Chapter 1 - Section 1.3 - Propositional Equivalences - Exercises - Page 36: 61

a) satisfiable b) unsatisfiable c) unsatisfiable

We draw truth table to check whether the following compound propositions are satisfiable or not. a) $\underline{p \quad q \quad p \lor \neg q \quad \neg p \lor q \quad \neg p \lor \neg q \quad compound statement}$ $ \quad T \quad T \quad T \quad\quad\quad\quad T \quad\quad\quad F\quad\quad\quad\quad \quad F$ $ \quad T \quad F \quad T \quad\quad\quad\quad F \quad\quad\quad T\quad\quad\quad\quad \quad F$ $ \quad F \quad T \quad F \quad\quad\quad\quad T \quad\quad\quad T\quad\quad\quad\quad \quad F$ $ \quad F \quad F \quad T \quad\quad\quad\quad T \quad\quad\quad T\quad\quad\quad\quad \quad T$ In the truth table we see that the compound statement is true if p is false and q is false, hence compound statement is satisfiable. b) $\underline{p \quad q \quad p\rightarrow q \quad p\rightarrow \neg q \quad \neg p\rightarrow q\quad \neg p\rightarrow \neg q \quad compound proposition} $ $T \quad T \quad T \quad\quad\quad F \quad\quad\quad\quad T\quad\quad\quad\quad T \quad \quad\quad\quad \quad \quad F$ $T \quad F \quad F \quad\quad\quad T \quad\quad\quad\quad T\quad\quad\quad\quad T \quad \quad\quad\quad \quad \quad F$ $F \quad T \quad T \quad\quad\quad T \quad\quad\quad\quad T\quad\quad\quad\quad F \quad \quad\quad\quad \quad \quad F$ $F \quad F \quad T \quad\quad\quad T \quad\quad\quad\quad F\quad\quad\quad\quad T \quad \quad\quad\quad \quad \quad F$ In the truth table we see that the compound statement is always false, hence, compound statement is unsatisfiable. c) $\underline{p \quad q \quad p \leftrightarrow q \quad \neg p \leftrightarrow q \quad compoundproposition}$ $\quad T \quad T \quad T \quad\quad\quad\quad F \quad\quad\quad\quad\quad F$ $\quad T \quad F \quad F \quad\quad\quad\quad T \quad\quad\quad\quad\quad F$ $\quad F \quad T \quad F \quad\quad\quad\quad T \quad\quad\quad\quad\quad F$ $\quad F \quad F \quad T \quad\quad\quad\quad F \quad\quad\quad\quad\quad F$ In the truth table we see that the compound statement is always false, hence, compound statement is unsatisfiable.