GATE | GATE CS 2019 | Question 43

Consider the following sets:

• S₁: Set of all recursively enumerable languages over the alphabet {0, 1}.
• S₂: Set of all syntactically valid C programs.
• S₃: Set of all languages over the alphabet {0, 1}.
• S₄: Set of all non-regular languages over the alphabet {0, 1}.

Which of the above sets are uncountable?
(A) S₁ and S₂
(B) S₃ and S₄
(C) S₁ and S₄
(D) S₂ and S₃

Answer: (B)
Explanation:

1. Recursively enumerable languages are countable.
2. Syntactically valid C program can be represented with CFG. CFG generates CFL, CFL is countable.
3. All languages over {0, 1} may not be countable, because they may also lie in the region of 2^Σ*.
4. Set of regular languages are countable, non-regular languages may not be countable.

Option (C) is true.
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