I.   In the following derivations, indicate the line(s) that each
            line is derived from, and the rule by which each line is
            derived.

           A.   Derive P from

               1.   ~P -> Q                 Premise
               2.   ~Q v ~Q                 Premise
               3.   ~(Q & Q)                2, DM
               4.   ~Q                      3, Taut. (or 2, Taut.)
               5.   ~~P                     1, 4, MT
               6.   P                       5, DN

           B.   Derive (P & Q) v (P & Q) from

               1.   R & Q                   Premise
               2.   S & (P & T)             Premise
               3.   Q & R                   1, Comm.
               4.   Q                       3, Simp.
               5.   (S & P) & T             2, Assoc.
               6.   (P & S) & T             5, Comm.
               7.   P & (S & T)             6, Assoc.
               8.   P                       7, Simp.
               9.   P & Q                   4, 8, Conj.
              10.   (P & Q) v (P & Q)       9, Taut.

           C.   Derive (P & Q) v (P & Q) from

               1.   R & Q                   Premise
               2.   S & (P & T)             Premise
               3.   S & (T & P)             2, Comm.
               4.   (S & T) & P             3, Assoc.
               5.   P & (S & T)             4, Comm.
               6.   P                       5, Simp.
               7.   P & (R & Q)             1, 6, Conj.
               8.   P & (Q & R)             7, Comm.
               9.   (P & Q) & R             8, Assoc.
              10.   P & Q                   9, Simp.
              11.   P & (Q v Q)             10, Taut.
              12.   (P & Q) v (P & Q)       11, Dist.

           D.   Derive P <-> Q from

               1.   ~P v Q                  Premise
               2.   ~P -> ~Q                Premise
               3.   P -> Q                  1, Impl.
               4.   Q -> P                  2, Trans.
               5.   (P -> Q) & (Q -> P)     3, 4, Conj.
               6.   P <-> Q                 5, Equiv.

           E.   Derive (P & Q) -> R from

               1.   P -> R                  Premise
               2.   (P -> R) v ~Q           1, Add.
               3.   ~Q v (P -> R)           2, Comm.
               4.   Q -> (P -> R)           3, Impl.
               5.   (Q & P) -> R            4, Exp.
               6.   (P & Q) -> R            5, Comm.