Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
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
or
This is in contrast to the statement of their equality, expressed by an equation
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
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.
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.
See also:
Last revised on December 27, 2022 at 08:28:48. See the history of this page for a list of all contributions to it.