Equality (mathematics)
Loosely, equality is the state of being quantitatively the same. In mathematical logic, equality is defined by axioms (e.g. the first few Peano axioms, or the axiom of extensionality in ZF set theory). It can also be viewed as a relation: the identity relation, or diagonal relation, the binary relation on a set X defined by
- .
The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary relations which are reflexive, symmetric, and transitive. The relation of equality is also antisymmetric. These four properties uniquely determine the equality relation on any set S and render equality the only relation on S that is both an equivalence relation and a partial order. It follows from this that equality is the smallest equivalence relation on any set S, in the sense that it is a subset of any other equivalence relation on S. An equation is simply an assertion that two expressions are related by equality (are equal).
The etymology of the word is from the Latin aequalis, meaning uniform or identical, from aequus, meaning "level, even, or just."
Read more about Equality (mathematics): Logical Formulations, Some Basic Logical Properties of Equality, Relation With Equivalence and Isomorphism
Famous quotes containing the word equality:
“It is the nature of our desires to be boundless, and many live only to gratify them. But for this purpose the first object is, not so much to establish an equality of fortune, as to prevent those who are of a good disposition from desiring more than their own, and those who are of a bad one from being able to acquire it; and this may be done if they are kept in an inferior station, and not exposed to injustice.”
—Aristotle (384–322 B.C.)