Consider the following belief space, where the set of players is N = {I, II}, the set of states of nature is S = {s1, s2}, the set of states of the world is Y = {ω1, ω2, ω3}, and the beliefs of the players are given by the following table:
(a) Prove that the state of the world ω3 is the only consistent state of the world in Y .
(b) Prove that at the state of the world ω2, Player II believes that the state of the world is consistent.
(c) Prove that at the state of the world ω1, both players believe that the state of the world is inconsistent.
(d) Prove that at the state of the world ω1, the fact that the state of the world is inconsistent is not common belief among the players.