Truthteller Or Liar?

On the some remote Island, there lived two kinds of people -- knights and knaves. The knights always tell the truth, but the knaves always tell a lie. John and Bill are residents of that Island.
Question 1: John says: We are both knaves. Who is who?
Question 2: John: If Bill is a knave then I'm a knight. Bill: We are different. Who is who?
Question 3: Logician: Are you both knights? John: Yes or No. Logician: Are you both knaves? John: Yes or No. Who is who?

Solution to Question 1:

We can use Boolean algebra to deduce who's who as follows:
Let J be true if John is a knight and let B be true if Bill is a knight. Now, either John is a knight and what he said was true, or John is not a knight and what he said was false. Translating that into Boolean algebra, we get:
(J^(J>^B>)) v (J>^(J>^B>)>) tautology
Simplification process:
(J^(J>^B>)) v (J>^(J>^B>)>)
false v (J>^(J>^B>)>); J^J> = contradiction
(J>^(J>^B>)>); contradiction v X=X
(J>^(J v B)); by de morgan theorem
((J>^J) v (J>^B))
(J>^B) = tautology

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Therefore John is a knave and Bill is a knight. Although most people can solve this puzzle without using Boolean algebra, the example still serves as a powerful testament of the power of Boolean algebra in solving logic puzzles.