From 5c955d99061fb0d505de7e86776f7ecedbf1cc74 Mon Sep 17 00:00:00 2001
From: joshuab <>
Date: Wed, 30 Jun 2010 19:05:49 +0000
Subject: Added Problems

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 Problem Set 1.page | 34 ++++++++++++++++++++++++++++++++++
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 create mode 100644 Problem Set 1.page

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+## 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 $  
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