$(20$ 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 Cady and

Karen$$s statements is equivalent to just Cady$$s statement.

Hint: This can be done without using the Distributive laws.

c$)(6$ points$)$ Translate $$Neither Gretchen nor Karen wrote the book$$ to propositional logic. Then

use the laws of logical equivalence to show that the conjunction of Cady$$s and Gretchen$$s statements

is equivalent to it$.$

Hint: Recall that you showed $$p$($p$$q$)$p$$q in Lab $1.$ You can use this as a logical equivalence,

just label the step as $$From Lab$.$ You will also want to use the $$Biconditional Laws$$ listed in

Section $1\mathrm{.}3$ Table $8$ in the textbook.

d$)(6$ points$)$ Translate $$If any of the three wrote the book, then only Cady wrote the book$$ into

propositional logic. Then prove using Logical equivalences that this is equivalent to the statement

from the previous part.

Hint: When you have large clauses, try to identify and group what they have in common.