Date: Mon, 28 Aug 2006 16:39:01 -0500 (CDT)
   From: Mark Jacobson <jacobson@math-cs.cns.uni.edu>
     To: 810-080-02-FALL@uni.edu
Subject: not (P implies Q)

Hi 080 students,

      What if you have the NOT of an If P Then Q ?????
      -------              ---       -----------

      After class, I answered a question about the

          NEGATION of an IMPLICATION...

     If you negate
                      A --------------> B

     you get:
                                         '
                      (A ------------> B)
          or
                      not (A implies B)

     There is nothing you can do immediately with the
     negation of an implication.

     DM (De Morgan's) laws only apply to NOT of an OR
                                   or to NOT of an AND

     However, by the implication equivalence or rule,

           every implies can be converted into an OR.
                 -------                          --

     So
        not (A -----> B)    is equivalent to

        not (A'  or   B)     where A' means not A

     Now, we can apply DM law to

        not (A'  or  B)    because it is a NOT of and OR, i.e.

                                   it is the NEGATION of a DISJUNCTION

        making it


         (A')'  and   B'         (using DM law)

                               which is just (after dn for double negation)

          A and B'

            so (A ----> B)'   is the same as  A and B

    1. (A -----> B)'     hyp (or given)   NEGATION of the A implies B
                                          ---------------------------

    2. ( A' or B )'      1, imp (imp = implication rule)

    3.   (A')' and B'    2, DM (De Morgan's laws)

    4. A and B'          2, dn (double negation)
                                -      -

    I will put this on the web page too, so you can read it there
    if you need to.

    http://www.cns.uni.edu/~jacobson/c080.html

    http://www.cns.uni.edu/~jacobson

    Mark