If you represent an antisymmetric relation as a matrix, this matrix, call it A, can have either 0 or 1 on the diagonal. Let A[i, j] denote the element of A in row i and column j. If i <> j, then one or both of A[i, j] and A[j, i] must be 0, so there are three possibilities for the pair of locations (i, j) and (j, i).