Rank relations between a {0, 1}-matrix and its complement

Let A be a {0, 1}-matrix and r(A) denotes its rank. The complement matrix of A is defined and denoted by Ac = J − A, where J is the matrix with each entry being 1. In particular, when A is a square {0, 1}-matrix with each diagonal entry being 0, another kind of complement matrix of A is defined and denoted by A = J − I − A, where I is the identity matrix. We determine the possible values of r(A) ± r(Ac) and r(A) ± r(A) in the general case and in the symmetric case. Our proof is constructive.