Exercises for the Week 11 part 3 lecture video

• For a language \( L \), let \( f( L ) = \{ x \cdot x^R \mid x \in L \} \). Is the family of regular languages closed under \( f \)? Why or why not?
  • show/hideNo. \( \{ 0, 1 \}^* \) is a regular language, but \( f( \{ 0, 1 \}^* ) \) is the set of palindromes, which is not regular.
• For a language \( L \), let \( g( L ) = \{ x \cdot \mathtt{a} \mid x \in L \} \). Is the family of regular languages closed under \( g \)? Why or why not?
  • show/hideYes. Given a regular expression \( r \) for \( L \), \( r \cdot \mathtt{a} \) is a regular expression for \( g( L ) \).