Theory-of-Computation

(a) Given a set
S = {x| there is an x-block of 5’s in the decimal expansion of π}
(Note: x-block is a maximal block of x successive 5’s)
Which of the following statements is true with respect to S? No reasons need to be given for the answer.

(i) S is regular
(ii) S is recursively enumerable
(iii) S is not recursively enumerable
(iv) S is recursive

(b) Given that a language L₁ is regular and that the language L₁ ∪ L₂ is regular, is the language L₂ always regular? Prove your answer.