Question 44

Let us determine the cardinality of each set:

• $S_1$ (Recursively enumerable languages): Every recursively enumerable language is accepted by a distinct Turing machine. Since every Turing machine can be encoded as a finite binary string, the set of all Turing machines is countably infinite. Hence, $S_1$ is countably infinite.

• $S_2$ (Syntactically valid C programs): Every valid C program is a finite sequence of ASCII characters. The set of all finite strings over any finite alphabet is countably infinite. A subset of a countable set is countable. Hence, $S_2$ is countably infinite.

• $S_3$ (All languages over $\{0, 1\}$): The set of all languages is the power set $\mathcal{P}(\Sigma^*)$. Since $\Sigma^*$ is countably infinite ($|\Sigma^*| = \aleph_0$), its power set has cardinality $|\mathcal{P}(\Sigma^*)| = 2^{\aleph_0} = \mathfrak{c}$, which is uncountably infinite.

• $S_4$ (All non-regular languages over $\{0, 1\}$): The set of all regular languages is countable (each regular language corresponds to a finite regular expression or DFA, which is countable). Since the set of all languages is uncountable and the set of regular languages is countable, their difference (the set of non-regular languages) must be uncountably infinite.

Therefore, sets $S_3$ and $S_4$ are uncountable (Option B).