nLab inequality

Idea

The common meaning of “an inequality” in mathematics is the statement that a given pair of expressions, $x, y$, with values in some strictly totally ordered set of numbers (such as natural, rational or real numbers) are in ordered relation to each other, such as

$$x \lt y$$

or

$$x \gt y \mathrlap{\,.}$$

This is in contrast to the statement of their equality, expressed by an equation

$$x = y \mathrlap{\,,}$$

whence the terminology. But, of course, there are other ways in which a pair of expressions can be “in-equal”; see at Inequality relations below.

On the other hand, in practice one also calls the non-strict total order

$$x \leq y$$

an “inequality”. Many famous inequalities are of this form (starting with the triangle inequality), often accompanied with statement of conditions when exactly the actual equality holds.

Examples

• triangle inequality

• Minkowski's inequality

• Cauchy-Schwarz inequality

• Young's inequality

• Hölder's inequality

• Cramer-Rao inequality

• Grothendieck's inequality

• Bell's inequality

Inequality relations

More generally, inequality may just be the statement that a pair of terms of any type are not equal.

In the foundations of mathematics, sometimes one talks about a particular relation called the inequality relation.

In classical mathematics, the inequality relation is defined as the negation ($\not$) of equality ($=$). However, in constructive mathematics, due to the lack of excluded middle, there are multiple different notions of inequality relation. The two most commonly used notions are the denial inequality relation and the tight apartness relation. Other relations which have been called “inequality relation” in the constructive mathematics literature are listed in irreflexive symmetric relation#ConstructiveMathematics.

More generally, any irreflexive relation $R(x, y)$ can be considered an “inequality” because by definition of irreflexive, $R(x, y)$ and equality $x = y$ are mutually exclusive for all $x$ and $y$, and the relation $R(x, y)$ gives rise to an irreflexive symmetric relation $x \nsim y \coloneqq R(x, y) \vee R(y, x)$. The equivalent for irreflexive relations of a tight irreflexive symmetric relation is a connected irreflexive relation.

 See also

• equality, equation

• denial inequality

References

See also:

• Wikipedia, Inequality (mathematics)