Question

Match LIST-I with LIST-II \[\begin{array}{|c|c|c|}\hline \text{ } & \text{LIST-I}  & \text{LIST-II} \\ \hline \text{A.} & \text{A Language L can be accepted by a Finite Automata, if and only if, the set of equivalence classes of $L$ is finite.} & \text{III. Myhill-Nerode Theorem} \\ \hline \text{B.} & \text{For every finite automaton M = $(Q, \Sigma, q_0, A, \delta)$, the language L(M) is regular.} & \text{II. Regular Expression Equivalence} \\ \hline \text{C.} & \text{Let, X and Y be two regular expressions over $\Sigma$. If X does not contain null, then the equation $R = Y + RX$ in R, has a unique solution (i.e. one and only one solution) given by $R = YX^*$.} & \text{I. Arden's Theorem} \\ \hline \text{D.} & \text{The regular expressions X and Y are equivalent if the corresponding finite automata are equivalent.} & \text{IV. Kleen's Theorem} \\ \hline \end{array}\] 

\[\text{Matching List-I with List-II}\]

Choose the correct answer from the options given below:

• A. A - I, B - IV, C - III, D - II
• B. A - I, B - III, C - D, D - IV
• C. A - I, B - III, C - II, D - IV
• D. A - III, B - IV, C - I, D - II

Hint:

When matching concepts, remember that regular expressions, finite automata, and theorems like Arden's and Kleen's are all connected through equivalence and properties of languages accepted by finite automata.

Answer 1

The Correct Option is C

Step 1: Concept Mapping. 
- \( A \): Relates to Arden's Theorem, concerning regular expressions and finite automata. Match: \( A \) - I. 

- \( B \): Pertains to the regularity of finite automaton languages, a concept within the Myhill-Nerode Theorem. Match: \( B \) - III. 

- \( C \): Corresponds to Regular Expression Equivalence, specifically the unique solution \( R = Y + RX \). Match: \( C \) - II. 

- \( D \): Addresses the equivalence of regular expressions \( X \) and \( Y \) when their finite automata are equivalent, as defined by Kleen's Theorem. Match: \( D \) - IV.

Step 2: Final Alignment. 
The established mapping is \( A - I, B - III, C - II, D - IV \), aligning with option (3).