I think $\{0,1\}^*$ represents all $0,1$-sequences, and $\{0,1\}^{\mathbb{N}}$ is the $0,1$-sequences with infinite length. So $\{0,1\}^{\mathbb{N}}$ is a subset of $\{0,1\}^*$.
$\{0,1\}^*$ is countable, while $\{0,1\}^\mathbb{N}$ is uncountable.
It's really strange, because I can count the whole set and cannot when it comes to its subset!
I don't know how to understand it. Who can save me? Thanks in advance.
Best Answer
The set of all $0$-$1$ sequences is uncountable. $\{0,1\}^*$ more commonly means the set of finite $0$-$1$ sequences, which is countable.