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author | luccul <luccul@gmail.com> | 2010-07-06 04:16:54 +0000 |
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committer | bnewbold <bnewbold@adelie.robocracy.org> | 2010-07-06 04:16:54 +0000 |
commit | f04dc6dfe61b4939d0cbacee34dcbf5f0e429daf (patch) | |
tree | bc1e0b77144aa844c0e1b4e517fe78f51001b5be | |
parent | 1fe1ef1eb46b08427c0039e03c9f72489c341572 (diff) | |
download | afterklein-wiki-f04dc6dfe61b4939d0cbacee34dcbf5f0e429daf.tar.gz afterklein-wiki-f04dc6dfe61b4939d0cbacee34dcbf5f0e429daf.zip |
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-rw-r--r-- | Problem Set 2.page | 2 |
1 files changed, 1 insertions, 1 deletions
diff --git a/Problem Set 2.page b/Problem Set 2.page index 2b9e5c6..5e41ea2 100644 --- a/Problem Set 2.page +++ b/Problem Set 2.page @@ -99,7 +99,7 @@ Proofs: -$\mathbb{Q}=\mathbf{N}$ by combining the bijections from $\mathbf{N}$ to $\mathbf{Z}$ and from $\mathbf{N}$ to $\mathbb{N} \times \mathbb{N}$. This provides a bijection from $\mathbf{N}$ to $\mathbb{Z} \times \mathbb{Z}$, and since every element of $\mathbb{Q}$ can be represented as the ratio of the two components of an ordered pair of integers, we have a bijection from $\mathbb{Z} \times \mathbb{Z}$ to $\mathbb{Q}$. --Josh explained Cantor's proof of the uncountability of the real numbers on the 28th; Wikipedia provides a good description thereof: [external](http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument). In a nutshell, you assume you can have an ordered list of the reals (i.e., a bijection to the naturals), then construct a real number not on that list by having its nth digit be different from the nth digit of the nth number on the list. +-Josh explained Cantor's proof of the uncountability of the real numbers on the 28th; [Wikipedia](http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument) provides a good description thereof. In a nutshell, you assume you can have an ordered list of the reals (i.e., a bijection to the naturals), then construct a real number not on that list by having its nth digit be different from the nth digit of the nth number on the list. -$(0,1)=\mathbf{R}$ and $[0,1]=\mathbf{R}$ under the same bijection: $n \mapsto \tan\left (\pi n - \pi/2 \right)$. |