One to One Correspondence Between SetsDate: 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/ |
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