Date: Mon, 22 Oct 2001 --- 9:22 PM Central Standard Time
   From: Mark Jacobson <jacobson@math-cs.cns.uni.edu>
     To: Wes.Montgomery@uni.edu
Subject: Re: HW #5 links...

The only set that is not a proper subset of

   A = {1,2,3,4,5,6,7,}

     is set A itself!           A is a subset of A, but
                                        A isn't a proper subset of A.
  In other words,
      {1,2,3,4,5,6,7,} is not a proper subset of {1,2,3,4,5,6,7,}

             A         is not a proper subset of        A.

   {2, 5) is a proper subset of  {1,2,3,4,5,6,7,}

   {1, 2, 4, 5, 6, 7} is a proper subset of {1,2,3,4,5,6,7,}

   { } is a proper subset of {1,2,3,4,5,6,7,}

   A proper subset is any subset of the set EXCEPT FOR the set itself!
                      ----------

   How many subsets would a set consisting of 7 elements have?

   I am thinking it would have twice as many as a set consisting of
      6 elements.  There is a formula on the class handout and on the
      class web page and many of the new sets hints/help pages that is to
      be studied and applied.

   The PowerSet( {a, b} )  will have 4 elements,
   the PowerSet({a, b, c}) will have 8 elements,

   How many of those subsets will NOT be proper subsets!
                     -------             --------------

   A PowerSet is just a set of all possible subsets of a set!

On Mon, 22 Oct 2001 Wes.Montgomery@uni.edu wrote:

> So for A = {1,2,3,4,5,6,7,} there are no proper subsets.  Or are the
> proper subsets {}, {1}..{7} sense in your example {1,2} could not be a
> proper subset. So ther are 8 proper subsets or am I still confused.

>  On Mon, 22 Oct 2001, Mark Jacobson wrote:
>
> >    Wes,
   
> >     I just sent out two examples of powersets.
> >
> >     A is not a proper subset of A, but everything else is.
> >
> >    If A = {a, b},
> >
> >       {} is proper subset of A,
> >       {a} is proper subset of A,
> >       {b} is proper subset of A,
> >       {a, b} is NOT a proper subset of A, i.e. A not proper subset of A.
> >
> >       {a, b} is a subset of {a, b} but its NOT a PROPER SUBSET of {a, b}.
> >
> >       Mark
> >
> > On Mon, 22 Oct 2001 Jewell.Alaska@uni.edu wrote:
> >
> > > I am having problems with number 8 on the first part of the homework.  
> > > A  =  {1,2,3,4,5,6,7}
> > > proper subsets of A, nonempty proper subsets of A?

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