[SOLVED] Basic Rules in Hypothesis

Late homework is not accepted, but remember that your lowest 3 scores will be dropped. Please turn it in on Gauchospace in .doc or .pdf format. Also please do not include your name on the assignment; it will be graded blindly.Note: On this assignment you can again use all of the 11 rules. You will need to use subproofs within subproofs on a number of occassions.Exercise 1. Use the basic rules to prove each of the following:a) P, (P ! Q) `? Q ! Rb) ? P ` P ! Qc) P ! Q `? P _ Qd) ? P _ Q ` P ! Q(Hint on d): This one is very di?cult. The way I did it was to use the same strategy from Exercises 3a and 3b on HW7; it will involve an application of _E and *a lot* of subproofs — I made 5 hypotheses total in the course of my derivation!)Exercise 2. Prove each of the following:a) ` P ! (Q ! (P & Q))b) ` ((P ! Q) & ? Q) !? Pc) `? (P $? P)d) ` (? P ! Q) ! (P _ Q)Note that on all of these your first move will have to be making some hypothesis — either for ?I or !I!1