I.   Solutions to the problems:

           A.   Derive (P v Q) <-> (P & Q) from

               1.   P -> Q                  Premise
               2.   P                       Premise
               3.       P v Q               Assumption
               4.       Q                   1, 2, MP
               5.       P & Q               2, 4, Conj.
               6.   (P v Q) -> (P & Q)      3-5, CP
               7.       P & Q               Assumption
               8.       P v Q               2, Add.
               9.   (P & Q) -> (P v Q)      7-8, CP
              10.   (P v Q) <-> (P & Q)     6, 9, Conj., Equiv.

           B.   Derive P -> (P -> (P & Q)) from

               1.   P -> Q                  Premise
               2.       P                   Assumption
               3.           P               Assumption
               4.           Q               1, 3, MP
               5.           P & Q           3, 4, Conj.
               6.       P -> (P & Q)        3-5, CP
               7.   P -> (P -> (P & Q))     2-6, CP

        or

               1.   P -> Q                  Premise
               2.       P                   Assumption
               3.       Q                   1, 2, MP
               4.       P & Q               2, 3, Conj.
               5.   P -> (P & Q)            2-4, CP
               6.   (P -> (P & Q)) v ~P     5, Add.
               7.   ~P v (P -> (P & Q))     6, Comm.
               8.   P -> (P -> (P & Q))     7, Impl.

           C.   Without using Trans., derive (P -> Q) <-> (~Q -> ~P) from
                scratch.

               1.       P -> Q              Assumption
               2.       ~P v Q              1, Impl.
               3.       Q v ~P              2, Comm.
               4.       ~~Q v ~P            3, DN
               5.       ~Q -> ~P            4, Impl.
               6.   (P -> Q) -> (~Q -> ~P)  1-5, CP
               7.       ~Q -> ~P            Assumption
               8.       ~~Q v ~P            7, Impl.
               9.       ~P v ~~Q            8, Comm.
              10.       P -> ~~Q            9, Impl.
              11.       P -> Q              10, DN
              12.   (~Q -> ~P) -> (P -> Q)  7-11, CP
              13.   (P -> Q) <-> (~Q -> ~P) 6, 12, Conj., Equiv.

           D.   Consider the following argument:

               1.   P -> Q                  Premise
               2.   ~P v Q                  1, Impl.
               3.   (~P v ~P) v Q           2, Taut.
               4.   ~P v (~P v Q)           3, Assoc.
               5.   P -> (~P v Q)           4, Impl.
               6.   P -> (P -> Q)           5, Impl.

                Write a 4-line proof in which you derive the conclusion of
                this proof from its premise by using CP--instead of the way
                I derived it here.

               1.   P -> Q                  Premise
               2.       P                   Assumption
               3.       P -> Q              1, R
               4.   P -> (P -> Q)           2-3, CP