GATE CS 1990 Q3-viii — Regular and Context-free Languages

Regular languages (or regular sets) are closed under standard boolean and algebraic operations. Let's analyze each option based on these closure properties:

• Option A: $R_1 \cap R_2$ is not regular.
  Incorrect. Regular languages are closed under intersection. If $R_1$ and $R_2$ are regular, then $R_1 \cap R_2$ is guaranteed to be regular.
• Option B: $R_1 \cup R_2$ is regular.
  Correct. Regular languages are closed under union. If $R_1$ and $R_2$ are regular, then $R_1 \cup R_2$ is regular.
• Option C: $\Sigma^* - R_1$ is regular.
  Correct. The set $\Sigma^* - R_1$ represents the complement of $R_1$ (often written as $\overline{R_1}$). Since regular languages are closed under complementation, $\Sigma^* - R_1$ is regular.
• Option D: $R_1^*$ is not regular.
  Incorrect. Regular languages are closed under the Kleene star operation. If $R_1$ is regular, then $R_1^*$ is also regular.

Thus, the correct statements are B and C.