BIG-THETA PROBLEM:  The find 3 numbers game.  Find c, d and m so that
                    c(g(n)) <= f(n) <= d(g(n)) is true for ALL n >= m 
                    -                  -                            -
                            to show 
                            function f is BigTheta function g, 
                      i.e. (f, g) element of the BigTheta binary relation.

9. a. Is f(n) = BigTheta( g(n) ) when f(n) =    3      2   and g(n) =    1    3
                                             2 n  + 3 n                ----- n
                                                                         10
      ---------------------------------------------
      Lets just look at keeping f(n) <= g(n) first:
      ---------------------------------------------
      Yes, the witnesses d and m that would work are:

                   d = 50 and m = 1 since f(n) <= (50)( g(n) ) for all n >= 1
                                                    d                       m
                                3
      Why choose 50?  To get 5 n  and thus keep
                             2 n cubed plus 3 n squared in line.....
                                                  from the very beginning,
                                                      n = 1 is the beginning.

      Other witnesses work too.  For example:

                 m = 10 and d = 23 are witnesses too.

                                23          3
       2,000 + 300 =  2,300 <=  --  times 10  = 2,300
                                10
            f(n)                     g(n)

      --------------------------------------------------
      Now we will add the g(n) <= f(n) part of BigTheta:
      --------------------------------------------------

9.b.  Is f(n) = BigTheta( g(n) )? 
     
      Yes, we can find values or witnesses c, d and m to testify to that
      fact.

              c g(n) <= f(n) <= d g<n) for all n >= m,
                                       using c = 1, d = 23, m = 10.
      Note that we could choose
              c = 20 but if the function is obviously <= without
                    an assist, we do not need to call in any expert witnesses,
                        c = 1 can step forward and be enough witness.
                  n
9.c.  For h(n) = 2 , is f(n) = BigTheta( h(n) )?

      Nope.  Cannot be done!  The two functions are NOT equivalent and
      there are not three witnesses in all the world of Natural numbers
      that can step forward and establish the fact!
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  Good review of binary relations and their properties.  Spring 1997 test.
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8.  isBrotherOf is transitive only.

                    John isBrotherOf Mary, but not (Mary isBrotherOf John)
                    is an example of why it is not symmetric.

                    It is not reflexive, in the usual sense of the term.
                    When you ask an only child who happens to be male,
                    if he has any brothers or sisters it would be quite
                    a surprise to hear, yes, just one brother, myself.
                        
    isFatherOf is not transitive, not reflexive, not symmetric.

    x isFatherOf y and y isFatherOf z does describe a binary
        relation, the isGrandFatherOf relation is what 
        ties x and z together here in a binary relationship.

    isEastOf on the set {Iowa, California, New York}

         is transitive (New York isEastOf Iowa and Iowa isEastOf California,
                        and indeed, New York isEastOf California),

         no symmetric relation property and no reflexive property for
         the isEastOf binary relation.

     You might want to review your definitions of antisymmetric and
       irreflexive for the final exam.  I will ask about all FIVE properties
       for any similar questions on the final.