Date: Mon, 11 Sep 2006 17:10:00 -0500 (CDT)
   From: Mark Jacobson <jacobson@math-cs.cns.uni.edu>
     To: 810-080-02-FALL@uni.edu
Subject: Read section 2.1

Hi 080 students,

     We just barely got started on chapter two, section 1 today at the
very end of class.

     I will hand out solutions to one or two of the homework problems that
were asked about today in class.  (The homework was NOT collected today,
as the Neko #4 cat not only did not win the race, it came in last)!

     Here is what we got to at the end of class.

Prove that the sum of an even and an odd number is odd, i.e.

Prove that
   If x is an even integer and y is an odd integer then the sum x + y is odd.
      --------------------------------------------      --------------------

   If x is even and y is odd then the sum of x and y is odd.

   1.  x is even              hyp
   2.  y is odd               hyp     GOAL to SHOW THAT x + y is odd
   3.  x = 2m for some
           integer m          1, definition of even

   4.  y = 2p + 1 for some    2, definition of odd
                 integer p

   5.  TO BE CONITINUED in class
       ON Wednesday....

   Note:  How will be know that

              x + y      is an odd integer?????

          When we have been able to show that

                x + y = 2t + 1 for some integer t.

   Note 2 divides even integers
        2 does NOT divide odd integers

   4 divides x means x = 4r for some integer r, i.e. x is a multiple of 4.

   4 does not divide y means x = 4s + t where t = 1, 2, or 3, i.e.

                     x can be expressed as 4 times some integer s plus
                       a remainder t
                                     where t > 0 and t < 4

   The even and odd numbers are special since when

         2 does NOT divide m, we know that the remainder is 1, i.e.

                m can be expressed as 2a + 1 for some integer a.

                There exists integer a, such that m = 2a + 1
                when m is an ODD integer.

   Mark