Question

Choose the regular set

• A. $\left\{ww^{R}ww^{R}ww \mid w \in \{a,b \}^+ \right\}$
• B. $\left\{ a^{2^n} a^{n!}a^{pq}  \mid p \text{ prime}, q\text{ not prime}, n \geq1\right\}$
• C. $\left\{ wXw^{R} \mid w, X \in \{a,b \}^+ \right\}$
• D. $\left\{a^{i}b^{j}a^{i}b^{j}a^{i}b^{j} \mid i,j \geq 1 \right\}$

Comments

its not visible clearly though it seems CSL coz of non linear power.

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C is regular ,a(a+b)⁺a +b(a+b)⁺b 

A and D is csl. 

Need some clarity about B, actually about the language (strings belong to L). [Yes it is nonregular and not cfl]

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Language is infinite.

A dfa can accept a infinite language iff it follows a certain pattern by forming a loop in DFA  According to PUMMPING LEMMA.

Since prime no doesnt follow any definite pattern so DFA for this is not possible

Answer 1

for any string in the language represented by option C

we can have a $X$ such that first and last symbol of the string are same.

so, due to the presence of that $X$ language is reduced to the language definition where the first and the last symbol are same

Hence, that can be accepted by a DFA. Therefore, the option C is Regular

Answer 2

C option is correct.