An infinite binary word a is a string (a₁a₂a₃ ...), where each aₙ ∈ {0,1}.

Fix a word s = (s₁s₂s₃ ...), where sₙ = 1 if and only if n is prime.

Let S = {a = (a₁a₂a₃ ...) | ∃m ∈ ℕ such that aₙ = sₙ, ∀n ≥ m}.

What is the cardinality of S?

Correct option is C

Understanding the Given Setup:

We are given an infinite binary word a = (a1 a2 a3 ...) where each $$a_n$$​ ∈ {0,1}.

A fixed sequence s = (s1 s2 s3 ...) is defined as:
$$s_n = 1$$ if and only if n is prime.
The set S is defined as:
S = { a = (a1 a2 a3 ...) | ∃ m ∈ ℕ such that $$a_n = s_n,$$​ ∀ n ≥ m }.
This means that each sequence in S must eventually stabilize to match s from some point onward.

Structure of S

Each element of S can be described as follows:
- The first (m-1) elements of a can be freely chosen from {0,1}.
- From some index m onward, the sequence must exactly match s.

Since m can take any natural number value, and for each m, there are $$2^{(m-1)}$$​​

possible choices for the initial segment before stabilization, we analyze the total count of such sequences.

 Determining Cardinality

1. For each fixed m, there are $$2^{(m-1)}$$​ different sequences.
2. Since m can be any natural number, the total number of sequences in S is equivalent to the number of finite binary sequences.
3. The set of all finite binary sequences is countably infinite.
Thus, the set S has countable infinity as its cardinality.

Final Answer:

The correct choice is (C) Countably infinite.