1. A set of propositions is consistent if there is an assignment of truth values to each of the variables in the propositions that makes each proposition true. Is the following set of propositions consistent? The system is in multiuser state if and only if it is operating normally. The system is operating normally only if the kernel is functioning. The kernel is not functioning, or the system is in interrupt mode. The system is in interrupt mode whenever it is not in multiuser state. The system is in interrupt mode. 2. Find a proposition with three variables p,q, and r that is: a) never true. b) never false. c) false, only when p=T,q=F and r=T. d) true, only when all variables are false. 4. Show that (qp)(pq) and pq are logically equivalent without using a truth table. 5. Find the Principal Disjunctive Normal Form and the Principal Conjunctive Normal Form for the following proposition: (qp)(pr)