worked on cardinality, showed that N=Z

1 files changed, 10 insertions, 0 deletions

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