"The set $\{0,1\}^{10^{10}}$ of all bit strings of length $10^{10}$ " meaning?

Question

I have a proof to do but I'm not sure about the meaning of the statement above, does it imply simply "The set of all 0s and 1s with the length of $10^{10}$ "?

Answer 1

Exactly! $X^n$ is the set $\underbrace{X \times X \cdots \times X}_{\text{n times}}$.

Equivalently, it is the set of functions $f : \{1,2,\ldots,n\} \to X$, but these are the same thing because we can identify a function $f$ with the tuple $(f(1), f(2), \ldots, f(n))$.

Either way, for $\{0,1\}^5$ this is the set of all $(b_1, b_2, b_3, b_4, b_5)$ with each $b_i \in \{0,1\}$. This is exactly the set of binary strings of length 5!

Similarly, in your case $\{0,1\}^{10^{10}}$ is exactly the set of strings of length $10^{10}$.