Cardinality between Open and Closed SetsDate: 09/20/2001 at 00:01:16 From: Phillip Brubaker Subject: Cardinality between open and closed sets I would like to know how to prove that the sets (0,1) and [0,1] have the same cardinality. I know that they have the same cardinality as the set of all reals. I can prove that each set by itself has the same cardinality of all reals, and since a has same cardinality as x and b has same cardinality as x, a and b have same cardinality. But I would like to prove it another way without using the set of all reals. Date: 09/20/2001 at 09:44:08 From: Doctor Rob Subject: Re: Cardinality between open and closed sets Thanks for writing to Ask Dr. Math, Phillip. Set up a one-to-one correspondence f between the two sets as follows. Let S = {1/(n+1): n in Z, n >= 1}, a countably infinite subset of (0,1). Define f(0) = 1/2, f(1) = 1/3, f(x) = 1/(2+1/x) = x/(2*x+1) if x in S, f(x) = x if x in (0,1) but not in S. Then f:[0,1] -> (0,1) is one-to-one and onto. A minor variation on this shows that any finite number k of points s[1], s[2], ..., s[k] can be added to an infinite set without changing its cardinality. (Replace the first two equations above with f(s[i]) = 1/(1+i) for i = 1, 2, ..., k, and change 2 to k in the third equation above.) What this proves is that in cardinal number arithmetic, N + k = N for any infinite cardinal number N and any finite cardinal number k. Actually, although I chose a specific S, any countably infinite subset of (0,1) can be used in a similar construction. A variation on this shows that any countable number of points can be added to an infinite set without changing its cardinality, or N + aleph_0 = N for any infinite cardinal number N. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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