Answer

Correct Option: (iv) $\{ w \mid w \text{ starts with 'a' and ends with 'b'} \}$

Explanation

Test each:

• (i) $\{a^n b^n\}$ → Not regular (needs stack, CFL).
• (ii) Equal number of $a$'s and $b$'s → $\{ w \mid \#_a = \#_b \}$ not regular (needs counting). Pumping fails.
• (iii) $\{a^n \mid n \text{ prime}\}$ → Not regular (non-periodic), also not CFL.
• (iv) Starts with a and ends with b → Regular.

DFA for (iv): 3 states

• $q_0$ start (no symbol yet, non-accepting)
• $q_1$ = last symbol a? Actually needs start a, so $q_0 \xrightarrow{a} q_1$, $q_0 \xrightarrow{b} q_{dead}$.
• $q_1$ = seen start a, last is a (non-accepting)
• $q_2$ = seen start a, last is b (accepting)
  Transitions: $q_1 \xrightarrow{a} q_1$, $q_1 \xrightarrow{b} q_2$, $q_2 \xrightarrow{a} q_1$, $q_2 \xrightarrow{b} q_2$.

Regular expression: $a(a+b)^*b$ (plus handle length 1? Actually $ab$ inclusive). More precisely $a\Sigma^*b$.

Hence regular.

Final Answer

Only (iv) is regular.