## Homework 3, Due Thursday, 02/18

1. Tautologies.

Use a truth table to show that the following proposition is a tautology. You must show the intermediate steps in the truth table. (Note: ⊕ means exclusive or.)

(~ (pq)) ⊕ ( (~p) → q)

2. Tautologies, again. [From Rosen 5/e.]

Show that the following proposition is a tautology using algebraic manipulation of logical equivalences (i.e., without using a truth table). Show all your work.

( ( pq) ∧ ( pr) ∧ ( qr) ) → r

You should use the logical equivalences in Theorem 2.1.1 of Epp 4/e (p. 35) or Theorem 1.1.1 of Epp 3/e (p. 14).

3. Logical Equivalences. [From Rosen 5/e.]

Show that ~ p → (qr) and q → (pr) are logically equivalent without using truth tables.

You should use the logical equivalences in Theorem 2.1.1 of Epp 4/e (p. 35) or Theorem 1.1.1 of Epp 3/e (p. 14).

4. Knights and Knaves. [From "A Whole Slew of Computer-Generated Knights and Knaves Puzzles" by Zac Ernst, 1999.]

Statements made by knights are true. Statements made by knaves are false. You meet three people: Xavier, Yolanda and Zain. You know that each is either a knight or a knave. This is what they said:

Xavier: "It is not the case that Zain is a knave."
Yolanda: "Zain and Xavier are both knights."
Zain: "Xavier is a knight or Yolanda is a knave."
Which of Xavier, Yolanda and Zain are knights? which are knaves? Show your reasoning. Follow the format in Example 2.3.14 in Epp 4/e (p. 60) or Example 1.3.16 in Epp 3/e (p. 39).