## Truth Tables for Statements, Logical Equivalence

The following is the list of problems for Sections 2.5 and 2.6 of the Book of Proof (pages 48, 51). There is a forum open at the end, so you can ask questions. It is a great way to interact with the instructor and with other students in your class, should you need some assistance with any question. Please, do not post solutions.

### Exercises for Section 2.5

Write a truth table for the logical statements in problems 1—9:

1. $$P \lor (Q \implies R)$$
2. $$(Q \lor R) \iff (R \land Q)$$
3. $$\lnot (P \implies Q)$$
4. $$\lnot (P \lor Q) \lor (\lnot P)$$
5. $$(P \land \lnot P) \lor Q$$
6. $$(P \land \lnot P) \land Q$$
7. $$(P \land \lnot P) \implies Q$$
8. $$P \lor (Q \land \lnot R)$$
9. $$\lnot ( \lnot P \lor \lnot Q)$$
10. Suppose the statement $$((P \land Q) \lor R) \implies (R \lor S)$$ is false. Find the truth values of $$P, Q, R$$ and $$S$$. (This can be done without a truth table.)
11. Suppose $$P$$ is false and that the statement $$(R \implies S) \iff (P \land Q)$$ is true. Find the truth values of $$R$$ and $$S$$. (This can be done without a truth table.)

### Exercises for Section 2.6

A. Use truth tables to show that the following statements are logically equivalent.

1. $$P \land (Q \lor R) = (P \land Q) \lor (P \land R)$$
2. $$P \lor ( Q \land R) = (P \lor Q) \land (P \lor R)$$
3. $$P \implies Q = (\lnot P) \lor Q$$
4. $$\lnot (P \lor Q) = (\lnot P) \land (\lnot Q)$$
5. $$\lnot (P \lor Q \lor R) = (\lnot P) \land (\lnot Q) \land (\lnot R)$$
6. $$\lnot (P \land Q \land R) = (\lnot P) \lor (\lnot Q) \lor (\lnot R)$$
7. $$P \implies Q = (P \land \lnot Q) \implies (Q \land \lnot Q)$$
8. $$\lnot P \iff Q = (P \implies \lnot Q) \land (\lnot Q \implies P)$$

B. Decide whether or not the following pairs of statements are logically equivalent.

1. $$P \land Q$$ and $$\lnot(\lnot P \lor \lnot Q)$$
2. $$(P \implies Q) \lor R$$ and $$\lnot((P \land \lnot Q) \land \lnot R)$$
3. $$(\lnot P) \land (P \implies Q)$$ and $$\lnot (Q \implies P)$$
4. $$\lnot (P \implies Q)$$ and $$P \land \lnot Q$$
5. $$P \lor (Q \land R)$$ and $$(P \lor Q) \land R$$
6. $$P \land (Q \lor \lnot Q)$$ and $$(\lnot P) \implies (Q \land \lnot Q)$$