In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions . It is often attached to a technical term to indicate that the exclusive meaning of the term is to be understood. The opposite is non-strict , which is often understood to be the case but can be put explicitly for clarity. In some contexts, the word "proper" can also be used as a mathematical synonym for "strict".
36-513: This term is commonly used in the context of inequalities — the phrase "strictly less than" means "less than and not equal to" (likewise "strictly greater than" means "greater than and not equal to"). More generally, a strict partial order , strict total order , and strict weak order exclude equality and equivalence. When comparing numbers to zero, the phrases "strictly positive" and "strictly negative" mean "positive and not equal to zero" and "negative and not equal to zero", respectively. In
72-410: A {\displaystyle a} and c {\displaystyle c} . Similar to equation solving , inequation solving means finding what values (numbers, functions, sets, etc.) fulfill a condition stated in the form of an inequation or a conjunction of several inequations. These expressions contain one or more unknowns , which are free variables for which values are sought that cause
108-399: A ≯ b , {\displaystyle a\ngtr b,} the symbol for "greater than" bisected by a slash, "not". The same is true for not less than , a ≮ b . {\displaystyle a\nless b.} The notation a ≠ b means that a is not equal to b ; this inequation sometimes is considered a form of strict inequality. It does not say that one
144-731: A 1 , a 2 , ..., a n we have where they represent the following means of the sequence: The Cauchy–Schwarz inequality states that for all vectors u and v of an inner product space it is true that | ⟨ u , v ⟩ | 2 ≤ ⟨ u , u ⟩ ⋅ ⟨ v , v ⟩ , {\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle }
180-529: A n d 0 ≠ j {\displaystyle i\neq 0~~\mathrm {and} ~~0\neq j} , which does not imply i ≠ j . {\displaystyle i\neq j.} Similarly, a < b > c {\displaystyle a<b>c} is shorthand for a < b a n d b > c {\displaystyle a<b~~\mathrm {and} ~~b>c} , which does not imply any order of
216-426: A and b : The transitive property of inequality states that for any real numbers a , b , c : If either of the premises is a strict inequality, then the conclusion is a strict inequality: A common constant c may be added to or subtracted from both sides of an inequality. So, for any real numbers a , b , c : In other words, the inequality relation is preserved under addition (or subtraction) and
252-898: A is strictly less than or strictly greater than b . Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: In the 17th and 18th centuries, personal notations or typewriting signs were used to signal inequalities. For example, In 1670, John Wallis used a single horizontal bar above rather than below the < and >. Later in 1734, ≦ and ≧, known as "less than (greater-than) over equal to" or "less than (greater than) or equal to with double horizontal bars", first appeared in Pierre Bouguer 's work . After that, mathematicians simplified Bouguer's symbol to "less than (greater than) or equal to with one horizontal bar" (≤), or "less than (greater than) or slanted equal to" (⩽). The relation not greater than can also be represented by
288-512: A ≤ b , then a + c ≤ b + c "). Sometimes the lexicographical order definition is used: It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c . 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
324-408: Is doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases. Inequation In mathematics , an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating
360-488: Is a binary relation ≤ over a set P which is reflexive , antisymmetric , and transitive . That is, for all a , b , and c in P , it must satisfy the three following clauses: A set with a partial order is called a partially ordered set . Those are the very basic axioms that every kind of order has to satisfy. A strict partial order is a relation < that satisfies: Some types of partial orders are specified by adding further axioms, such as: If ( F , +, ×)
396-673: Is a field and ≤ is a total order on F , then ( F , +, ×, ≤) is called an ordered field if and only if: Both ( Q , + , × , ≤ ) {\displaystyle (\mathbb {Q} ,+,\times ,\leq )} and ( R , + , × , ≤ ) {\displaystyle (\mathbb {R} ,+,\times ,\leq )} are ordered fields , but ≤ cannot be defined in order to make ( C , + , × , ≤ ) {\displaystyle (\mathbb {C} ,+,\times ,\leq )} an ordered field , because −1
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#1732765457729432-420: Is a field , but it is impossible to define any relation ≤ so that ( C , + , × , ≤ ) {\displaystyle (\mathbb {C} ,+,\times ,\leq )} becomes an ordered field . To make ( C , + , × , ≤ ) {\displaystyle (\mathbb {C} ,+,\times ,\leq )} an ordered field , it would have to satisfy
468-464: 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 by their size. The main types of inequality are less than (<) and greater than (>). There are several different notations used to represent different kinds of inequalities: In either case, a is not equal to b . These relations are known as strict inequalities , meaning that
504-508: Is an inequality containing terms of the form a , where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises. Examples: 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: The set of complex numbers C {\displaystyle \mathbb {C} } with its operations of addition and multiplication
540-448: Is described in constraint programming ; in particular, the simplex algorithm finds optimal solutions of linear inequations. The programming language Prolog III also supports solving algorithms for particular classes of inequalities (and other relations) as a basic language feature. For more, see constraint logic programming . Usually because of the properties of certain functions (like square roots), some inequations are equivalent to
576-524: 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
612-489: Is given, that is to be minimized or maximized by an optimal solution. For example, is a conjunction of inequations, partly written as chains (where ∧ {\displaystyle \land } can be read as "and"); the set of its solutions is shown in blue in the picture (the red, green, and orange line corresponding to the 1st, 2nd, and 3rd conjunct, respectively). For a larger example. see Linear programming#Example . Computer support in solving inequations
648-447: Is greater than the other; it does not even require a and b to be member of an ordered set . 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 . This implies that the lesser value can be neglected with little effect on the accuracy of an approximation (such as the case of ultrarelativistic limit in physics). In all of
684-471: Is said to be sharp if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ , if ψ ⇒ φ holds, then ψ ⇔ φ also holds. For instance, the inequality ∀ a ∈ R . a ≥ 0 is sharp, whereas the inequality ∀ a ∈ R . a ≥ −1 is not sharp. There are many inequalities between means. For example, for any positive numbers
720-410: 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: A (non-strict) partial order
756-781: Is the inner product . Examples of inner products include the real and complex dot product ; In Euclidean space R with the standard inner product, the Cauchy–Schwarz inequality is ( ∑ i = 1 n u i v i ) 2 ≤ ( ∑ i = 1 n u i 2 ) ( ∑ i = 1 n v i 2 ) . {\displaystyle {\biggl (}\sum _{i=1}^{n}u_{i}v_{i}{\biggr )}^{2}\leq {\biggl (}\sum _{i=1}^{n}u_{i}^{2}{\biggr )}{\biggl (}\sum _{i=1}^{n}v_{i}^{2}{\biggr )}.} A power inequality
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#1732765457729792-484: Is the square of i and would therefore be positive. Besides being an ordered field, R also has the Least-upper-bound property . In fact, R can be defined as the only ordered field with that quality. 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
828-420: 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 languages such as Python . 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. An inequality
864-556: The additive inverse states that for any real numbers a and b : If both numbers are positive, then the inequality relation between the multiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbers a and b that are both positive (or both negative ): All of the cases for the signs of a and b can also be written in chained notation , as follows: Any monotonically increasing function , by its definition, may be applied to both sides of an inequality without breaking
900-534: The cases above, any two symbols mirroring each other are symmetrical; a < b and b > a are equivalent, etc. Inequalities are governed by the following properties . 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 functions . The relations ≤ and ≥ are each other's converse , meaning that for any real numbers
936-441: The chain is shorthand for which also implies that 0 < b {\displaystyle 0<b} and a < 1 {\displaystyle a<1} . In rare cases, chains without such implications about distant terms are used. For example i ≠ 0 ≠ j {\displaystyle i\neq 0\neq j} is shorthand for i ≠ 0
972-451: The condition to be fulfilled. To be precise, what is sought are often not necessarily actual values, but, more in general, expressions. A solution of the inequation is an assignment of expressions to the unknowns that satisfies the inequation(s); in other words, expressions such that, when they are substituted for the unknowns, make the inequations true propositions. Often, an additional objective expression (i.e., an optimization equation)
1008-510: The context of functions, the adverb "strictly" is used to modify the terms "monotonic", "increasing", and "decreasing". On the other hand, sometimes one wants to specify the inclusive meanings of terms. In the context of comparisons, one can use the phrases "non-negative", "non-positive", "non-increasing", and "non-decreasing" to make it clear that the inclusive sense of the terms is being used. The use of such terms and phrases helps avoid possible ambiguity and confusion. For instance, when reading
1044-431: 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 example, the defining condition of a zigzag poset is written as a 1 < a 2 > a 3 < a 4 > a 5 < a 6 > ... . Mixed chained notation
1080-421: The following two properties: Because ≤ is a total order , for any number a , either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ − a ). In either case 0 ≤ a ; this means that i > 0 and 1 > 0 ; so −1 > 0 and 1 > 0 , which means (−1 + 1) > 0; contradiction. However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if
1116-404: The inequality relation (provided that both expressions are in the domain 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
Strict - Misplaced Pages Continue
1152-419: The phrase " x is positive", it is not immediately clear whether x = 0 is possible, since some authors might use the term positive loosely to mean that x is not less than zero. Such an ambiguity can be mitigated by writing " x is strictly positive" for x > 0, and " x is non-negative" for x ≥ 0. (A precise term like non-negative is never used with the word negative in
1188-466: The real numbers are an ordered group under addition. The properties that deal with multiplication and division state that for any real numbers, a , b and non-zero 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 field . For more information, see § Ordered fields . The property for
1224-467: 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
1260-428: The specific inequality relation. Some examples of inequations are: In some cases, the term "inequation" can be considered synonymous to the term "inequality", while in other cases, an inequation is reserved only for statements whose inequality relation is " not equal to " (≠). A shorthand notation is used for the conjunction of several inequations involving common expressions, by chaining them together. For example,
1296-578: The wider sense that includes zero.) The word "proper" is often used in the same way as "strict". For example, a " proper subset " of a set S is a subset that is not equal to S itself, and a " proper class " is a class which is not also a set. This article incorporates material from strict on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License . Inequality (mathematics) In mathematics , an inequality
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