Symbolize each of the following arguments (both of which are intended to be valid) and use indirect derivations.

1.  If Alice studies then she will get most of the answers correct, and if she gets most of the answers correct then she will pass the test.  Therefore, if Alice does not pass the test then she did not study.  [use S,A,P]

S -> A, A -> P | ~P -> ~S

1.  S -> A
2.  A -> P           \ ~P -> ~S
    3.  ~(~P -> ~S)       HIP
    4.  ~(P v ~S)           3,  Impl
    5.  ~P & S              4,  DM
    6.  S -> P                1,2,  HS
    7.  S                       5,  Simp
    8.  P                       6,7,  MP
    9.  ~P                     5,  Simp
    10.  P & ~P            8,9,  Conj
 11.  ~P -> ~S             3-10,  IP


2.  Knowing that Alice studies only if she is not working and she is not working only if she does not have a job proves that unless she is not studying she does not have a job.   [use S,W,J]

|-  [(S -> ~W) & (~W -> ~J)] -> (~S v ~J)

    1.  ~{ [(S -> ~W) & (~W -> ~J)] -> (~S v ~J)}     HIP
    2.  ~{~[(S -> ~W) & (~W -> ~J)] v (~S v ~J)}      1,  Impl
    3.  [(S -> ~W) & (~W -> ~J)] & ~(~S v ~J)         2,  DM    
    4.  (S -> ~W) & (~W -> ~J)                              3,  Simp
    5.  S -> ~J                                                      4,  HS      (yes, here I am condensing what would be redundant steps by not first simplifying the expression)
    6.  ~(~S v ~J)                                                  3,  Simp
    7.  ~(S -> ~J)                                                  6,  Impl
    8.  (S -> ~J) &  ~(S -> ~J)                                5,7, Conj
9.   [(S -> ~W) & (~W -> ~J)] -> (~S v ~J)             1-8,  IP