# MATH 300 Fall 2018 Assignment 09

## 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:

- \( P \lor (Q \implies R) \)
- \( (Q \lor R) \iff (R \land Q) \)
- \( \lnot (P \implies Q) \)
- \( \lnot (P \lor Q) \lor (\lnot P) \)
- \( (P \land \lnot P) \lor Q \)
- \( (P \land \lnot P) \land Q \)
- \( (P \land \lnot P) \implies Q \)
- \( P \lor (Q \land \lnot R) \)
- \( \lnot ( \lnot P \lor \lnot Q) \)
- 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.)
- 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.

- \( P \land (Q \lor R) = (P \land Q) \lor (P \land R) \)
- \( P \lor ( Q \land R) = (P \lor Q) \land (P \lor R) \)
- \( P \implies Q = (\lnot P) \lor Q \)
- \( \lnot (P \lor Q) = (\lnot P) \land (\lnot Q) \)
- \( \lnot (P \lor Q \lor R) = (\lnot P) \land (\lnot Q) \land (\lnot R) \)
- \( \lnot (P \land Q \land R) = (\lnot P) \lor (\lnot Q) \lor (\lnot R) \)
- \( P \implies Q = (P \land \lnot Q) \implies (Q \land \lnot Q) \)
- \( \lnot P \iff Q = (P \implies \lnot Q) \land (\lnot Q \implies P) \)

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

- \( P \land Q \) and \( \lnot(\lnot P \lor \lnot Q) \)
- \( (P \implies Q) \lor R \) and \( \lnot((P \land \lnot Q) \land \lnot R) \)
- \( (\lnot P) \land (P \implies Q) \) and \( \lnot (Q \implies P) \)
- \( \lnot (P \implies Q) \) and \( P \land \lnot Q \)
- \( P \lor (Q \land R) \) and \( (P \lor Q) \land R \)
- \( P \land (Q \lor \lnot Q) \) and \( (\lnot P) \implies (Q \land \lnot Q) \)