In the image you provided, there are two mathematical problems:

Problem 3: "Express $\mathbb{R}^2 \setminus \{(0, 0)\}$ as a union of pairwise disjoint family of sets indexed by $\mathbb{R}$."

Problem 4: "Let $x \in \mathbb{R}$. Express $\{x\}$ as an intersection of a family of open intervals indexed by $\mathbb{N}$."

Problem 3:

To express $\mathbb{R}^2 \setminus \{(0, 0)\}$ as a union of pairwise disjoint sets, one possible approach is to use polar coordinates. Polar coordinates $(r, \theta)$ map every point in $\mathbb{R}^2 \setminus \{(0, 0)\}$ except the origin. Thus, we can define the sets as:

$A_r = \{ (r \cos \theta, r \sin \theta) : \theta \in [0, 2\pi) \}$

for each $r > 0$. These sets $A_r$ form a disjoint union, and each one corresponds to a unique radius $r$, thus indexed by $\mathbb{R}^+$. So, the union becomes:

$\mathbb{R}^2 \setminus \{(0, 0)\} = \bigcup_{r > 0} A_r$

where the sets $A_r$ are pairwise disjoint because each corresponds to a different radius.

Problem 4:

To express $\{x\}$ as an intersection of a family of open intervals, consider the fact that you can construct smaller and smaller open intervals around $x$. Let’s take a sequence of intervals:

$I_n = \left( x - \frac{1}{n}, x + \frac{1}{n} \right)$

for $n \in \mathbb{N}$. Each interval $I_n$ is open, and as $n \to \infty$, these intervals shrink and converge to $\{x\}$. Thus, we can write:

$\{x\} = \bigcap_{n=1}^{\infty} I_n$

This intersection of open intervals gives us the point $x$.

Would you like more details or clarifications on these problems?

Relative Questions:

1. How do polar coordinates help in converting Cartesian coordinates?
2. Why is $\mathbb{R}^2 \setminus \{(0, 0)\}$ difficult to represent directly in Cartesian coordinates?
3. What are the properties of disjoint sets, and why is this relevant in set theory?
4. How does the concept of open intervals relate to the definition of single points in real analysis?
5. Can you generalize the intersection method for expressing finite sets of points?

Tip:

When converting geometric sets or points in $\mathbb{R}^2$, utilizing polar coordinates can simplify the expression of sets that exclude the origin.