Chapter 1.3, Problem 66E

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.)

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

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

b) $\left(\neg p\vee \neg q\vee r\right)\wedge \left(\neg p\vee q\vee \neg s\right)\wedge \left(p\vee \neg q\vee \neg s\right)\wedge \left(\neg p\vee \neg r\vee \neg s\right)\wedge \left(p\vee q\vee \neg r\right)\wedge \left(p\vee \neg r\vee \neg s\right)$

c) $\left(p\vee q\vee r\right)\wedge \left(p\vee \neg q\vee \neg s\right)\wedge \left(q\vee \neg r\vee s\right)\wedge \left(\neg p\vee r\vee s\right)\wedge \left(\neg p\vee q\vee \neg s\right)\wedge \left(p\vee \neg q\vee r\right)\wedge \left(\neg p\vee \neg q\vee s\right)\wedge \left(\neg p\vee \neg r\vee s\right)$