If only one of these conditions is strict, then the resultant inequality is non-strict. If the inequality is strict ( a b) and the function is strictly monotonic, then the inequality remains strict. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function. However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. Applying a function to both sides Īny monotonically increasing function, 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 of that function). If a ≤ b and c > 0, then ac ≤ bc and a/ c ≤ b/ c. The properties that deal with multiplication and division state that for any real numbers, a, b and non-zero c: If either of the premises is a strict inequality, then the conclusion is a strict inequality: The transitive property of inequality states that for any real numbers a, b, c: If a ≤ b and b ≤ c, then a ≤ c. The relations ≤ and ≥ are each other's converse, meaning that for any real numbers a and b:Ī ≤ b and b ≥ a are equivalent. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities () and - in the case of applying a function - monotonic functions are limited to strictly monotonic functions. Inequalities are governed by the following properties. 2 Formal definitions and generalizations.In all of the cases above, any two symbols mirroring each other are symmetrical a a are equivalent, etc. 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). The notation a ≫ b means that a is much greater than b.The notation a ≪ b means that a is much less than b.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. 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. The notation a ≠ b means that a is not equal to b this inequation sometimes is considered a form of strict inequality.
The same is true for not less than and a ≮ b. The relation not greater than can also be represented by a ≯ b, the symbol for "greater than" bisected by a slash, "not". 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 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).
In contrast to strict inequalities, there are two types of inequality relations that are not strict: These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b.
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