One to One Correspondence Between Sets

Date: 04/04/97 at 00:24:47
From: Anonymous
Subject: One to One Correspondence Between Sets

Dr. Math,

I am trying to find the one-to-one correspondence between the set of 
natural numbers and the following set "S":

     N= { 1    2     3       4     5   . . . . . . }
     S= {1/3  1/6   1/12   1/24  1/48  . . . . . . }   
      
What I have done is to make the corresponding element in set S a 
fraction of 1 over something.  This is where I begin to have my 
problem; I can not find a correlation between the natural numbers and 
the denominators of set S.  I've tried squaring and cubing "n" and 
then adding, subtracting, or multiplying different variations of n but 
nothing seems to work.  Should I get rid on the 1 over something and 
treat this correspondence to n as a fraction and use the rules of 
division regarding fractions?

Please help.

Thanks
Cathy Nicholas

Date: 04/04/97 at 01:53:43
From: Doctor Mike
Subject: Re: One to One Correspondence Between Sets

Dear Cathy, 
  
The key idea is seeing the relationship between one denominator
and the next one.  The first denominator is three, but then each
larger one is 2 times the one before.  So you can make a table:
  
       Denominator          Multiplication for
        Number               the denominator
       -----------          -------------------
           1                  3
           2                  3*2
           3                  3*2*2
           4                  3*2*2*2
           5                  3*2*2*2*2
           6                  3*2*2*2*2*2
           7                  3*2*2*2*2*2*2
          ...                ...
           
The general connection is the number of factors in the denominator
that are 2.  For instance, for denominator 7 there are six 2's.  The
number of 2 factors is always one less than the denominator.  That
is even the case for the first denominator, where there are zero 2's.
When you have the same number multiplied together several times you
use the exponent (also called power) notation.  Here that would be
2 with a superscript written above and to the right.  In e-mails we
usually write it with the "^" symbol, like 2^N for 2 to the power N.

Back to your specific question.  The denominator for the fraction
number N would be 1 over 3*2^(N-1).  I'll write it the other way too.

                   1     
               ----------
                     N-1   
                3 * 2      
   
By the way, when N is 1 (the first fraction) you get 2 to the zero
power in the formula.  We are often asked why a number to the zero
power is said to equal 1.  One very good reason that mathematicians
define use of the zero power this way is that it works out so
naturally in many formulas like this.  If we had said that 2^0 is
something else, like = 0 maybe, then we would have to say that the
formula here holds for all of your fractions EXCEPT the first one. 
  
I hope this helps.

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/