Equality (mathematics)

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."