# Problem Set 2

1. Easy
Let $$\mathbf{u},\mathbf{v}$$ be two vectors in $$\mathbb{R}^{3}$$. Compute $$\mathbf{u}\times\mathbf{v}$$, $$\mathbf{u}\cdot\mathbf{v}$$ for the following choices of $$\mathbf{u}$$ and $$\mathbf{v}$$, and state whether or not $$\mathbf{u}$$ and $$\mathbf{v}$$ are either orthogonal or colinear.
1. $$\mathbf{u} = (6,0,-2)$$, $$\mathbf{v} = (0,8,0)$$
2. $$\mathbf{u} = (1,1,-1)$$, $$\mathbf{v} = (2,4,6)$$

2. Easy
Let $$\vec{a},\vec{b},\vec{c},\vec{d} \in \mathbb{R}^{3}$$ be vectors. Determine which of the following expressions are meaningless:
1. $$(\vec{a}\cdot\vec{b})\cdot\vec{c}$$
2. $$\vert\vec{a}\vert(\vec{b}\cdot\vec{c})$$
3. $$\vec{a}\cdot(\vec{b}+\vec{c})$$
4. $$(\vec{a}\cdot\vec{b})\vec{c}$$
5. $$\vec{a}\cdot\vec{b}+c$$
6. $$\vec{a}\cdot(\vec{b}\times\vec{c})$$
7. $$(\vec{a}\cdot\vec{b})\cdot(\vec{c}\cdot\vec{d})$$
8. $$(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})$$

3. Easy
If $$S$$ is not closed, then there exists $$x \in \overline{S}$$ but $$x \notin S$$

4. Medium
Prove that for any $$x,y \in \mathbb{R}^{n}$$, $$\vert x - y \vert \geq \big\vert \vert x \vert - \vert y \vert \big\vert$$. This is commonly called, the reverse triangle inequality''. It is extremely useful when you want to prove inequalities like $$\vert (x-a)^{-1} \vert \leq M$$ for $$x$$ in some fixed closed set.

5. Medium
If $$U$$ and $$V$$ are open (resp. closed) then $$U\cup V$$ is open (resp. $$U\cap V$$ is closed). If $$\left\{U_{i}\right\}_{i\in I}$$ is a countable collection of open sets, must $$\bigcap_{i\in I} U_{i}$$ be open? Provide a proof or counterexample. Similarly, if $$\left\{A_{i}\right\}_{i\in I}$$ is an infinite collection of closed sets, must $$\bigcap_{i\in I} A_{i}$$ be closed?

6. Medium
Prove that the following sets are open
1. $$\mathbb{R}^{n}$$
2. $$B(r,x)$$
3. $$\{ (x,y) \in \mathbb{R}^{2}: x > 0\}$$
4. $$\{(x,y) \in \mathbb{R}^{2}: x > 1\,\text{and}\, y > 0\}$$ (Hint: You could do this by brute force, or notice that the set can be written as an intersection of two other sets)
5. $$\{(x,y) \in \mathbb{R}^{2}:x \notin \mathbb{Z}\}$$ (This one is a bit harder)

7. Medium
Determine and prove whether the following sets are open, closed, or neither open nor closed
1. $$\{(x,y)\in\mathbb{R}^{2} : x \in \mathbb{Z}\}$$
2. $$\{(x,y) \in \mathbb{R}^{2}: x^2+y^2 = 1\}$$
3. $$\{(x,y) \in \mathbb{R}^{2}: x^2+y^2 \leq 1\}$$
4. $$\{(x,y) \in \mathbb{R}^{2}: x^2+y^2 \leq 1, (x,y) \neq (0,0)\}$$
5. $$\{(x,y) \in \mathbb{R}^{2}: x > 0, y=\sin(1/x)\}$$
6. $$\{ (x,y) \in \mathbb R^2 : 0<x<1, 0<y<1, x,y \in \mathbb Q \}$$
7. $$\{ (x,y) \in \mathbb R^2: y > x^2 \}$$

8. Hard
Construct an open set of arbitrarily small "size" which contains the rational numbers $$\mathbb{Q}$$, but which is a proper subset of $$\mathbb{R}$$. In this context we define the size of $$(a,b)$$ for $$a < b$$ to be $$b-a$$. We define the size of a union of open intervals to be the sum of their sizes.

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