CPS 341 HOMEWORK 1 Due in class, at class time, Thursday, 26 January, 2023 These guidelines apply to all the homework - Feel free to use any result from a previous lecture/homework/test in this class by simply stating it. - You must write your solutions clearly. Show all the work so that partial credit is possible. Your proofs must be in proper form to receive credit. Read the relevant material from the textbook before you attempt the problems. (1) (5 pts) Consider p(q - ). Fill the first blank with a logical connective and the second blank with one of r,r so that the following is true: in the truth table for the resulting wff, out of the 8 rows, in exactly 3 rows the value of the wff is true. Therefore, in 5 rows the value of wff is false, and in the remaining 3 rows its value is true. Justify your answer. (2) (5 pts) Is (pq) the negation of ((pq)(qp)) ? Prove your answer. (3) (5 pts) Is p(qr) logically equivalent to r(pq) ? Prove your answer. (4) (15 pts) Consider the following arguments. If an argument is valid, then present a proof sequence; otherwise, prove that the argument is invalid. You are forbidden to use truth tables to justify your answers (but, you may use them otherwise). (4a)((pq)r)((pr)(qr)) (4b)((pq)r)((pr)(qr)) (4c) ((p(qr))(rs)(st))(pt)