# nLab inequality

## In higher category theory

#### Constructivism

intuitionistic mathematics

# Contents

## Idea

The common meaning of “an inequality” in mathematics is the statement that a given pair of expressions, $x, y$, with values in some linearly 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 relation

$x \leq y$

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.

## 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, the latter of which is used to define inequality spaces. Other relations which have been called “inequality relation” in the constructive mathematics literature are listed in irreflexive symmetric relation#ConstructiveMathematics.