1. For statements pq and r, use a truth table to show that each of the following pairs of statements is logically equivalent. a. (p Aq) p and p-q. b. p(q Vr) and -q-(-p Vr). c. (p vq) rand (p-q) ^(qr). d. p(q vr) and (r)(pq). e. p(q vr) and ((-) Ap)-q. 2. For statements p.q, and r, show that the following compound statements are tautology. a. p (p va). b. (p ^(p-q))-q. L c. ((p-q) ^(q-r)) (pr). 3. For statements p and q, show that (p Aq) (p Aq) is a contradiction. 4. Write the contrapositive and the converse of the following conditional statements. a. If it is cold, then the lake is frozen. b. If Solomon is healthy, then he is happy. c. If it rains, Tigist does not take a walk. 5. Let p and q be statements. Which of the following implies that p vq is false? a. p V-q is false. d. p-q is true. b. p Vq is true. e. p Aq is false. c. p A-q is true. 6. Suppose that the statements p,q,r, and s are assigned the truth values T,F,F, and T, respectively. Find the truth value of each of the following statements. a. (p va) vr. f. (p Vr)( A-s). b. p v(q vr). g. (Sp) (p Vs). h. (q A-s) (pes). c. (SAP). d. p(s). i. (r As)-(p-(-q Vs)). e. p-(r vs). j. (DV-a) Vr-(s A-s). 7. Suppose the value of pq is T; what can be said about the value of pлq-p vq? 8. a. Suppose the value of pq is T; what can be said about the values of p and-p-q? b. Suppose the value of pq is F; what can be said about the values of p and-p-q? 9. Construct the truth table for each of the following statements. a. p-(p-q). d. (p q)-(-p va). b. (p Va)-(q Vp). e. (p (q Ar)) V(-p ^q). c. p(q Ar). f. (p Aq)-((q Aq)-(r^q)). 10. For each of the following determine whether the information given is sufficient to decide the truth value of the statement. If the information is enough, state the truth value. If it is insufficient, show that both truth values are possible. a. (p-q)-r, where r -T. b. p A(q-r), where qr -T. c. p V(q-r), where q-r -T. d. (p va)-(-p A-q), where p Vq -T. e. (pq)-(-q-p), where q -T. f. (p An in Vel where n-T and F Tools Mobile View