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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, ${-2,-1,0,1,2} \mapsto {4,2,1,3,5}$.
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