From d4e4095fe79a0b7dee17316129f1ae1fe09a23a3 Mon Sep 17 00:00:00 2001
From: joshuab <>
Date: Wed, 30 Jun 2010 20:26:13 +0000
Subject: created

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 Problem Set 2.page | 37 +++++++++++++++++++++++++++++++++++++
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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 $n,m \in \mathbb{Z}$
+    - $\mathbb{R}$
+    - The open interval $(0,1)$
+    - The closed interval $[0,1]$
+    - $2^{\mathbb{N}}$ = 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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