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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