba > 1. This result was generalized by R. Ozols in 2002 who proved that if a1, a2, ..., an are any real positive numbers, then a_1^{a_2}+a_2^{a_3}+...+a_n^{a_1}>1 (result is published in Latvian popular-scientific quarterly The Starry Sky, see references).

Well-known inequalities

See also list of inequalities. This page lists Wikipedia articles about particular mathematical inequalities. ...


Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names: Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ...

In probability theory, Azumas inequality gives a concentration result for the values of martingales that have bounded differences. ... In real analysis, Bernoullis inequality is an inequality that approximates exponentiations of 1 + x. ... In probability theory, Booles inequality, named after George Boole, (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. ... In mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Bunyakovski–Schwarz inequality, named after Augustin Louis Cauchy, Viktor Yakovlevich Bunyakovsky and Hermann Amandus Schwarz, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in... In probability theory, Chebyshevs inequality (also known as Tchebysheffs inequality, Chebyshevs theorem, or the Bienaymé-Chebyshev inequality), named after Pafnuty Chebyshev, who first proved it, states that in any data sample or probability distribution, nearly all the values are close to the mean value, and provides a... In probability theory, Chernoffs inequality, named after Herman Chernoff, states the following. ... In statistics, the Cramér-Rao inequality, named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, expresses an upper bound on the precision of a statistical estimator, based on Fisher information. ... Hoeffdings inequality, named after Wassily Hoeffding, is a result in probability theory that gives an upper bound on the probability for the sum of random variables to deviate from its expected value. ... In mathematical analysis, Hölders inequality, named after Otto Hölder, is a fundamental inequality relating Lp spaces: let S be a measure space, let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, let f be in Lp(S) and g be in Lq(S). ... 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... In mathematics, Jensens 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. ... Kolgomorovs inequality is an inequality which gives a relation among a function and its first and second derivatives. ... In probability theory, Markovs 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. ... In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. ... In mathematics, Nesbitts inequality states that for positive real a, b and c we have: Proof Starting from Nesbitts inequality we transform the left hand side: Now this can be transformed into: Division by 3 and the right factor yields: Now on the left we have the arithmetic... In geometry, Pedoes inequality, named after Dan Pedoe, states that if a, b, and c are the lengths of the sides of a triangle with area f, and A, B, and C are the lengths of the sides of a triangle with area F, then with equality if and... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...

Mnemonics for students

Young students sometimes confuse the less-than and greater-than signs, which are mirror images of one another. A commonly taught mnemonic is that the sign represents the mouth of a hungry alligator that is trying to eat the larger number; thus, it opens towards 8 in both 3 < 8 and 8 > 3.[1] Another method is noticing the larger quantity points to the smaller quantity and says, "ha-ha, I'm bigger than you." For other uses, see Alligator (disambiguation). ...


Also, on a horizontal number line, the greater than sign is the arrow that is at the larger end of the number line. Likewise, the less than symbol is the arrow at the smaller end of the number line (<---0--1--2--3--4--5--6--7--8--9--->). A number line, invented by John Wallis, is a one-dimensional picture in which the integers are shown as specially-marked points evenly spaced on a line. ...


The symbols may also be interpreted directly from their form - the side with a large vertical separation indicates a large(r) quantity, and the side which is a point indicates a small(er) quantity. In this way the inequality symbols are similar to the musical crescendo and decrescendo. The symbols for equality, less-than-or-equal-to, and greater-than-or-equal-to can also be interpreted with this perspective. “Fortissimo” redirects here. ...


Complex numbers and inequalities

By introducing a lexicographical order on the complex numbers, it is a totally ordered set. However, it is impossible to define ≤ so that mathbb{C},+,*,≤ becomes an ordered field. If mathbb{C},+,*,≤ were an ordered field, it has to satisfy the following two properties: In mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al) product), is a natural order structure of the Cartesian product of two ordered sets. ... A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1] Every complex number can be... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, an ordered field is a field together with an ordering of its elements. ... In mathematics, an ordered field is a field together with an ordering of its elements. ...

  • if ab then a + cb + c
  • if 0 ≤ a and 0 ≤ b then 0 ≤ a b

Because ≤ is a total order, for any number a, a ≤ 0 or 0 ≤ a. In both cases 0 ≤ a2; this means that i2 > 0 and 12 > 0; so 1 > 0 and − 1 > 0, contradiction. In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ...


However ≤ can be defined in order to satisfy the first property, i.e. if ab then a + cb + c. A definition which is sometimes used is the lexicographical order:

  • a ≤ b if Re(a) < Re(b) or (Re(a) = Re(b) and Im(a)Im(b))

It can easily be proven that for this definition ab then a + cb + c


See also

In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... For technical reasons, :) and some similar combinations starting with : redirect here. ... Fourier–Motzkin elimination is a mathematical algorithm for solving a system of linear inequalities. ... In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... In computer programming languages, a relational operator symbol or a relational operator name is a lexical or syntactic unit that denotes a relation, for example, equality or greater than, among two or more domains, the members of which are typically denoted by further expressions. ...

References

  • Hardy, G., Littlewood J.E., Polya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0-521-05206-8. 
  • Beckenbach, E.F., Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0-394-01559-2. 
  • Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. Springer-Verlag. ISBN 0-387-98404-6. 
  • Murray S. Klamkin. ""Quickie" inequalities" (PDF).
  • Harold Shapiro (missingdate). "Mathematical Problem Solving". The Old Problem Seminar. Kungliga Tekniska högskolan.
  • "3rd USAMO".
  • . "The Starry Sky".

External links

Ed Pegg, Jr. ...

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