What is the cardinal all the finite & infinite binary sequences that don't contain '10'?

Question

What is the cardinal all the finite & infinite binary sequences that don't contain '10'?

I know I can make a function $f(z, o)$ where $z, o \in \mathbb{N}$ that builds a sequence that starts with $z$ zeros and ends with $o$ ones. But that only gives me the cardinal of the finite sequences. How should I continue?

Answer 1

You have the right idea. Such a sequence is determined by:

• The length of the sequence, which takes values in $\mathbb{N} \cup \{ \infty \}$;
• The position (counting from $1$) at which the sequence switches from $0$ to $1$, which takes values in $\mathbb{N} \cup \{ \infty \}$, where a value of $0$ corresponds to a sequence which is constantly zero, and a value of $\infty$ (or a value greater than the length of a finite sequence) corresponds to a sequence which is constantly one.

So you can surject $(\mathbb{N} \cup \{ \infty \}) \times (\mathbb{N} \cup \{ \infty \})$ onto your set, by mapping $(\ell, p)$ to the sequence with length $\ell$, which has $0$ in all positions up to and including position $p$ and $1$ in all positions afterwards.

(P.S. This post uses the convention that $0 \in \mathbb{N}$.)