Unique representation bi-basis for rational numbers field

For (Asubseteq {mathbb {Q}}), (alpha in {mathbb {Q}}), let (r_{A}(alpha )=#{(a_{1}, a_{2})in A^{2}: alpha =a_{1}+a_{2}, a_{1}le a_{2}},) (delta _{A}(alpha )=#{(a_{1}, a_{2})in A^{2}: alpha =a_{1}-a_{2} }.) In this paper, we construct a set (Asubset {mathbb {Q}}) such that (r_{A}(alpha )=1) for all (alpha in {mathbb {Q}}) and (delta _{A}(alpha )=1) for all (alpha in {mathbb {Q}}setminus {{0}}).