Solve the following knights always tell the truth and knaves

Solve the following knights (always tell the truth) and knaves (always lie) puzzle, and a

write a formal proof of your solution:

You meet five inhabitants: Bozo, Alice, Zippy, Marge, and Zoey.

Bozo says that Zippy and Zoey are knaves.
Alice says, “Zoey is a knave and I am a knight.”
Zippy says that at least one of the following is true: that Marge is a knight or that Alice is a knight.
Marge says that it\'s false that Bozo is a knave.
Zoey says that at least one of the following is true: that Marge is a knight or that Bozo is a knave.

Solution

Say,

Statement A: Bozo is knight

Statement B: Alice is knight

Statement C: Zippy is knight

Statement D: Marge is knight

Statement E: Zoey is knight

Now,

Bozo says, ~C^~E

Alice says, E^B

Zippy says, D or B

Marge says, ~A

Zoey says, D or (~A)

Case1:

Assume Marge is a Knave .

Then statement made by Marge is false and Bozo is knave.

If Bozo is knave, the statement made by Zoey is true and Zoey is a knight.

If zoey is knight statement made by Alice is false and Alice is a Knave.

Then Zippy\'s statement is false. Zippy is knave.

Case 2:

Assume Marge is a knight.

Then, he tells truth. Therefore Bozo is a knight.

If Bozo is knight, Zippy is knave and Zoey is knave.

Since Zippy is knvae, Zippy\'s statement is false, Marge is knave. But we started wuth the assumption that Marge is knight.

Hence the first case hold true.