I know that the cardinality of all binary words is $2^{\aleph_0}$, but in my assignment I need to find out what is the cardinality of all finite and infinite binary words excluding those that contain the sequence $01$.
I think there are just 3 options:
$(1,1,0,0,0,\ldots)$ finite sequence of $1$'s and then infinite sequence of $0$'s
$(0,0,0,0,0,\ldots)$
$(1,1,1,1,1,\ldots)$
Since the first option is countable the answer should be $|R^N| =$ $2^{\aleph_0}$ Am I right?
$\aleph$ is not a cardinal. It is $\aleph_0$.Next, youare right for your three cases. But then, it means that the cardinality you seek is $\aleph_0$ (and not $2^{\aleph_0}$). Indeed, you have an surjection from $\mathbb N \times \mathbb N $ to the set of finite and infinite binary words excluding those that contain the sequence 01. Namely, to $(m,n)$ we assign the sequence with $m$ ones and $n-2$ zeros if $n \ge 2$, $m$ ones and infinitely many $0$ if $n=1$ and infinitely many $m$ otherwise.