Chapter 1.3, Problem 65E

We now present a group of exercises that involve the logical operatorsNANDandNOR. The propositionp NAND q, is true when eitherporq, or both, are false; and it is false when bothpandqare true. The propositionp NOR qis true when bothpandqare false, and it is false otherwise. The propositionsp NAND qandp NOR qare denoted by $p|q$

$p\downarrow q$ , respectively. (The operators |and $\downarrow$ are called theSheffer strokeand thePeirce arrowafter H. M. Sheffer and C. S. Peirce, respectively.)

65. Determine whether each of these compound propositions is satisfiable.

a) $\left(p\vee \neg q\right)\wedge \left(\neg p\vee q\right)\wedge \left(\neg p\vee \neg q\right)$

b) $\left(p\to q\right)\wedge \left(p\to \neg q\right)\wedge \left(\neg p\to q\right)\wedge \left(\neg p\to \neg q\right)$

c) $\left(p\leftrightarrow q\right)\wedge \left(\neg p\leftrightarrow q\right)$