Explanation of Notation Regarding Power Set of a Set In Grimmet and Strizaker's Probability and Random Processes it states section 1.2

The power set of $\Omega$, which is written $\{0,1\}^{\Omega}$ and contains all subsets of $\Omega$....

It goes on to use this in an example as follows:

A die is thrown once. We can take $\Omega$ = $\{1,2,3,4,5,6\}$, $F = \{0,1\}^{\Omega}$

Here $F$ is the set of all subsets of the sample space $\Omega$ which are of interest. I don't understand this notation to specify that set. How does it generate/enumerate all the subsets of interest?
 A: This is actually quite a profound aspect of set theory.
The powerset of $A$, or $\mathcal{P}(A)$, is defined to be $\{x : x \subseteq A\}$.
What is a subset of $A$? I suggest that, in some sense, a subset of $A$ is really just a function from $A$ to $\{0, 1\}$, where $0$ represents False and 1 represents True.
A function $f : A \to \{0, 1\}$ determines a subset of $A$ by $\{x \in A : f(x) = 1\}$.
Conversely, a subset $X \subseteq A$ determines a function $f : A \to \{0, 1\}$ by defining $f(a) = 1$ if $a \in X$, and $f(a) = 0$ otherwise.
Thus, with 0 representing False and 1 representing True, $f(a)$ tells us whether the statement $a \in X$ is true or false.
So strictly speaking, elements of $\mathcal{P}(A)$ are not functions $f : A \to \{0, 1\}$. But there is a bijective correspondence. Thus, we often write $\mathcal{P}(A) = \{0, 1\}^A$.
A: The Cartesian product
$$\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}$$ denotes the set of all $6$-tuples made of zeroes and ones. There are $2^6$ of them. You could write this as
$$\{0,1\}^6.$$
Now if you interpret a $0$ as "leave" and a $1$ as "take", every tuple corresponds to a subset. E.g.$(1,1,0,1,0,0)\equiv\{1, 2, 4\}$. In this context, the notation
$$\{0,1\}^\Omega$$ makes sense, though it is purely conventional. Another common notation is
$$2^\Omega.$$
