Here are the 8 different logic rules that you should memorize the name
or the abbreviations for.  

    Equivalence rules                Inference rules
    -----------------                ---------------
    dist (Distributive)              mp (modus ponens)
    DM (De Morgan's)                 mt (modus tollens)
    contra (contrapositive)          ds (disjunctive syllogism)
    imp (implication)
    dn (double negative)

Without using any notes, using the letters P, Q and R, give the minimal
  number of numbered steps to illustrate each of the 3 inference rules, 
  and give the minimum number of letters and logical operators to 
  illustrate each of the 5 equivalence rules.

  1.  dist                         2.  mp


  3.  mt                           4.  DM


  5.  imp                          6.  dn


  7.  ds                           8.  contra

Write down the 8 abbreviations on a separate sheet of paper, then write
   the name of the inference rule or the equivalence rule that each
   is an abbreviation or acronym for.  Don't look at this sheet, or
   the exercise will not do you any good.

For each of the inference rules, write the shortest possible proof 
  involving A and B that proves B as the goal.  Note that sometimes
  you will need to have A' or B' along with A, B and whatever 
  operators are needed.

  1.  mt


  2.  mp


  3.  ds

Now, go back an do each of the 3 different proofs using the other two
  rules instead of the rule you used.  Each one will be longer, and you
  will have to bring in other rules, perhaps equivalence rules such as 
  imp or contra, etc.

For example, for the mt (modus tollens) proof from 1. mt above:

   Do 1. mt using mp instead of mt.

   Do 1. mt using ds instead of mt.

For 2. mp proof you developed above:

    Do 2. mp using mt instead of mp.

    Do 2. mp using ds instead of mp.

For 2. ds proof you developed above:

    Do 2. ds using mp instead of ds.

    Do 2. ds using mt instead of ds.

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If you can do this without looking at your notes or using the textbook
  and handouts, you are learning the rules and getting comfortable with
  the logical equivalence and logical inference rules and the required
  moves of the "game" of mathematical reasoning.
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12.  Prove that [ not (p and q) and q and (not r --> p) ] --> r

     a.  How many premises or hyp will this proof have?

     b.  What is the goal of the proof?

     c.  Do the proof and number each step, as well as stating the
         reasons for each step and the justifications for each step,
         as we have always done in the classroom examples.

21.  Simplify [ (A and B) or (A and C) ] and (not C) and (not B)

     (NOTE:  The NOT operator has the highest precendence of all the 
             operators, so writing

                not R ---> S       means the same as  (not R) ---> S
  
                              and
 
                not S and not T    means the same as   (not S) and (not T)

      In other words, the parenthesis are NOT needed, because the
         not operator has a higher precedence than AND, OR and IMPLIES).