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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?

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? Correct Answer S3 and S4

  • Every Recursively enumerable language has a Turing Machine, and a set of all Turing Machine's is countable. Therefore, the set of all Recursively enumerable languages is countable.
  • There exists a one to one equivalence for all valid C programs and all valid TM encodings. Since the set of all valid TM encodings is countable, it means that the set of all syntactically valid C programs is also countable.
  • Set of all languages is uncountable and the set of all Regular Languages is countable. So, the set of all non-regular languages should be uncountable as the union of two countable sets cannot make an uncountable set.


S3 and S4 are uncountable sets.

Important Points:

Every TM can be encoded with 0's and 1's, that is, every TM can be represented by a unique binary number.

Σ = {0,1} then set of all binary strings = Σ∗. Σ∗ is countable → Set of all TM's is countable.

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