BijectionsDate: 04/28/2003 at 13:08:03 From: Mandy Subject: Bijections Find a bijection from (0, 1) to (0, 1]. (Be sure to prove that your function has the proper properties.) Date: 04/28/2003 at 16:52:15 From: Doctor Rob Subject: Re: Bijections Thanks for writing to Ask Dr. Math, Mandy. The trick here is to pick a countably infinite subset S of (0,1). Let S = {x[1],x[2],x[3],...}. Then define your function f:(0,1)->(0,1] by f(x[1]) = 1, f(x[n+1]) = x[n] for all n >= 1, f(x) = x for all x in (0,1) but not in S. Then prove that f is one-to-one and onto (that is, a bijection). Feel free to write again if I can help further. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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