$(20$ points$)$ Principal Duvall has found the $$Burn Book,$$ a book containing cruel

gossip about almost every student at North Shore High School. Determined to get to the bottom

of who has written the book, he narrows the suspects down to the only three students who aren$$t

listed in the book: Cady, Karen, and Gretchen. He questions the three of them and gets the

following responses:

$1.$ Cady says: $$At least one of Gretchen or Karen didn$$t write the book.$$

$2.$ Gretchen says: $$I wrote it if and only if Karen wrote it$.$

$3.$ Karen says: $$We didn$$t all write the book.$$

Let C$,$ K$,$ and G represent that Cady, Karen, and Gretchen wrote the Burn Book respectively.

You need to help Principal Duvall deduce who wrote the Burn Book.

a$)(2$ points$)$

Convert their sentences into propositional formulas.

b$)(6$ points$)$ Use the laws of logical equivalences to show that the conjunction of both