Model Answers for Section 3-2

a)   B v ~A                      b)   (A & B) v C
     ~A v B         CM                C v (A & B)        CM
     ~A v ~~B       DN, STE           (C v A) & (C v B)  D
     ~(A & ~B)      DM                (A v C) & (B v C)  C, SLE

c) A & (~~C v B)         d)  ~((A&~B) v (C & ~B))
   A & (C v B)     DN, SLE   ~(A& ~B) & ~ (C & ~B)     DM
   (A & C) v (A & B) D       (~A v ~~B) & (~C v ~~B)   DMx2, SLE
                             (~A v B) & (~C v B)       DN, SLE 
                             (B v ~A) & (B v ~C)       CMx2, SLE
                             B v (~A & ~C)             D 
                             (~A & ~C) v B             CM

e)   (A v B) & (C v D)   
     ((A v B) & C) v ((A v B) & D)              D
     (C & (A v B)) v (D & (A v B))              CMx2, SLE
     ((C & A) v (C & B)) v ((D & A) v (D & B))  Dx2, SLE
     ((A & C) v (B & C)) v ((A & D) v (B & D))  CMx4, SLE
     (A & C) v (B & C) v (A & D) v (B & D)      A

NOTE: THIS LAST STATEMENT IS AN ABBREVIATION FOR :

      (((A & C) v (B & C)) v (A & D)) v (B & D)

f)   (A & B) v (C & D)
     ((A & B) v C) & ((A & B) v D)              D
     (C v (A & B)) & (D v (A & B))              CMx2, SLE
     ((C v A) & (C v B)) & ((D v A) & (D v B))  Dx2, SLE
     ((A v C) & (B v C)) & ((A v D) & (B v D))  CMx4, SLE
     (A v C) & (B v C) & (A v D) & (B v D)      A


g)   ((C & A) v (B & C)) v (C & ~(~B & ~A))
     ((C & A) v (C & B)) v (C & ~(~B & ~A))     CM, SLE
     (C & (A v B)) v (C & ~(~B & ~A))           D, SLE
     (C &  (A v B)) v (C & (~~B v ~~A))         DM, SLE
     (C & (A v B)) v (C & (B v A))              DN x 2, SLE
     (C & (A v B))  v (C & (A v B))             CM, SLE
     C & (A v B)                                RD


h)   IN THE FOLLOWING PROOFS WE WORK UPSIDE DOWN, THAT IS, BEGIN
WITH THE TARGET, AND WORK BACKWARDS TO THE ORIGINAL STATEMENT.

     C & (~A v ~(~C v A))
     C & (~Av (~~C & ~A))          DM, SLE
     C & (~A v (C&~A))             DN, SLE
     (C & ~A) v (C & (C & ~A))     D
     (C & ~A) v ((C & C) & ~A)     A, SLE
     (C & ~A) v (C & ~A)           R, SLE
     C &~A                         R


i)   C & (~(Av~B) v (B & ~(~C v A)))
     C & ((~A & ~~B) v (B & (~~C & ~A)))     DMx2, SLE
     C & ((~A & B) v (B & (C & ~A)))         DNx2, SLE
     C & ((~A & B) v ((B & C) & ~A))         A, SLE
     (C & (~A & B)) v (C & ((B & C) & ~A))   D
     (C & (~A & B) v ((C & (B & C)) & ~A)    A, SLE
     (C & (~A & B) v (((B & C) & C) & ~A)    CM, SLE
     (C & (~A & B) v ((B & (C & C)) & ~A)    A, SLE
     (C& (~A & B)) v ((B & C) & ~A)          R, SLE
     ((~A & B) & C) v (~A & (B & C))         CM, SLE
     ((~A & B) & C) v ((~A & B) & C)         A, SLE
     (~A & B) & C                            RD