It does not say that one is greater than the other, or even that they can be compared in size.
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. (For example, the ultrarelativistic limit in physics.)
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 strictlymonotonic functions.
If both numbers are positive, the relation between the multiplicative inverses us the opposite of the relation between the original number. more generally, for any non-zero real numbers a and b that are both positive or both negative:
If a ≤ b, then 1/a ≥ 1/b.
All of the cases for the signs of a and b can also be written in chained notation as follows:
If 0 < a ≤ b, then 1/a ≥ 1/b > 0.
If a ≤ b < 0, then 0 > 1/a ≥ 1/b.
If a < 0 < b, then 1/a < 0 < 1/b.
Applying a function to both sides
The graph of y = ln x
Any monotonically increasing function may be applied to both sides of an inequality, provided they are in the domain of that function, and it will still hold. Applying a monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. 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. The rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.
A few examples of this rule are:
Exponentiating both sides of an inequality by n > 0 when a and b are positive real numbers:
a ≤ b ⇔ an ≤ bn.
a ≤ b ⇔ a−n ≥ b−n.
Taking the natural logarithm to both sides of an inequality when a and b are positive real numbers:
a ≤ b ⇔ ln(a) ≤ ln(b).
a < b ⇔ ln(a) < ln(b).
This is true because the natural logarithm is a strictly increasing function.
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. other axioms that exist for other definitions of orders on a set P include:
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, a1 ≤ a2 ≤ ... ≤ an means that ai ≤ ai+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to ai ≤ aj 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 4x < 2x + 1 ≤ 3x + 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 of the inequalities between adjacent terms. 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.
An inequality 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. a2 ≥ 0 is sharp, whereas the inequality ∀a ∈ ℝ. a2 ≥ −1 is not sharp.
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 ≤ a2; this means that and ; so and , which means ; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b, then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:
, if or and
It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.
Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors (meaning that and , where and are real numbers for ), we can define the following relationships:
, if for .
, if for .
, if for and .
, if for .
Similarly, we can define relationships for , , and . This notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).
The trichotomy property (as stated above) is not valid for vector relationships. For example, when and , there exists no valid inequality relationship between these two vectors. Also, a multiplicative inverse would need to be defined on a vector before this property could be considered. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.
General existence theorems
For a general system of polynomial inequalities, one can find a condition for a solution to exist. Firstly, any system of polynomial inequalities can be reduced to a system of quadratic inequalities by increasing the number of variables and equations (for example, by setting a square of a variable equal to a new variable). A single quadratic polynomial inequality in n − 1 variables can be written as
where X is a vector of the variables , and A is a matrix. This has a solution, for example, when there is at least one positive element on the main diagonal of A.
Systems of inequalities can be written in terms of matrices A, B, C, etc., and the conditions for existence of solutions can be written as complicated expressions in terms of these matrices. The solution for two polynomial inequalities in two variables tells us whether two conic section regions overlap or are inside each other. The general solution is not known, but such a solution could be theoretically used to solve such unsolved problems as the kissing number problem. However, the conditions would be so complicated as to require a great deal of computing time or clever algorithms.