Question 46
The parliament of some country consists of several persons, some of them are friends. Computer prints all the lists of the members of the parliament such that no two persons in one list are friends. The number of these lists (including empty list) equals $M$. The members of the parliament want to constitute two committees — the senior committee $A$ and the minor committee $B$, such that:
1) committees can be empty, they can intersect or even coincide;
2) the presence of friends inside each committee are allowed, but no person of committee $A$ has a friend in committee $B$ (we assume that nobody is a friend of himself);
3) in the joint sessions of committees $A$ and $B$ all the persons can be split into two groups with no friends inside each group.
Prove that the number of ways to choose these committees equals $M^2$.
1) committees can be empty, they can intersect or even coincide;
2) the presence of friends inside each committee are allowed, but no person of committee $A$ has a friend in committee $B$ (we assume that nobody is a friend of himself);
3) in the joint sessions of committees $A$ and $B$ all the persons can be split into two groups with no friends inside each group.
Prove that the number of ways to choose these committees equals $M^2$.