What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them

Question

As the title suggests, the question is :

What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them ?

I'll tell you where im stuck, let's say f is a function that recieves a natural number and creates a sequence of just ones, i.e : f(6) = (1,1,1,1,1,1), f(2) =(1.1). Uniting this countable sets togheter will generate a countable set, so lets call the orinignal set(the one we need to find his cardinality) A, so A>= 0א because this set is a subset of A.

Also, A is a subset of all binary sqeunces (infinte and finite) and all the binary sqeunces cardinaliy is א, so A <= א.

So right now i have א0 <= A <= א, which does not help me a lot because i need to find out if its א or א0.

Thanks in advance !

Answer 1

If $0$ appears in the sequence, then all the following terms must be $0$ so that $01$ does not appear in the sequence. So the sequences are: $$000\ldots\\ 10000\ldots\\ 11000\ldots\\$$

and so on. The sequence $111\ldots$ is also included. This is clearly a countable set.

Answer 2

If the sequence $01$ cannot appear, then any sequence in your set must be either an infinite sequence of $1$s, or be of the form $$\underbrace{11 \cdots 1}_{m\ \text{times}}\underbrace{00\cdots}_{n\ \text{times}}$$ where $0 \le m < \omega$ and $0 \le n \le \omega$.

Why? If a $0$ appears at any point in the string then $1$ can't appear at any point later in the string. Thus there is an evident bijection between your set and the set $$\{ (\omega, 0) \} \cup \{ (m, n) : 0 \le m < \omega,\ 0 \le n \le \omega \}$$ What is the cardinality of this set?