intersection (respectively, union) of any two of its members contains as a subset (respectively, is contained as a subset of) some third member. In symbols, a family of sets is directed with respect to (respectively, ) if and only if
:for all there exists some such that and (respectively, and ) or equivalently,
:for all there exists some such that (respectively, ).

Many important examples of directed sets can be defined using these partial orders. For example, by definition, a [[prefilter]] or filter base is a non-empty