Consider the two lists List-I and List-II given below:

List-I:
(i) Context free languages
(ii) Recursive languages
(iii) Regular languages

List-II:
(a) Closed under union
(b) Not closed under complementation
(c) Closed under intersection

For matching of items in List-I with those in List-II, which of the following option(s) is/are CORRECT?

The correct matching options are:
(i) – (b), (ii) – (a), and (iii) – (c)
(i) – (b), (ii) – (c), and (iii) – (a)

Let us analyze the closure properties of each language class from List-I against the operations listed in List-II to understand why these matchings are correct.

1. Context-Free Languages (i):
Context-Free Languages (CFLs) are closed under union and Kleene star, but they are not closed under intersection or complementation.
Therefore, item (i) matches with (b) Not closed under complementation.

2. Recursive Languages (ii):
Recursive languages are closed under all standard boolean operations, including:
• Union (a)
• Complementation
• Intersection (c)
Therefore, item (ii) can match with either (a) Closed under union or (c) Closed under intersection.

3. Regular Languages (iii):
Regular languages are the most restrictive class in the Chomsky hierarchy and are closed under all standard operations, including:
• Union (a)
• Complementation
• Intersection (c)
Therefore, item (iii) can also match with either (a) Closed under union or (c) Closed under intersection.

Combining the Matches:
Since Context-Free Languages (i) must match with (b), we look at the possible assignments for (ii) and (iii) using the remaining properties (a) and (c):
• If we match Recursive languages (ii) with (a) Closed under union, then Regular languages (iii) matches with (c) Closed under intersection. This gives the valid combination: (i) – (b), (ii) – (a), and (iii) – (c).
• If we match Recursive languages (ii) with (c) Closed under intersection, then Regular languages (iii) matches with (a) Closed under union. This gives the valid combination: (i) – (b), (ii) – (c), and (iii) – (a).