2.4 APPLICATION: DIGITAL LOGIC CIRCUITS 79 c. Lefty: Muscles was shooting craps with Socko 42. a. PV q when Sharky was knocked off. b. q -r d. Muscles: Lefty didn't kill Sharky. C. prs-t Who did kill Sharky? de ~r In 41-44 a set of premises and a conclusion are given. Use e. ~q ->uAS the valid argument forms listed in Table 2.3.1 to deduce the f. .. t conclusion from the premises, giving a reason for each step as in Example 2.3.8. Assume all variables are statement variables. 43. a. ~p-rA~s 44. a. P-> q D. t - s b. rVS 41. a. ~p v q -r C. u - ~p C. ~ s - ~t b. sv ~ q ~W d ~ q Vs C. ~t e. u VW e. ~S d p-t f. .. ~t f. ~p Ar- u e. ~p Ar- ~s g. wVt f. .. ~q h. .uAwStep 1: Apply Premise e (s) in Premise From s, we can substitute into Premise (s t), which gives us t (i.e., t is false). Step 2: Apply t is false in Premise g Since t is false, we can deduce from Premise g (w V t) that w must be true (i.e., w). Step 3: Apply Premise e (s) in Premise d Substitute s into Premise d (g V s), which simplifies to g, meaning q is false. Step 4: Apply q is false in Premise a From p g (Premise a), and knowing q is false, we can deduce p is false (contrapositive: if q is false, then p is false). Step 5: Apply p is false and 7 from Premise b Premise b states r \\V s. Since s is false from Premise e, we can deduce 7 is true. Step 6: Apply p is false and 7 is true in Premise f From Premise f ((p A r) w), since p is false (so p is true) and r is true, we deduce u is true. Step 7: Conclude u A w Since we have deduced that both u (from Step 6) and w (from Step 2) are true, we can conclude u A\\ w. Conclusion: u N\\ w Thus, the conclusion u A w follows logically from the premises