equality (mathematics)

Two mathematical objects are equal if and only if they are precisely the same in every way. The complementary notion is distinctness. This defines a binary relation, equality, denoted by the sign of equality "=" in such a way that the statement "x = y" means that x and y are equal.Equality is the paradigmatic example of the more general concept of equivalence relations on a set: those binary relations that are reflexive, symmetric, and transitive. It goes beyond the other equivalence relations by also being  antisymmetric. In fact, 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.
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