The exercises and theorems referred to by number come from Chapter 3 of the Gries & Schneider text. In proving a numbered theorem, you may make use of only lower-numbered theorems (or metatheorems). Note that some of the exercises may have hints that appear in the book, but not here. In exercises involving conjunction, expect to apply the Golden Rule (3.35) quite frequently and (3.32) occasionally. Always respect the relative precedences of operators, which can be found on the textbook's inside front cover (in the hardback version at least) and in a web page to which there is a link from the course web page.
For each scenario described, make the strongest statement you can regarding the status —knight or knave— of each of Bill and Carol (or the relationship between their statuses). In other words, your answer should end with a statement such as "Both Bill and Carol are knights" or "If Bill is a knave, then Carol is a knight", or "Bill is a knight, but Carol could be either a knight or a knave".
Let propositional variable b stand for the statement that Bill is a knight, and, similarly, let c stand for the statement that Carol is a knight.
7. Bill said, "Carol said that she (Carol) is a knight."
10. Bill said, "At least one among Carol and me is a knave."
11. Bill said, "Carol said that if she is a knave, so am I."