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| -rw-r--r-- | Problem Set 1.page | 22 |
1 files changed, 11 insertions, 11 deletions
diff --git a/Problem Set 1.page b/Problem Set 1.page index a68109f..eb2c6e7 100644 --- a/Problem Set 1.page +++ b/Problem Set 1.page @@ -2,15 +2,15 @@ 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. + 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! @@ -21,8 +21,8 @@ Cook up other examples and post them on the wiki! 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) + 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} $ |