This is more of a graph problem. You are given set A={1,2,..,n}. You have to create sets B(i) for all i=1,2,..,n, such that i $\notin$ B(i). The constraint on set are the following:

1. If $j \in B(i)$ then $ i \notin B(j)$.

2. For all i $\in$ B(j), {B(j) $\setminus$ i }$\not \subset$ B(i).

3. For all i $\notin$ B(k), B(i) $\not \subset$ {k $\cup$ B(k)}

Can such subsets be created. I can prove it for n=4,5,6 but I am not able to generalise it. I have an opinion that such subsets cannot be created.