From c17d9d136c0d00f861689001aef4e1c315acf5e1 Mon Sep 17 00:00:00 2001
From: joshuab <>
Date: Wed, 30 Jun 2010 20:44:35 +0000
Subject: worked on cardinality, showed that N=Z

---
 Problem Set 2.page | 10 ++++++++++
 1 file changed, 10 insertions(+)

(limited to 'Problem Set 2.page')

diff --git a/Problem Set 2.page b/Problem Set 2.page
index 9ea5399..ab8a149 100644
--- a/Problem Set 2.page	
+++ b/Problem Set 2.page	
@@ -35,3 +35,13 @@ $\int_0^{2\pi} \sin^4(x) dx = \frac{3 \pi}{4}$
 $a_n = \frac{1}{\sqrt 2\pi} \int_0^{2\pi} \sin^2(x) e^{-inx} dx$  
 and use this to show that  
 $\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$  
+
+# Solutions
+
+## Countability
+
+Cardinality of the natural numbers (countable): $\mathbf{N}$,$\mathbf{Z}$
+Cardinality of the real numbers (continuum): $\mathbf{R}$
+
+Proofs:
+- $\mathbf{Z}=\mathbf{N}$ under the bijection $n \mapsto 2n+1$ for nonnegative $n$ and $n \mapsto 2|n|$ for negative $n$. For example, ${-2,-1,0,1,2} \mapsto {4,2,1,3,5}$.
\ No newline at end of file
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