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In mathematics, an inequality is a statement about the relative size or order of two objects. (See also: equality) The notation a < b means that a is less than b and the notation a > b means that a is greater than b. These relations are known as strict inequality; in contrast a ≤ b means that a is less than or equal to b and a ≥ b means that a is greater than or equal to b. 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 mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ...
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...
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 "unconditonal" 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. The sense of an inequality is not changed if both sides are increased or decreased by the same number, or if both sides are multiplied or divided by a positive number; the sense of an inequality is reversed if both members are multiplied or divided by a negative number. A negative number is a number that is less than zero, such as −3. ...
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 a hungry alligator that is trying to eat the larger number; thus, it opens towards 8 in both 3 < 8 and 8 > 3.[1] Species Alligator mississippiensis Alligator sinensis An alligator is a crocodile in the genus Alligator of the family Alligatoridae. ...
The notation a >> b means that a is "much greater than" b. What this means exactly can vary, meaning anything from a factor of 100 difference to a ten order of magnitude difference. It is used in relation to equations in which a much greater value will cause the output of the equation to converge on a certain result.
Properties
Inequalities are governed by the following properties: 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 any real numbers, "a" and "b", only one of the following is true:
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Transitivity The transitivity of inequalities states: - 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 are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Reversal The inequality relations are mirror images in the sense that: - For any real numbers, "a" and "b":
- If a > b then b < a
- If a < b then b > a
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Addition and subtraction The properties which deal with addition and subtraction states: 3 + 2 with apples Addition is the most basic operation of arithmetic. ...
5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ...
- For any real numbers, "a", "b", "c":
- If a > b; then a + c > b + c and a − c > b − c
- If a < b; then a + c < b + c and a − c < b − c
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Multiplication and division The properties which deal with multiplication and division state: In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ...
In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. ...
- For any real numbers, "a", "b", and "c":
- If c is positive and a > b; then a × c > b × c and a / c > b / c
- If c is positive and a < b; then a × c < b × c and a / c < b / c
- If c is negative and a > b; then a × c < b × c and a / c < b / c
- If c is negative and a < b; then a × c > b × c and a / c > b / c
A negative number is a number that is less than zero, such as −3. ...
Applying a function to both sides Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold. In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...
Partial plot of a function f. ...
Chained notation The notation a < b < c stands for a < b and b < c 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, eg. a < b + e < c is equivalent to a − e < b < c − e.
This notation can be generalized to any number of terms: for instance, a1 ≤ a2 ≤ ... ≤ an means that ai ≤ ai+1 for i = 1, 2, ..., n−1. By the transitivity property, this condition is equivalent to ai ≤ aj for any 1 ≤ i ≤ j ≤ n.
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 > c ≤ d means that a < b, b > c, and c ≤ d. In addition to rare use in mathematics, this notation exists in a few programming languages such as Python. AND Logic Gate In mathematics, logical conjunction (usual symbol and) is a logical operator that results in false if either of the operands is false. ...
Computer code (HTML with JavaScript) in a tool that uses Syntax highlighting (colors) to help the developer see the function of each piece of code. ...
Python is an interpreted programming language created by Guido van Rossum in 1990. ...
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: This article is in need of attention from an expert on the subject. ...
In probability theory, Azumas inequality gives a concentration result for the values of martingales that have bounded differences. ...
Bernoullis inequality in real analysis states that for every integer n ≥ 0 and every real number x ≥ −1. ...
In probability theory, Booles 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. ...
In mathematics, the Cauchy-Schwarz inequality, also known as the Schwarz inequality, or the Cauchy-Bunyakovski-Schwarz inequality, named after Augustin Louis Cauchy, Hermann Amandus Schwarz and Viktor Yakovlevich Bunyakovsky, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to...
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 accuracy 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. ...
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, the triangle inequality states that the distance from A to B to C is never shorter than going directly from A to C. The triangle inequality is a theorem in spaces such as the real numbers, Euclidean space, Lp spaces (p ≥ 1) and in all inner product spaces...
See also In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ...
In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...
In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. ...
References - Hardy, G., Littlewood J.E., Polya, G. (1999). Inequalities, Cambridge Mathematical Library, Cambridge University Press. ISBN 0521052068.
- Beckenbach, E.F., Bellman, R. (1975). Introduction to Inequalities, Random House Inc. ISBN 0394015592.
- Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering, Springer-Verlag. ISBN 0387984046.
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