The following problems come from Smullyan's book, Chapter 16. As in the above section, construct a rigorous formal proof of your answer. The Island of G A certain island, G, is inhabited exclusively by knights who always tell the truth and knaves who always lie. in addition, some of the knights are called "established knights" (these are knights who in a certain sense have "proved themselves") and certain knaves are called "established knaves." Now, the inhabitants of this island have formed various clubs. It is possible that an inhabitant may belong to more than one club. Given any inhabitant XX and any club CC, either XX claims that he is a member of CC or he claims that he is not a member of CC. We are given that the following four conditions, E1,E2,C,GE1,E2,C,G hold.

• E1E1: The set of all established knights forms a club.
• E2E2: The set of all established knaves forms a club.
• CC (The Complementation Condition): Given any club CC, the set of all inhabitants of the island who are not members of CC form a club of their own. (This club is called the complement of CC and is denoted by CC.)
• GG (The Go Delian Condition): given any club CC, there is at least one inhabitant of the island who claims that he is a member of CC. (Of course this claim might be false: he could be a knave.)

1. Proving that there is at least one unestablished knight on the island.
2. Proving that there is at least one unestablished knave on the island.
3. Does the set of all knaves on the island form a club?
4. Does the set of all knights on the island form a club?