This problem is awesome!

First, the $\LaTeX$ syntax for $\mathcal{P}(\mathbb{N})$ is

`[[$\mathcal{P}(\mathbb{N})$]]`

Let $X$ be set of all sequences of 0's and 1's. Each element of $X$ is of the form $(a_i)_{i=1}^{\infty}$, where each $a_i$ equals 0 or 1 (but not both!). In order to show that $\mathcal{P}(\mathbb{N})$ and $X$ have the same cardinality, we need to show that there is a 1-1 correspondence between the two sets. Our job is to construct a function between these two sets and then show that it is (1) 1-1, and (2) onto. The hardest part is figuring out what the function should be and actually writing it down nicely. I'll do that part for you now.

Define $f: \mathcal{P}(\mathbb{N}) \to X$ via

(1)where $A\subseteq \mathbb{N}$ and $(a_i)$ is defined by

(2)The $\LaTeX$ syntax for the above definition of $a_i$ is

```
[[math]]
a_i=\begin{cases}
0, & \text{if } i\notin A\\
1, & \text{if } i\in A
\end{cases}.
[[/math]]
```

All that remains to do is to show that $f$ is 1-1 and onto.