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diff --git a/Problem Set 1.page b/Problem Set 1.page new file mode 100644 index 0000000..a68109f --- /dev/null +++ b/Problem Set 1.page @@ -0,0 +1,34 @@ +## Countability + +1. Group the following sets according to their cardinality: + + - $\mathbb{N} = \{ 1,2,3,4,\dots \}$ + - $\mathbb{Z} = \{ \dots, -2, -1,0,1,2, \dots \}$ + - $\mathbb{N} \times \mathbb{N}$ + - $\mathbb{Q}$ = Set of all fractions $\frac{n}{m}$ where $n,m \in \mathbb{Z}$ + - $\mathbb{R}$ + - $(0,1)$ + - $2^{\mathbb{N}}$ = Set of all subsets of $\mathbb{N}$. + - $2^{\mathbb{R}}$ = Set of all subsets of $\mathbb{R}$. + - $\mathbb{R}^{\mathbb{R}}$ = Set of all functions from $\mathbb{R}$ to itself. + +Cook up other examples and post them on the wiki! + +2. Let $X$ be any set. Show that the cardinality of $2^{X}$ is larger than the cardinality of $X$. Hint: Let $f: X \to 2^X$ be a bijection. Consider the set of all elements $x \in X$ such that $x$ is not an element of $f(x)$. + + +## Fourier Series + + +1. Compute the Fourier Series of the following functions. Do both the exponential and sin/cos expansions. +- $f(x) = sin^3(3x)cos^2(4x)$ +- $g(x) = x(x-2\pi)$ (Hint: Use integration by parts) + +2. Show that +$ \int_0^{2\pi} sin^4(x) dx = \frac{3 \pi}{4} $ +(Hint: write out the exponential fourier expansion of $sin^4(x)$.) + +3. Compute the exponential Fourier coefficients of $sin^2(x)$: +$ 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 $ |