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

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

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.

If you can figure out the path by drawing a picture, I can help you scan it and post it here.

What does Theorem 7.3.4 tell us about $N'$? Next, define the function $g: N'\to X$ via $g(n_x)=x$. Is $g$ onto? How about 1-1? What does Theorem 7.3.3 tell us?

1. $C$ finite,
2. $C$ countably infinite.

In case 1, there exists a 1-1 correspondence $g: \{1,2,\dots,k\}\to C$ for some $k$ (by definition of $C$ being finite). Consider the composition $f\circ g: \{1,2,\dots, k\}\to B$ and take a look at Theorem 5.2.3. Conclude that $B$ is finite and hence countable.

In case 2, there exists a 1-1 correspondence $g: \mathbb{N}\to C$ (since $\mathbb{N}$ and $C$ have same cardinality). Again consider the composition $f\circ g$ and argue similar to case 1, except you will conclude that $B$ is countably infinite.

[[$ insert LaTeX equation here $]]

Between the dollar signs is where you place specific $\LaTeX$ commands for whatever math symbol(s) you want. All $$\LaTeX$ commands begin with the \-symbol. For example, if you wanted to typeset "square root of x squared plus y squared", you would type the following in the page editor.

[[$\sqrt{x^2+y^2}$]]

The result of the above is code is:

$\sqrt{x^2+y^2}$

Also, notice that when you are in the page editor, among the many buttons above the text box are two labelled $\sqrt{x}$ and $x/2$, respectively. The first of these inserts initial code for a mathematical expression or equation that you want to display on its own line and centered. The second button inserts the initial code for an inline mathematical expression. Using these buttons can save you a bit of typing.

Just try it out and see what happens. The beauty of the wiki is that you can go back and fix and/or edit things at any later date. It doesn't have to be perfect on the first go. Let me know if you have more questions.