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## Countability
1. Group the following sets according to their cardinality:
a. $\mathbb{N} = \{ 1,2,3,4,\dots \}$
- $\mathbb{Z} = \{ \dots, -2, -1,0,1,2, \dots \}$
- $\mathbb{N} \times \mathbb{N}$
- $\mathbb{Q}$ = 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.
a. $f(x) = \sin^3(3x)\cos^2(4x)$
- $g(x) = x(x-2\pi)$
(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
$\int_0^{2\pi} |\sin^2(x)|^2 dx = \sum |a_n|^2.$
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