1. A statement is logically necessary if it is true regardless of the
   premises.

2 -> Rule states that:

 p->Q
 P
---
 Q

3. A sound argument is one that is valid and whose premises are all
   true.

4. A Sentence that can be made true on some logical grounds is called
   logically possible.

5 The completeness theorem of F states that:
 If S is a tautological consequence of T then T |- S (ie. S can be
 proven from T)

6. All logical necessities are tautologies  [False]
7. All logically equivalent statements are also tautologically
equivalent [False].

8 All sound arguments are valid [True]
9. A -> B
   B
   ---
   A
  [false]

10. The sentence S is a tautological consequence of T iff T U {-S} is
    not tt-satisfiable [True]


11. proove that (A ^ B) V C |--t B V C

  1. (A ^ B) V C
  ---------------
    |A ^ B
    |------
    | B              ^ Elim 2
    | B V C          V intro 3
    
    | C
    |-----
    |B V C            V intro 5
  
     B V C             
 


13. resolution
   (C V A)  ^ -C ^ -(A V B)
      
   {C, A}  {-C} {-A} {-B}
   -----------
	{A}     {-A} {-B}
        ------------
             []      {-B}   # you can stop here
	     ------------
		  []  #redundant step



18.  Either P or Q or both   = P V Q (since V is an inclusive or it
     takes care of the 'both')

      an alternate answer is  (P V Q) V (P ^ Q) but you can easily see
      that this statement reduces to P V Q

      observe
      (P V Q) V (P ^ Q)
      
      ==> [(P V Q V P) ^ (P V Q V Q)] distribution over V
      ==> (P V P V Q) ^ (P V Q V Q)
      ==> (P V Q) ^ (P V Q)
      ==> P V Q                  qed.