# Dense order

In mathematics, a partial order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z in X such that x < z < y.

## Example

The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. On the other hand, the ordinary ordering on the integers is not dense.

## Generalizations

Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y are R-related.[citation needed] Formally:

$\forall x\ \forall y\ xRy\Rightarrow (\exists z\ xRz \land zRy).$

Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation.