inequality (mathematics)

TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the
number line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as abstraction for real numbers, denoted by \mathbb. Every point of a number line is assumed to correspond to a real number, and ever ...

by their size. There are several different notations used to represent different kinds of inequalities: * The notation ''a'' < ''b'' means that ''a'' is less than ''b''. * The notation ''a'' > ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b''). The relation not greater than can also be represented by ''a'' ≯ ''b'', the symbol for "greater than" bisected by a slash, "not". The same is true for not less than and ''a'' ≮ ''b''. The notation ''a'' ≠ ''b'' means that ''a'' is not equal to ''b''; this ''
inequation In mathematics, an inequation is a statement that an inequality (mathematics), inequality or a non-equality holds between two values. It is usually written in the form of a pair of expression (mathematics), expressions denoting the values in ques ...

'' sometimes is considered a form of strict inequality. It does not say that one is greater than the other; it does not even require ''a'' and ''b'' to be member of an
ordered set Image:Hasse diagram of powerset of 3.svg, upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparab ...
. In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several
orders of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually ten, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic ...
. * The notation ''a'' ≪ ''b'' means that ''a'' is much less than ''b''. * The notation ''a'' ≫ ''b'' means that ''a'' is much greater than ''b''. This implies that the lesser value can be neglected with little effect on the accuracy of an
approximation An approximation is anything that is intentionally similar but not exactly equal Equal or equals may refer to: Arts and entertainment * Equals (film), ''Equals'' (film), a 2015 American science fiction film * Equals (game), ''Equals'' (game), a ...
(such as the case of
ultrarelativistic limitIn physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Phys ...
in physics). In all of the cases above, any two symbols mirroring each other are symmetrical; ''a'' < ''b'' and ''b'' > ''a'' are equivalent, etc.

# Properties on the number line

Inequalities are governed by the following
properties Property (''latin: Res Privata'') in the abstract is what belongs to or with something, whether as an attribute or as a component of said thing. In the context of this article, it is one or more components (rather than attributes), whether phys ...
. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and — in the case of applying a function — monotonic functions are limited to ''strictly''
monotonic function Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given orde ...
s.

## Converse

The relations ≤ and ≥ are each other's
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a categorical or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical ...
, meaning that for any
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s ''a'' and ''b'': : ''a'' ≤ ''b'' and ''b'' ≥ ''a'' are equivalent.

## Transitivity

The transitive property of inequality states that for any
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s ''a'', ''b'', ''c'': : If ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c''. If ''either'' of the premises is a strict inequality, then the conclusion is a strict inequality: : If ''a'' ≤ ''b'' and ''b'' < ''c'', then ''a'' < ''c''. : If ''a'' < ''b'' and ''b'' ≤ ''c'', then ''a'' < ''c''.

A common constant ''c'' may be to or from both sides of an inequality. So, for any
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s ''a'', ''b'', ''c'': : If ''a'' ≤ ''b'', then ''a'' + ''c'' ≤ ''b'' + ''c'' and ''a'' − ''c'' ≤ ''b'' − ''c''. In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an
ordered group In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order Image:Hasse diagram of powerset of 3.svg, 250px, The Hasse diagram of the power set, set of all subsets of a three-element set , ordered by inclus ...

## Multiplication and division

The properties that deal with
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary Operation (mathematics), mathematical operations of arithmetic, with the ...

and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
state that for any real numbers, ''a'', ''b'' and non-zero ''c'': : If ''a'' ≤ ''b'' and ''c'' > 0, then ''ac'' ≤ ''bc'' and ''a''/''c'' ≤ ''b''/''c''. : If ''a'' ≤ ''b'' and ''c'' < 0, then ''ac'' ≥ ''bc'' and ''a''/''c'' ≥ ''b''/''c''. In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More generally, this applies for an
ordered fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

The property for the
additive inverse In mathematics, the additive inverse of a is the number that, when to , yields . This number is also known as the opposite (number), sign change, and negation. For a , it reverses its : the additive inverse (opposite number) of a is negative, ...
states that for any real numbers ''a'' and ''b'': : If ''a'' ≤ ''b'', then −''a'' ≥ −''b''.

## Multiplicative inverse

If both numbers are positive, then the inequality relation between the
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

s is opposite of that between the original numbers. More specifically, for any non-zero real numbers ''a'' and ''b'' that are both
positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, in optics * Plus sign, the sign "+" used to indicate a positive number * Positive (electricity), a po ...
(or both negative): : If ''a'' ≤ ''b'', then ≥ . All of the cases for the signs of ''a'' and ''b'' can also be written in
chained notation Chained may refer to: * ''Chained'' (1934 film), starring Joan Crawford and Clark Gable * ''Chained'' (2012 film), a Canadian film directed by Jennifer Lynch * ''Chained'' (2020 film), a Canadian film directed by Titus Heckel * ''Chained'', a 20 ...
, as follows: : If 0 < ''a'' ≤ ''b'', then ≥ > 0. : If ''a'' ≤ ''b'' < 0, then 0 > ≥ . : If ''a'' < 0 < ''b'', then < 0 < .

## Applying a function to both sides

Any
monotonic Figure 3. A function that is ''not'' monotonic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
ally increasing
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, by its definition, may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function. If the inequality is strict (''a'' < ''b'', ''a'' > ''b'') ''and'' the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a ''strictly'' monotonically decreasing function. A few examples of this rule are: * Raising both sides of an inequality to a power ''n'' > 0 (equiv., −''n'' < 0), when ''a'' and ''b'' are positive real numbers: : 0 ≤ ''a'' ≤ ''b'' ⇔ 0 ≤ ''an'' ≤ ''bn''. : 0 ≤ ''a'' ≤ ''b'' ⇔ ''a''−''n'' ≥ ''b''−''n'' ≥ 0. * Taking the
natural logarithm The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natura ...
on both sides of an inequality, when ''a'' and ''b'' are positive real numbers: : 0 < ''a'' ≤ ''b'' ⇔ ln(''a'') ≤ ln(''b''). : 0 < ''a'' < ''b'' ⇔ ln(''a'') < ln(''b''). : (this is true because the natural logarithm is a strictly increasing function.)

# Formal definitions and generalizations

A (non-strict) partial order is a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
≤ over a set ''P'' which is reflexive, antisymmetric, and
transitive Transitivity or transitive may refer to: Grammar * Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects * Transitive verb, a verb which takes an object * Transitive case, a grammatical case to mark arg ...
. That is, for all ''a'', ''b'', and ''c'' in ''P'', it must satisfy the three following clauses: # ''a'' ≤ ''a'' ( reflexivity) # if ''a'' ≤ ''b'' and ''b'' ≤ ''a'', then ''a'' = ''b'' (
antisymmetry In linguistics, antisymmetry is a theory of syntax, syntactic linearization presented in Richard Kayne's 1994 monograph ''The Antisymmetry of Syntax''. The crux of this theory is that hierarchical structure in natural language maps universally onto ...
) # if ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c'' ( transitivity) A set with a partial order is called a
partially ordered set upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not. In mathem ...
. Those are the very basic axioms that every kind of order has to satisfy. Other axioms that exist for other definitions of orders on a set ''P'' include: # For every ''a'' and ''b'' in ''P'', ''a'' ≤ ''b'' or ''b'' ≤ ''a'' (
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X: # a \ ...
). # For all ''a'' and ''b'' in ''P'' for which ''a'' < ''b'', there is a ''c'' in ''P'' such that ''a'' < ''c'' < ''b'' (
dense orderIn mathematics, a partial order or total order < on a Set (mathematics), set $X$ is said to be dense if, for all $x$ and $y$ in $X$ for which $x < y$, there is a $z$ in $X$ ...
). # Every non-empty
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of ''P'' with an
upper bound In mathematics, particularly in order theory Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...
has a ''least'' upper bound (supremum) in ''P'' (
least-upper-bound property In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
).

## Ordered fields

If (''F'', +, ×) is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
and ≤ is a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X: # a \ ...
on ''F'', then (''F'', +, ×, ≤) is called an
ordered fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
if and only if: * ''a'' ≤ ''b'' implies ''a'' + ''c'' ≤ ''b'' + ''c''; * 0 ≤ ''a'' and 0 ≤ ''b'' implies 0 ≤ ''a'' × ''b''. Both (Q, +, ×, ≤) and (R, +, ×, ≤) are
ordered fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s, but ≤ cannot be defined in order to make (C, +, ×, ≤) an
ordered fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, because −1 is the square of ''i'' and would therefore be positive. Besides from being an ordered field, R also has the
Least-upper-bound property In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. In fact, R can be defined as the only ordered field with that quality.

# 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''. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, ''a'' < ''b'' + ''e'' < ''c'' is equivalent to ''a'' − ''e'' < ''b'' < ''c'' − ''e''. This notation can be generalized to any number of terms: for instance, ''a''1 ≤ ''a''2 ≤ ... ≤ ''a''''n'' means that ''a''''i'' ≤ ''a''''i''+1 for ''i'' = 1, 2, ..., ''n'' − 1. By transitivity, this condition is equivalent to ''a''''i'' ≤ ''a''''j'' for any 1 ≤ ''i'' ≤ ''j'' ≤ ''n''. When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance, to solve the inequality 4''x'' < 2''x'' + 1 ≤ 3''x'' + 2, it is not possible to isolate ''x'' in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved 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 In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...
of the inequalities between adjacent terms. For example, the defining condiction of a
zigzag poset In mathematics, a fence, also called a zigzag poset, is a partially ordered set in which the order relations form a path graph, path with alternating orientations: :''a'' < ''b'' > ''c'' < ''d'' > ''e'' < ''f'' > ''h'' < ''i'' . ...

is written as ''a''1 < ''a''2 > ''a''3 < ''a''4 > ''a''5 < ''a''6 > ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤. For instance, ''a'' < ''b'' = ''c'' ≤ ''d'' means that ''a'' < ''b'', ''b'' = ''c'', and ''c'' ≤ ''d''. This notation exists in a few
programming language A programming language is a formal language In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calcu ...

s such as
Python PYTHON was a Cold War contingency plan of the Government of the United Kingdom, British Government for the continuity of government in the event of Nuclear warfare, nuclear war. Background Following the report of the Strath Committee in 1955, the ...
. In contrast, in programming languages that provide an ordering on the type of comparison results, such as C, even homogeneous chains may have a completely different meaning.

# Sharp inequalities

An inequality is said to be ''sharp'', if it cannot be ''relaxed'' and still be valid in general. Formally, a
universally quantified Universality most commonly refers to: * Universality (philosophy) * Universality (dynamical systems) Universality principle may refer to: * In statistics, Random matrix#Bulk statistics, universality principle, a property of systems that can be ...
inequality ''φ'' is called sharp if, for every valid universally quantified inequality ''ψ'', if holds, then also holds. For instance, the inequality is sharp, whereas the inequality is not sharp.

# Inequalities between means

There are many inequalities between means. For example, for any positive numbers ''a''1, ''a''2, …, ''a''''n'' we have where :

# Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality states that for all vectors ''u'' and ''v'' of an
inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
it is true that : $, \langle \mathbf,\mathbf\rangle, ^2 \leq \langle \mathbf,\mathbf\rangle \cdot \langle \mathbf,\mathbf\rangle,$ where $\langle\cdot,\cdot\rangle$ is the
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
. Examples of inner products include the real and complex
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
; In
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
''R''''n'' with the standard inner product, the Cauchy–Schwarz inequality is : $\left\left(\sum_^n u_i v_i\right\right)^2\leq \left\left(\sum_^n u_i^2\right\right) \left\left(\sum_^n v_i^2\right\right).$

# Power inequalities

A "power inequality" is an inequality containing terms of the form ''a''''b'', where ''a'' and ''b'' are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.

## Examples

* For any real ''x'', :: $e^x \ge 1+x.$ * If ''x'' > 0 and ''p'' > 0, then :: $\frac \ge \ln\left(x\right) \ge \frac.$ : In the limit of ''p'' → 0, the upper and lower bounds converge to ln(''x''). * If ''x'' > 0, then :: $x^x \ge \left\left( \frac\right\right)^\frac.$ * If ''x'' > 0, then :: $x^ \ge x.$ * If ''x'', ''y'', ''z'' > 0, then :: $\left\left(x+y\right\right)^z + \left\left(x+z\right\right)^y + \left\left(y+z\right\right)^x > 2.$ * For any real distinct numbers ''a'' and ''b'', :: $\frac > e^.$ * If ''x'', ''y'' > 0 and 0 < ''p'' < 1, then :: $x^p+y^p > \left\left(x+y\right\right)^p.$ * If ''x'', ''y'', ''z'' > 0, then :: $x^x y^y z^z \ge \left\left(xyz\right\right)^.$ * If ''a'', ''b'' > 0, then :: $a^a + b^b \ge a^b + b^a.$ * If ''a'', ''b'' > 0, then :: $a^ + b^ \ge a^ + b^.$ * If ''a'', ''b'', ''c'' > 0, then :: $a^ + b^ + c^ \ge a^ + b^ + c^.$ * If ''a'', ''b'' > 0, then :: $a^b + b^a > 1.$

# Well-known 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: * Azuma's inequality * Bernoulli's inequality * Bell's inequality * Boole's inequality * Cauchy–Schwarz inequality * Chebyshev's inequality * Chernoff's inequality * Cramér–Rao inequality * Hoeffding's inequality * Hölder's inequality * Inequality of arithmetic and geometric means * Jensen's inequality * Kolmogorov's inequality * Markov's inequality * Minkowski inequality * Nesbitt's inequality * Pedoe's inequality * Poincaré inequality * Samuelson's inequality * Triangle inequality

# Complex numbers and inequalities

The set of complex numbers ℂ with its operations of addition and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary Operation (mathematics), mathematical operations of arithmetic, with the ...

is a field (mathematics), field, but it is impossible to define any relation ≤ so that becomes an
ordered fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. To make an
ordered fieldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, it would have to satisfy the following two properties: * if , then ; * if and , then . Because ≤ is a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X: # a \ ...
, for any number ''a'', either or (in which case the first property above implies that ). In either case ; this means that and ; so and , which means (−1 + 1) > 0; contradiction. However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if , then "). Sometimes the lexicographical order definition is used: * , if ** , or ** and It can easily be proven that for this definition implies .

# Vector inequalities

Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors $x, y \in \mathbb^n$ (meaning that $x = \left(x_1, x_2, \ldots, x_n\right)^\mathsf$ and $y = \left(y_1, y_2, \ldots, y_n\right)^\mathsf$, where $x_i$ and $y_i$ are real numbers for $i = 1, \ldots, n$), we can define the following relationships: * $x = y$, if $x_i = y_i$ for $i = 1, \ldots, n$. * $x < y$, if $x_i < y_i$ for $i = 1, \ldots, n$. * $x \leq y$, if $x_i \leq y_i$ for $i = 1, \ldots, n$ and $x \neq y$. * $x \leqq y$, if $x_i \leq y_i$ for $i = 1, \ldots, n$. Similarly, we can define relationships for $x > y$, $x \geq y$, and $x \geqq y$. This notation is consistent with that used by Matthias Ehrgott in ''Multicriteria Optimization'' (see References). The trichotomy property (as stated #Ordered fields, above) is not valid for vector relationships. For example, when $x = \left(2, 5\right)^\mathsf$ and $y = \left(3, 4\right)^\mathsf$, there exists no valid inequality relationship between these two vectors. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.

# Systems of inequalities

Systems of linear inequalities can be simplified by Fourier–Motzkin elimination. The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is double exponential function, doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.

*Binary relation *Bracket (mathematics), for the use of similar ‹ and › signs as brackets *Inclusion (set theory) *Inequation *Interval (mathematics) *List of inequalities *List of triangle inequalities *Partially ordered set *Relational operators, used in programming languages to denote inequality

# Sources

* * * * * * * * * * *